The Modal Logic of Potential Infinity, with an Application to Free Choice

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The Modal Logic of Potential Inﬁnity, With an Application to Free Choice Sequences

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By Ethan Brauer, B.A.

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Graduate Program in Philosophy The Ohio State University 2020

Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract

This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument—which is due to Linnebo and which I partially endorse—that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo’s result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity. I argue that there is an important range of potential infinities, which I call infinities with branching possibilities, for which S4 is the right modal logic. I further argue that the usual operator is not sufficiently expressive for reasoning about branching potential infinities. A new operator is needed, which I call the inevitability operator. (In tense logic, the same operator has been called the strong future tense). I show that first-order S4 with the inevitability operator is

ii unaxiomatizable. This makes it improbable that there can be a faithful translation of non-modal mathematical discourse into modal language. I suggest that this means standard mathematical theories of infinite sequences are thus committed to actual infinities. In Part II, I apply the modal account of potential infinity to develop a theory of free choice sequences. Free choice sequences are a concept from intuitionistic mathematics. They can be thought of as potentially infinite sequences of natural numbers whose values are freely chosen one at a time by an (idealized) mathematician. They figure prominently in results of intuitionistic mathematics that contradict classical mathematics. I propose an axiomatic theory of free choice sequences in a modal extension of classical second-order arithmetic, called MC, with the aim of providing modal analogues of key ideas and results from the intuitionistic analysis. In this theory I define the temporal-potential continuum, which serves as an ersatz intuitionistic continuum in my modal theory. I show that the temporal-potential continuum exhibits many characteristically intuitionistic properties. The main results are these: it is not the case that every real number is determinately rational or irrational; the natural order on real numbers is not linear; a bounded monotone sequence of rationals need not be Cauchy; if two disjoint sets A and B decompose the temporal-potential continuum, then both A and B are topologically open; finally, I introduce a notion of sharp discontinuity and show there is no function on the temporal-potential continuum which is sharply discontinuous.

iii Acknowledgments

Many thanks are due to my dissertation committee, Stewart Shapiro, Neil Tennant, Chris Miller, and Chris Pincock. They have been most helpful in discussing and commenting on multiple drafts of papers, including several which did not end up in this dissertation. The resulting work would have been much the worse if not for their help. I especially appreciate that Neil and Stewart have spent so much time discussing the rough and vague ideas that I have at one time or another thought about pursuing. A good portion of this dissertation was written in the summer of 2019 during a stay at the University of Oslo. Thanks to Øystein Linnebo, who graciously spent many hours talking about this material as I worked on it. I presented the material in Chapter 3 at the 2020 Cambridge Graduate Conference on the Philosophy of Math- ematics and Logic. Thanks to Wes Wrigley for comments on that occasion and to Robin Solberg for helpful conversation. Material from chapters 4 to 6 was presented to the Ohio State Logic Seminar and a University of Connecticut Logic Colloquium in the fall of 2019; thanks to audiences on both occasions for helpful questions and comments, especially Tim Carlson, Neil DeBoer, Ivo Herzog, Marcus Rossberg, and Lionel Shapiro. Many teachers have been inﬂuential in my growth as a philosopher, and without a doubt they have had an impact on this work, even if it is only felt indirectly. In this connection I want to thank especially Ben Caplan, Jeﬀ Dunn, Marcia McKelligan, and Declan Smithies. A debt is also owed to my fellow graduate students at Ohio State

iv and I would be remiss not to mention the members of the Logic or Language Society from whom I have learned much: Andr´eCurtis-Trudel, Steven Dalglish, Teresa Kouri Kissel, Giorgio Sbardolini, and Damon Stanley. Equally important are those whose company over the last several years has been as likely to involve beer as philosophy: Rachel Harris, Preston Lennon, Erin Mercurio, Daniel Olson, and Evan Woods. The largest, if least tangible, debt is due to my family. Thanks to my parents Tom and Janice, my brother Matt and sister-in-law Hallie, and ﬁnally to my grandmother Mary to whom this work is dedicated.

v Vita

2014 ...... B.A. Physics and Philosophy, DePauw University 2014-2020 ...... Graduate Teaching Associate, Dept. of Philosophy, Ohio State University

Publications

“Second-order Logic and the Power Set,” 2018. Journal of Philosophical Logic Vol. 47 No. 1, pp. 123-142.

“Relevance for the Classical Logician,” 2018. Review of Symbolic Logic, forthcoming.

Fields of Study

Major Fields: Philosophy

vi Contents

Abstract ...... ii

Acknowledgments ...... iv

Vita ...... vi

List of Figures ...... ix

1 Introduction ...... 1 1.1 Modal Logic...... 7 1.2 Free Choice Sequences...... 13

I The Modal Logic of Potentialism ...... 18

2 Convergent Possibilities ...... 19 2.1 The Role of the Mirroring Theorems...... 19 2.2 The Need for Free Logic...... 26 2.3 Classical Free Logic...... 28 2.4 Intuitionistic Free Logic...... 34 2.5 Conclusion...... 38

3 Branching Possibilities ...... 40 3.1 Hamkins on Potentialism...... 42 3.2 Branching vs. Convergent Possibilities...... 44 3.3 An Incompleteness Theorem...... 50 3.4 Axioms for Inevitability...... 58 3.5 The Philosophical Upshot...... 59 3.6 A Remark on Models and Potentialism...... 62 3.7 Conclusion...... 65

II A Modal Theory of Free Choice Sequences ...... 66

vii 4 Free Choice Sequences ...... 67 4.1 The goal...... 70 4.2 Free choice sequences...... 72 4.3 The modal approach...... 75 4.3.1 Notational preliminaries...... 76 4.3.2 Conceptions of choice sequences...... 77

5 Lawless Sequences ...... 83 5.1 The Intuitionistic Theory of Lawless Sequences...... 83 5.2 The Modal Theory of Lawless Sequences...... 87 5.2.1 A Kripke Model for MCLS ...... 92 5.2.2 Translations into MCLS ...... 95 5.2.3 Basic Properties of MCLS ...... 101 5.2.4 Continuity Principles...... 105 5.2.5 The Bar Theorem...... 113 5.3 Conclusion...... 116

6 Non-Lawless Sequences and Real Numbers ...... 118 6.1 A Kripke Model for MC ...... 123 6.2 Kripke’s Schema...... 126 6.3 Bar Induction and the Fan Theorem...... 128 6.4 Real numbers...... 131 6.5 Continuity...... 143 6.6 Decomposing the Continuum...... 148 6.7 Conclusion...... 164

References ...... 166

Appendices ...... 173

A Frame Conditions For Intuitionistic S4.2 ...... 173

B The Need for Axiom S8 ...... 175

C More Stability Properties ...... 178

viii List of Figures

5.1 Sketch of a Kripke model for MCLS ...... 94

6.1 A simple decomposition...... 158 6.2 A general decomposition...... 159

B.1 Sketch of a Kripke model for MCLS without S8...... 177

ix Chapter 1

Introduction

The infinite is inescapable in mathematics: there are infinitely many natural numbers, there are infinitely many points on a line, infinite sequences lie at the foundations of calculus, infinite sets are a mainstay of contemporary mathematics. One could go on. The concept of infinity, however, has not always had an easy time. Since antiquity, paradoxes have arisen in connection with the concept of infinity. Among the oldest and best-known are Zeno’s paradoxes. Consider Achilles trying to run some length from A to B. To do this, he must first run half the distance from A to B. Then he must run half the distance from this point to B, and then he must run half the distance from that point to B, and so on. Thus there are an infinite number of distances that Achilles must traverse before reaching B. Considering each traversal as a separate task, we can rephrase this as saying that Achilles must perform an infinite number of tasks to reach B. Since an infinite number of tasks can never be completed, we infer that Achilles can never reach B. But of course, this is absurd. Achilles can easily reach point B. Another famous example is known as Galileo’s paradox. This paradox begins with the observation that we can put the natural numbers in one-one correspondence with the perfect squares by associating each n with n2. If two collections can be put into one-one correspondence, then it is quite reasonable to say that they are the same size. So the collection of natural numbers must be the same size as the collection of perfect

1 squares. But on the other hand, it is also quite reasonable to say that no collection can be the same size as any of its subcollections. After all, most numbers are not perfect squares, and if we remove most of the members of some collection, how can be left with a collection the same size as the one we started with? A traditional resolution of such paradoxes appeals to Aristotle’s distinction between the potentially infinite and the actually infinite. The actually infinite is what it sounds like: an object or collection is actually infinite if it actually has a magnitude or quantity which is infinite. The potentially infinite, by contrast, is not an infinite magnitude or quantity which is ever achieved, but is an unboundedness: a quantity is potentially infinite when, for any portion of that quantity that we have ‘taken’, there remains a part of that quantity outside what we have ‘taken’. Hilbert described the distinction between actual and potential infinity when he claimed analysis only requires the potentially infinite, while set theory requires actual infinities. (In Chapter 3 I will cast some doubt on the idea that analysis requires only potential infinities). He wrote:

in analysis we deal with the infinitely large or the infinitely small only as a limit notion—as something that is becoming, coming to be, being produced—that is, as we say, with the potential infinite. But this is not the real infinite itself. That we have when we consider the totality of numbers 1, 2, 3, 4 ... as a completed entity, or when we regard the points of a line segment as a totality that is actually given and complete. This kind of infinite is called the actual infinite.(Hilbert, 1925, 373)

If for Hilbert potential infinity was a mere ersatz infinity, for Aristotle it was the only infinity. Aristotle held that only the potentially infinite truly existed and there

2 is no such thing as an actually inﬁnite magnitude:

[T]he infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. (Physics 206a27-29)1

If we successively ‘take’ one thing after another and can do so without limit, then the collection whose members we are taking (or perhaps creating) cannot be finite, and so must be infinite. But at every stage in this process we have only a finite (albeit growing) collection, so the collection cannot be actually infinite—thus we arrive at the idea of the potentially infinite. If we follow Aristotle in denying the actual infinite, we can resolve Zeno’s paradox as follows. We can grant that the length from A to B is potentially infinitely divisible; that is, there are a potential infinity of divisions that we can acknowledge between A and B. But we cannot regard these divisions as an actual infinity. Thus, as Moore describes it:

If Achilles runs straight from A to B, ... which he can certainly do, then he does not actually perform inﬁnitely many tasks or pass over inﬁnitely many points. It is just that, however many divisions we recognize in his journey, we can always recognize more. (Moore, 2001, 42)

This strategy can also be used to resolve Galileo’s paradox. The heart of this paradox, recall, was the tension between the claim that the collection of natural numbers was the same size as its subcollection of perfect squares and the claim that

1See also Physics 207a6. The Physics can be found in Aristotle(1941). See also (Hintikka, 1973, ch. 6) and Lear(1980) for more thorough interpretations of Aristotle’s theory of inﬁnity. The account of Aristotle I am oﬀering here is something like the conventional account among non-experts.

3 no collection can be the same size as any of its subcollections. The Aristotelian view that there are no actual infinities neatly resolves this tension by denying that there is any such completed collection of natural numbers. This Aristotelian view of infinity persisted throughout the middle ages and into the modern period, so that we find Gauss saying: “I protest first of all against the use of an infinite quantity as a completed one, which is never permissible in mathematics. The infinite is only a fa¸conde parler, where one is really speaking of limits to which certain ratios come as close as one likes while others are allowed to grow without restriction.”2 With the advent of set theory, however, this began to change. Cantor thought the traditional view got things exactly backwards, and that the notion of a potential infinity requires the existence of an actual infinity. In brief (and in somewhat anachro- nistic terms), Cantor’s basic idea is that the notion of potential infinity requires the use of variables, and that variables must range over some domain, and that domain must itself be infinite. In Cantor’s own words:

For since there can be no doubt but what we cannot do without variable magnitudes in the sense of the potential infinite, then the necessity for the actual infinite can be proved as follows: In order for such variable magnitudes to be capable of evaluation in a mathematical investigation, the ‘range’ of their variation must be precisely known by means of a prior definition; but this ‘range’ cannot itself be in turn something variable, for otherwise every fixed support for the investigation would give way; hence, this ‘range’ is a definite actually infinite set of values. Thus every potential

2Gauss(1900). See also Waterhouse(1979) for discussion of the historical context of this quota- tion.

4 inﬁnity, if it is to be employable rigorously in mathematics, presupposes an actual inﬁnity. (Cantor(1962), as quoted in (Hart, 1976, 249).)

As the theory of transfinite sets became familiar and common in mathematics, Cantor’s view that actually infinite collections exist and are in perfectly good order became the dominant view. The concept of potential infinity accordingly crept to the background. The concept of potential infinity is not given much attention in mathematical contexts—except for its historical interest—and is sometimes even the subject of skepticism. For instance, (Niebergall, 2014, 256) writes: “Personally, I simply have no ordinary understanding of [the phrase ‘potentially infinite’], and I do not find much help in the existing literature.” He goes on to diagnose the issue:

The reason is that those philosophers who are interested in the theme of the potentially infinite are usually drawn to it because they regard it as desirable to avoid assumptions of infinity (i.e., of the actual infinity), yet do not want to be restricted to a mere finitist position. An assumption of merely the potentially infinite seems to be a way out of this quandary: it seems to allow you to have your cake and eat it too. (257, footnote omitted.)

It is hard not to feel some sympathy to Niebergall’s objection, too. Explanations of the potentially infinite inevitably appeal to some such metaphors as a collection being ‘given’ or ‘taken’ or of the potential infinity as a limit we can ‘approach’ but never ‘reach’. Evocative as these metaphors are, without a deeper explication of potential infinity it is reasonable to worry that these metaphors are indeed an excuse to have our cake and eat it too.

5 Recently, however, Linnebo and Shapiro(2019) have taken up the defense of potential infinity by invoking concepts and tools from modal logic.3 To illustrate the idea, let us consider Aristotle’s example of the infinite divisibility of a given length, say a stick. This infinite divisibility is merely potential because the stick has not actually been divided into infinitely many pieces. Although we can always divide the stick into more parts, we will never actually be able to divide the stick infinitely many times. The infinite divisions are a limit towards which one can always work, but never reach. As Linnebo and Shapiro observe, it is natural to formalize the temporal language here in modal terms: if P xy means that x is a proper part of y, then we can express the infinite divisibility of our stick s as:

∀x(P xs → ♦∃yP yx)

This contrasts with the stronger claim we could assert if the divisions of the stick formed an actual inﬁnity: ∀x(P xs → ∃yP yx)

Similarly, if we thought the natural numbers were potentially inﬁnite we could express it as follows, using Succ(n, m) to mean that n is succeeded by m:

∀n♦∃mSucc(n, m) 3Linnebo and Shapiro had some precedents in using modal logic to analyze potential infinity, see Parsons(1971) and Hart(1976). Niebergall(2014) had also suggested that modal resources might provide resources for avoiding his skepticism of potential infinity. Building off work from Parsons (1977), Linnebo(2013) also used modal logic to give a potentialist account of the set-theoretic heirarchy.

6 And to say that the numbers are not actually inﬁnite is to say that it is not possible to have all of them at once, or that it is not possible for every number to have a successor:

¬♦∀n∃mSucc(n, m)

Thus, (Linnebo and Shapiro, 2019, 8) write, “provided we are willing to use the resources of modal logic, there is no problem whatsoever in distinguishing the merely potential infinite from the actual infinite.” This suggests a program of using modal logic to develop formal theories of various potentially infinite domains of objects. This raises two questions: what is the right modal logic to use in developing such theories, and to which domains of objects can this approach be fruitfully applied? These are the questions I address in this dissertation. Part I is about the right modal logic. Part II applies the modal approach to a particular domain of objects, namely free choice sequences. In the remainder of this introduction I explain the motivation behind the two parts of this dissertation, and give a brief overview of their contents.

1.1 Modal Logic

The modal framework for developing theories of potentially infinite domains is just that, a framework. An essential part of any such theory is to give a philosophical explanation of the nature of the modal in question. Still, the general study of the modal logic of potential infinity is useful for isolating important structural and logical features of potential infinity. And there is a general and intuitive picture behind the modal analysis of potential infinity that is helpful for isolating the correct modal axioms. This general picture can be reached by thinking of the objects in a potentially

7 infinite domain as being constructed. Then taking a model-theoretic view of modal logic, we can think of the possible worlds in a Kripke model as corresponding to states in which various objects have been constructed. As a simple example, consider geometric figures as forming a potentially infinite domain. The various possible worlds correspond to which geometric figures have been constructed. (The geometric case is convenient in part because we can construe talk of constructing objects literally). Since figures are only ever added and never destroyed, one world will be accessible from another just in case its domain is an extension of the latter world’s domain. With this picture guiding us, we can isolate some unproblematic principles of modal logic and postulate them as axioms. The first pair of assumptions is just that our modality respects logic, so that if φ is a logical truth then it must be necessary, and that the necessary truths should be closed under modus ponens. Formally, this gives us one rule and one axiom:

Nec If ` φ then ` φ

K (φ → ψ) → (φ → ψ)

The next three principles are more specifically justified by our intuitive guiding picture. The first of these is that if φ is actual, then φ is possible. Equivalently, if φ is necessarily true, then φ is true. Formally, this gives us the axiom:

T φ → φ

The next principle is that if φ is necessary, then φ is necessarily necessary. Intuitively, if we realize some possibilities and then go on realize some more possibilities, that is itself a way of realizing the latter set of possibilities from the initial starting point. So if φ holds no matter which possibilities we realize, then even if we realize some

8 possibilities it will still be true that φ holds no matter what possibilities we realize. This gives us the axiom:

4 φ → φ

The final unproblematic principle is that when we realize new possibilities, no objects cease to exist. Accordingly, if it is necessarily true that every existing object satisfies φ, then it is true that every actually existing object necessarily satisfies φ.

CBF ∀xφ(x) → ∀xφ(x)

This gives us the modal logic S4 with the Converse Barcan Formula as a basis. Now we face a choice point. Should the G axiom be included in our modal logic?

G ♦φ → ♦φ

The result of adding G to S4 is known as S4.2. The G axiom is valid in a Kripke frame just in case the accessibility relation satisﬁes the following convergence property:

If wRu and wRv, then there is some s such uRs and vRs

(Thus the accessibility relation in a frame for S4.2 must be transitive, reﬂexive, and convergent). So the choice facing us is whether the various possibilities that we may realize are convergent or branching. In other words, are there are any pairs of possible objects that we might construct which are mutually incompatible? Or can any pair of possible constructions be joined in some further more inclusive possible construction? For some domains, convergence is appropriate, while for other domains convergence is not appropriate and branching is preferable. For instance, convergence is a good assumption for the domain of geometrical constructions, since no construction that can be performed would ever prevent you

9 from subsequently performing another construction. For instance, consider the two possibilities where I respectively construct a triangle and a square. Then there is a third possibility accessible from each of these first two possibilities in which I have constructed both a triangle and a square. On the other hand, say I am constructing a potentially infinite sequence by choosing its values one at a time. If I am currently choosing the third value, then it is possible that I choose the value 0 and it is possible that I choose the value 1. But once I have chosen the value 0, it is no longer possible that the sequence have value 1, and vice versa. So there is no mutual extension of these two possibilities. In this case we have non-convergent branching possibilities. Chapter 2 is about the modal logic of convergent possibilities, and Chapter 3 is about the logic of non-convergent, or branching, possibilities. In Chapter 2 I discuss the mirroring theorems that are available in the context of convergent possibilities—so-called because they show that, in a suitable sense, entailments in the modal theory mirror the entailments in the non-modal theory. The basic idea is that, for the potentialist, to say that φ holds of all objects really means that φ holds of all objects no matter what objects there are. In other words, φ holds of all potential objects. Likewise, for the potentialist, to say that there is an object such that φ should really be understood as saying that there is a potential object such that φ. In other words, there could be an object such that φ. For instance, when our geometrizing theorist says that every line segment has a bisector, they really mean that for any possible line segment that you construct, you could also construct a bisector for that segment. Other than this, however, the potentialist can accept the ordinary meaning of all the logical and non-logical symbols. Thus we can define the potentialist translation φ♦ of φ to be the result of replacing every occurrence of ∀ in

10 φ with ∀ and every occurrence of ∃ with ♦∃. The mirroring theorems state that

♦ ♦ ♦ any non-modal sentences φ1, ..., φn entail ψ just in case φ1 , ..., φn entail ψ . There are two versions of the mirroring theorem, corresponding to the two options of taking the consequence relation to be either classical or intuitionistic. In this way, we have a partial vindication of the Gaussian claim that infinity is merely a fa¸conde parler. If we interpret quantifiers potentially—ranging over potential objects, as it were—then we can accept all the same mathematical entailments as the standard mathematical theory would have it. Thus accepting the potentially infinite need not commit us to any deviant mathematics, and conversely accepting standard mathematics may not after all require us to accept the existence of actual infinities. Of course, this will depend to some extent on which pieces of standard mathematics we accept. While the ♦ interpretation allows us to make perfectly good sense of arithmetic without accepting actual infinities, perhaps accepting even the potential existence of infinite sets requires the existence of actual infinities. Even leaving aside such interesting questions, though, there is a problem with appealing to the mirroring theorems to interpret mathematical discourse from the potentialist point of view. In their basic form, the mirroring theorems have been proved by Linnebo(2013) and Linnebo and Shapiro(2019), but Linnebo and Shapiro’s versions assume non-free logics. As a result, all terms are assumed to denote actual objects. In this setting, the arithmetic potentialist could not use any numerical terms, since any numerical term would have to actually (not merely potentially) denote. Assuming as usual that we have a term for 0 and a symbol for the successor function, there would be a term for every natural number and hence every natural number would have to actually exist rather than merely potentially exist. This obstacle precludes us

11 from using the mirroring theorems to interpret mathematical discourse potentially.4 What we want are versions of the mirroring theorems that hold in the context of a free logic. And that is exactly what Chapter 2 provides. The mirroring theorems rely essentially on the presence of the G axiom, however. So when we turn our attention to non-convergent possibilities in Chapter 3, we can

no longer use the ♦ translation to achieve a satisfactory potentialist interpretation

of mathematical discourse. In fact, even the motivation for the ♦ translation looks questionable in the context of branching possibilities. In the context of convergent possibilities, the background space of possibilities does not change based on what one

does. So ♦∃x behaves just like an ordinary existential quantiﬁer over the background space of possibilities. Indeed, the mirroring theorems turn essentially on this fact. But this is no longer the case when possibilities can branch. If the objects that you

construct can preclude the construction of other objects, then ♦∃x only expresses the fact that it is currently open that there exist an x such that ... . And this is not how a potentialist should want to interpret ordinary mathematical discourse. For instance, consider the construction of a potentially inﬁnite sequence σ of natural numbers. When the ordinary mathematician says that the sequence σ has a billionth value—i.e. ∃x(σ(1, 000, 000, 000) = x)—the potentialist will interpret this to mean that σ potentially has a billionth value. This is naturally understood not just as saying that there could or might be a billionth value for σ; the desired reading is that σ will eventually have a billionth value. In other words, no matter how the future progresses, there will come a time when σ has a billionth value. This, however, is not

expressible using just and ♦. Thus, to make explicit all the modal concepts that the 4At least, it precludes us from using the mirroring theorems to interpret mathematical discourse potentially unless we engage in some contortions to re-interpret numerical terms as predicates.

12 potentialist might avail themselves of, we need to add a new modal operator to our logic. In Chapter 3 I introduce an operator I, which can be interpreted as inevitably, and which has the semantics such that, informally, Iφ is true at a possibility w just in case on every path through possibilities accessible from w there is some possibility u such that φ is true at u. (This will be made formally precise in Chapter 3). The logic that results from adding this operator to S4 is called S4+ below. That this is a genuine increase in expressive strength follows from the main theorem of Chapter 3, which is that S4+ is not axiomatizable. The results of Chapters 2 and 3 thus provide something of a dichotomy. In the case of convergent possibilities, infinities can be interpreted potentially and all the familiar mathematics can be systematically recaptured within this context. On the other hand, when the possibilities in question are not convergent, new expressive resources are needed to capture all the modal concepts that the potentialist might avail themselves of. Moreover, these new expressive resources make our logic highly incomplete. Thus in the context of branching possibilities, taking infinities to be potential rather than actual makes a significant difference to the mathematics. Part II explores the mathematics of one such theory of a potentially infinite domain with branching possibilities.

1.2 Free Choice Sequences

Part II is an application of the modal account of potential inﬁnity to free choice sequences. Free choice sequences are a concept from intuitionistic mathematics; they are potentially inﬁnite sequences of natural numbers whose values are chosen more or less freely by the mathematician. From the preceding discussion it is clear that

13 this concept is ripe for a modal analysis. A sequence that is being constructed by a mathematician as time progresses is exactly the sort of object that the modal approach to potential infinity was intended to capture. Chapter 4 introduces the idea of a free choice sequence and the objections that have been raised against them. These objections can be grouped into three main points: first, free choice sequences introduce a temporal aspect into mathematics, second, they introduce a subjective aspect into mathematics, and third, they lead to results that contradict classical mathematics. Taken together, these objections can make free choice sequences look like the bizarre—and even questionably coherent—stepchild of Brouwerian intuitionism. My aim throughout Part II is to dispel these objections; the strategy is to provide a modal theory of free choice sequences in a background classical theory. This strategy blunts the force of the first objection, since the temporal aspect of free choice sequences is given a mathematical treatment via modal logic. Modal logic is a well-understood and mathematically sensible framework, and its use in a theory of free choice sequences should be no more problematic than in, say, logics of program verification or provability.5 The second objection is also mooted by this strategy. Once we have laid down a set of axioms governing free choice sequences, there is no role for a mathematical subject or agent in the theory. It is true that the intuitive picture of a mathematician engaged in a certain type of activity is how we motivate the choice of axioms for our theory. But the same thing could be said of Euclid’s geometrical postulates (e.g. any straight line segment can be extended) or Turing’s definition of a computing machine.

5See, for instance, Emerson(1990) for an overview of modal and temporal logic in theoretical computer science, and see Boolos(1993) for an overview of provability logic.

14 This vestigial role for the mathematical subject looks innocuous, and certainly does not introduce any novel or troublesome subjective element into mathematics. The third objection—namely, that free choice sequences lead to results that contradict classical mathematics—is directly refuted by the fact that the theory of free choice sequences I develop is itself a classical theory. The remainder of Chapter 4 gives a brief overview of the Brouwerian idea of free choice sequences and various precisiﬁcations of this idea that can be found in the intuitionistic literature. I show in each case how the resources of modal logic (with second-order arithmetic in the background) allow us to describe these conceptions of free choice sequences in modal-temporal terms. In Chapter 5, I develop in more detail a modal theory of so-called lawless sequences. These are free choice sequences which satisfy no constraints, rules, or laws (hence the name ‘lawless’). Lawless sequences form an important core of the intuitionistic theory of free choice sequences, manifesting the most distinctive features of free choice sequences, and serving as a building block on which more complicated theories of free choice sequences can be developed. I begin by describing the intuitionistic theory of lawless sequences. I then deﬁne a modal theory of lawless sequences, called MCLS. I argue that the axioms of MCLS adequately capture the concept of a lawless sequence. This claim is further bolstered by showing that for each of the axioms of the intuitionistic theory of lawless sequences, if we were to recast the axiom in modal-logical terms (rather than in intuitionistic logic), the result is provable in MCLS. In other words, each of the axioms of the intuitionistic theory of lawless sequences has a modal-logical analogue which is a theorem of MCLS. Lawless sequences exhibit some interesting metamathematical properties, but they do not have enough structure to develop any interesting mathematics. Thus to capture

15 any results of intuitionistic mathematics in the modal theory of free choice sequences, it is necessary to postulate the existence of free choice sequences that are not lawless, that do satisfy some constraints. In Chapter 6 new axioms are postulated that allow the definition of new free choice sequences by projection from already existing free choice sequences. For instance, if we already have a free choice sequence α, we can define a new choice sequence β by setting each β(x) to be double the value of α(x). Or we could define β by setting β(x) = 1 when α(x) is even and β(x) = 0 when α(x) is odd. And so on. The resulting modal theory of free choice sequences is called MC. It is routine to cast the familiar definitions of Cauchy sequences and related notions into this modal framework, allowing us to mimic the intuitionistic development of real number generators. Real number generators are free choice sequences that eventually converge to, or pin down, a unique point on the continuum. Real number generators form the basis of the intuitionistic analysis. Once we have defined real number generators in MC, standard notions of addition, subtraction, multiplication, and the multiplicative inverse (for non-zero values) can be defined and shown to satisfy the usual properties of commutativity and so forth. The more interesting aspect of MC, however, is that once we have given modal definitions of real number generators and the operations on them, we are able to prove several of the anti-classical theorems from the intuitionistic theory of real numbers. For instance, the natural order < on real number generators is not a linear order, and there exist bounded monotone sequences of rationals that do not necessarily converge. The use of real number generators in weak Brouwerian counterexamples can similarly be mimicked in MC. Finally, I show that if two sets A and B decompose the continuum, then A and B are both open in the sense that every point of A is internal in A and every point of B is internal in B. This allows us to prove a weak

16 version of the intuitionistic result that every function on the reals is continuous.

17 Part I

The Modal Logic of Potentialism

18 Chapter 2

Convergent Possibilities

Recently, Linnebo and Shapiro(2019) have argued that the use of modal notions can provide an account of potential inﬁnity. The technical side of their account turns on two mirroring theorems. Each mirroring theorem shows, roughly, that under a suit-

able translation ♦ of non-modal language into modal language, non-modal sentences

♦ ♦ ♦ φ1, ..., φn entail ψ just in case φ1 , ..., φn entail ψ . The two versions of mirroring theorem correspond to the two options of taking the consequence relation to be either classical or intuitionistic. In neither case is the logic allowed to be free; but there are reasons to prefer using a free logic in their analysis of potential infinity. In this chapter I will argue for the importance of using a free logic, and then prove the appropriate mirroring theorems for a free logic. In §1 I outline how the mirroring theorems fit into Linnebo and Shapiro’s account of potential infinity and state their theorems precisely. In §2 I explain why a free logic would be more appropriate for their project. In §3 I prove the mirroring theorem for classical free logic, and in §4 I prove the intuitionistic version.

2.1 The Role of the Mirroring Theorems

The basic idea behind the modal account of potential inﬁnity is that by using modal resources we can distinguish between actual and potential inﬁnities, as explained in

19 Chapter 1. Once we are able to draw these distinctions, a further goal of the account of potential infinity is to assess what the right logic is to use when reasoning about a potentially infinite domain. Of particular interest is the question of what the right logic is for arithmetic if we take the natural numbers to be merely potentially infinite. What should arithmetic look like if we assume that the natural numbers are merely potentially infinite? As Linnebo and Shapiro note, contemporary mathematical language is non-modal. So if we want to apply the modal analysis of potential infinity to, say, arithmetic, we need some translation of non-modal mathematical language into modal language. This leads to two questions in applying the modal picture of potential infinity to mathematics. First, what is the right modal logic of potential infinity? And second, what is the right translation of the non-modal mathematical language into a modalized mathematical language? There is a general and intuitive picture behind the modal analysis of potential infinity that is helpful for isolating the correct modal logic. This general picture can be reached by thinking of the objects in a potentially infinite domain as being constructed. Then taking a model-theoretic view of modal logic, we can think of the possible worlds in a Kripke model as corresponding to states in which various objects have been constructed. As a simple example, consider geometric figures as forming a potentially infinite domain. The various possible worlds correspond to which geometric figures have been constructed. Since figures are only ever added and never destroyed, one world will be accessible from another just in case its domain is an extension of the latter world’s domain. (In general, the latter world’s domain need not be a proper extension of the former world’s domain. Given the stability assumption explained below, however, there would be no loss of generality in requiring

20 this extension to be proper). With this picture guiding us, we can isolate some unproblematic principles of modal logic and postulate them as axioms. The ﬁrst pair of assumptions is just that our modality respects logic, so that if φ is a logical truth then it must be necessary, and that the necessary truths should be closed under modus ponens. Formally, this gives us one rule and one axiom:

Nec If ` φ then ` φ

K (φ → ψ) → (φ → ψ)

The next three principles are more specifically justified by our intuitive guiding picture. The first of these is that if φ is true no matter which objects have been constructed, then φ is true of the objects that have been constructed thus far. Equiv- alently, if φ is necessarily true, then φ is true. Formally, this gives us the axiom:

T φ → φ

The next principle is that if φ is necessary, then φ is necessarily necessary. Intuitively, if φ holds no matter which objects we construct, then if we construct some objects it will still be true that φ holds no matter which further objects we go on to construct. In other words, the accessibility relation is transitive. This gives us the axiom:

4 φ → φ

CBF ∀xφ(x) → ∀xφ(x)

21 This gives us the modal logic S4 with the Converse Barcan Formula as a basis. To get S4.2 (with the CBF), we need to add another axiom:

G ♦φ → ♦φ

The rationale for this assumption is easier to appreciate from the model-theoretic point of view. The G axiom corresponds to the assumption that the frame of our Kripke model satisﬁes the following convergence property:

wRv1 ∧ wRv2 ⇒ ∃u(v1Ru ∧ v2Ru)

In justifying this assumption, (Linnebo and Shapiro, 2019, 13) write:

This principle ensures that, whenever we have a choice of mathematical objects to generate, the order in which we choose to proceed is irrelevant. Whichever object(s) we choose to generate ﬁrst, the other(s) can always be generated later. Unless [R] is convergent, our choice whether to extend

the ontology of w0 to that of [v1] or that of [v2] might have an enduring eﬀect.

In other words, if we have a choice between constructing some objects a and b, constructing a should not preclude us from then going on to construct b. For instance, given a triangle ABC you can construct the bisector of either the side AB or the side BC; but whichever bisector you choose to draw ﬁrst, you can then draw the other one next. The stage where you construct the bisector of AB and the stage where you construct the bisector of BC have a common extension where you have constructed both bisectors. As we saw in Chapter 1, we should not always assume that the accessibility relation is convergent—or, informally, that the possibilities in question

22 are convergent as opposed to branching—when giving an theory of a potentially infinite domain. However, in the case of arithmetic it does seem appropriate. And more generally, it is perfectly reasonable to study the logic of convergent possibilities in the abstract. In sum, then, the right modal logic for potential infinities with convergent possibilities is S4.2 with the Converse Barcan Formula. For classical S4.2, the axioms and rules are the axioms and rules of classical first-order logic, plus: Nec If ` φ then ` φ K (φ → ψ) → (φ → ψ) T φ → φ 4 φ → φ G ♦φ → ♦φ

(I have omitted reference to the Converse Barcan Formula because it is derivable from the above axioms in the context of the usual rules governing universal and existential generalization in a non-free logic.) The most important difference between classical and intuitionistic modal logic, at least deductively, is that in intuitionistic modal logic ♦ cannot be defined as ¬¬. Thus we need additional axioms to govern the behavior of ♦. For intutionistic S4.2 we start with intuitionistic first-order logic, include the axioms and rules above, and then also add:

• (φ → ψ) → (♦φ → ♦ψ)

•¬ ♦⊥

• ♦(φ ∨ ψ) → (♦φ ∨ ♦ψ)

23 • (♦φ → ψ) → (φ → ψ)

• φ → ♦φ

• ♦♦φ → ♦φ

(See Simpson(1994) and references therein for the framework that Linnebo and Shapiro work in.) If this is the modal logic, how does it relate to the non-modal language of ordinary mathematics? The core idea is that when we say ∀xφ, we really mean that φ holds no matter what objects there are. In other words, the potentialist will regard ∀xφ

as really meaning ∀xφ. Similarly, to assert the existence of an object is really to assert the potential existence of that object. For instance, when the potentialist about arithmetic says every number has a successor, they don’t mean that every number actually has a successor, but that every number potentially has a successor or could

have a successor. Thus for the potentialist ∃xφ really means ♦∃xφ. Following Linnebo and Shapiro, we will call the strings ‘∀’ and ‘♦∃’ modalized quantifiers. Let φ♦ denote the result of replacing every quantifier in φ with the corresponding modalized quantifier. There is one more component we need in order to translate between non-modal mathematics and the modalized mathematics that is intended to make explicit the conception of some mathematical domain as potentially infinite. When we construct new objects new relational properties might obtain among our objects, but the intrinsic properties of the objects already constructed should not change when we construct

new objects. For instance, if we have a circle C1 centered on a point and draw a larger

circle C2 centered on the same point, it becomes true that C1 is contained in C2. But

it does not cease to be true C1 has a radius of r, say. When a sentence ‘never changes

24 its mind’1 we can call it stable. Formally, we will say φ is stable if the following conditionals hold (or the necessitations of their universal closures, if φ has free variables):

φ → φ ¬φ → ¬φ

To capture the idea that once a property or relation holds no construction of new mathematical objects can make it cease to hold, we will require that atomic sentences be stable. We are now in a position to appreciate the potentialist view of mathematics by systematically translating ordinary mathematical discourse into a modal discourse that makes explicit the modality inherent in potential infinity. As we have seen, the potentialist regards the true meaning of quantifiers as captured by their modalized counterparts, sees S4.2 (plus the CBF, a qualification I will henceforth take as implicit) as the right modal logic, and takes atomic sentences to be stable. Thus, to appreciate the view of the potentialist, we should investigate not ordinary mathematics of theories closed under a deduction relation `, but rather the potentialist translation of the mathematical axioms with the addition of stability axioms for atomic formulas

and closed under a deductive relation `♦ of S4.2. The two statements of the following mirroring theorem provide a systematic way of understanding what mathematics the potentialist will regard as acceptable:2

Theorem 2.1. Let ` be classical deducibility and `I intuitionistic deducibility in a

♦ ♦ non-modal language L and let ` and `I be classical and intuitionistic deducibility in the corresponding modal language plus the axioms of S4.2 and atomic stability axioms. Then for any formulas φ1, ..., φn, ψ ∈ L: 1In the words of (Linnebo and Shapiro, 2019, 14). 2The mirroring theorem is so-called because it shows that the modal and non-modal consequence relations mirror each other.

25 ♦ ♦ ♦ ♦ 1. φ1, ..., φn ` ψ iﬀ φ1 , ..., φn ` ψ ,

♦ ♦ ♦ ♦ 2. φ1, ..., φn Ì ψ iff φ1 , ..., φn Ì ψ .

Statement (1) is proved in Linnebo(2013), and (2) is proved in Linnebo and Shapiro(2019). As can be seen by inspecting the proofs below, the G axiom—or equivalently, the requirement that the accessibility relation be convergent—is only needed for the left-to-right direction of these results. What Theorem 2.1 shows is that, although the potentialist regards ordinary mathematics as less-than-explicit about its inﬁnitary commitments, she is nonetheless free to accept the canon of classical or constructive mathematics under a suitable and straightforward interpretation of its non-modal language. Whether the potentialist adopts classical or constructive mathematics will turn on whether she takes her modal logic to be classical or intuitionistic. Linnebo and Shapiro emphasize that this latter choice may be motivated by principles of potentialism rather than the anti-realist arguments of, say, Brouwer(1913), Heyting(1971), or Dummett(1975). These considerations do not bear on the present concerns, however, so I will simply refer the interested reader to Linnebo and Shapiro(2019).

2.2 The Need for Free Logic

The mirroring theorems stated in the previous section hold for non-free classical or intuitionistic logic. There are reasons, however, to prefer a free logic.3 Consider a natural model of the potentialist’s modal language: let there be an ω- sequence of worlds hwiii∈N, with domains D(wi) = {0, ..., i − 1} and wiRwj iﬀ i ≤ j. 3(Linnebo and Shapiro, 2019, n. 17) acknowledge this point in passing.

26 Intuitively, the ﬁrst world represents the initial stage where 0 exists and the nth world is the stage where we have constructed the numbers up to n − 1. Now consider the following entailment in PA:

∀x(0 + x = x) |= 0 + s0 = s0

Applying the potentialist translation,

∀x(0 + x = x) |=♦ 0 + s0 = s0

Take the first world in that model, where 0 exists. Since ∀x(0 + x = x) is an axiom of potentialized PA, we want it to hold in that world. Thus, it should also hold in that world that 0+s0 = s0. But since we are using a non-free logic, the terms in this true atomic sentence must denote. It follows that 1 also exists in this world. And likewise for every other number. So the numbers are not a potential infinity with new ones being added in each world. Instead, the natural numbers are an actual infinity, with all of them present in each world. So to capture the potentialist view, we need to allow for the possibility of non- denoting terms. That is, the modal logic should be free. Linnebo and Shapiro avoid this problem by using a successor relation rather than a successor function. While arithmetic can be developed using relational predicates in place of function symbols, this is not typical.4 And unlike modalizing quantifiers and adding axioms for atomic stability, which are well-motivated aspects of the potentialist translation, there seems to be no intrinsic reason for the potentialist to

4See, e.g., (H´ajekand Pudl´ak, 1993, 86-9) for arithmetic with no function symbols.

27 paraphrase away function symbols in terms of relations. It is just a matter of technical convenience. It would be nice if the potentialist could remain more faithful to the mathematical practice of using functions rather than relations. Thus, it would be good if we could eliminate the seeming necessity of paraphrasing away function symbols in terms of relations. I do this in the following sections by establishing the mirroring theorems for free logic.

2.3 Classical Free Logic

In this section I prove the mirroring theorem for classical logic when the modal logic is a negative free logic. My approach will be model-theoretic. While one could alternatively modify the proof-theoretic approach of Linnebo(2013), I think it is instructive to see how models of the non-modal logic and the modal logic can be obtained from one another. As above, we take the modal logic we use to be S4.2 with a growing domain.

Deﬁnition 2.2. A model for S4.2 with a growing domain is a quadruple M = hW, R, D, V i, where:

• W is a non-empty set (intuitively, the set of worlds)

• R is a relation on W that is reﬂexive, transitive, anti-symmetric and convergent,

i.e. if wRv1 and wRv2, then there is a u such that v1Ru and v2Ru (intuitively, R is the accessibility relation)

• D is a function assigning to each element of W a set of objects, such that if wRv, then D(w) ⊆ D(v)

28 • V is a valuation function assigning to each world w and n-place relation symbol a subset of D(w)n as extension and to each world w and n-place function symbol the graph of a (not necessarily total) function D(w)n → D(w).

We can treat individual constants as 0-place function symbols. Since function symbols may denote partial functions, this is consistent with individual constants not denoting. Since we are using a negative free logic, we will let atomic sentence with non-denoting terms be false and then give the usual recursive semantic clauses for non-atomic sentences. For simplicity, I will assume a substitutional interpretation of quantiﬁers; if o ∈ D, let o be an individual constant such that V (o) = o.

Deﬁnition 2.3. Let M be an S4.2 model, and let w ∈ W . Deﬁne M, w |= φ recursively:

• M, w |= P a1, ..., an iﬀ all of V (a1), ..., V (an) are deﬁned and hV (a1), ..., V (an)i ∈ V (w, P )

• M, w |= f1(a1, ..., an) = f2(b1, ..., bm) iﬀ for some o ∈ D(w), hV (a1), ..., V (an), oi ∈

V (w, f1) and hV (b1), ..., V (bm), oi ∈ V (w, f2)

• M, w |= ¬φ iﬀ M, w 6|= φ

• M, w |= φ ∧ ψ iﬀ M, w |= φ and M, w |= ψ

• M, w |= φ ∨ ψ iﬀ M, w |= φ or M, w |= ψ

• M, w |= ∀xφ(x) iﬀ for all o ∈ D(w), M, w |= φ(o)

• M, w |= ∃xφ(x) iﬀ for some o ∈ D(w), M, w |= φ(o)

• M, w |= φ iﬀ for all u ∈ W such that wRu, M, u |= φ

29 • M, w |= ♦φ iﬀ for some u ∈ W such that wRu, M, u |= φ

Stability has the same meaning as above, but of more interest to us here is what we will call existential stability. Let E be an existence predicate (either primitive or deﬁned). Then a sentence φ is existentially stable if it satisﬁes the necessiation of the universal closures of the following conditionals, where the ti’s are the terms occurring in φ: ^ Eti → (φ → φ) i and ^ Eti → (¬φ → ¬φ) i

As above, the potentialist translation φ♦ is the result of replacing in φ every occurrence of a quantifier by the corresponding modalized quantifier. In the mirroring theorems below, we will include axioms of existential stability for all atomic sentences φ. An important consequence of the existential stability axioms is that they require the interpretation of predicates and function symbols to be growing. In other words, if wRv then for any relation symbol P and function symbol f, V (w, P ) ⊆ V (v, P ) and V (w, f) ⊆ V (v, f). We will assume we have a fixed language. For technical convenience, we will assume in what follows that we have a term in the language for every object that exists at any world. This will allow us to avoid talk about variable assignments and satisfaction. Obviously it is not essential to the results below, however, since we can always expand a language by adding individual constants and run through the reasoning below. To apply the results to a language that does not have terms for every object, we then consider the reduct of our model to the original unexpanded

30 language. (The assumption that there is a term for every object is also a natural one to make in the context of arithmetic, since every number has a numeral.)

Lemma 2.4. If all atomic formulas are existentially stable, then for any non-modal formula φ, its translation φ♦ is existentially stable.

Proof. Straightforward induction on complexity of φ.

Now, we might like to claim the following:

♦ ♦ ♦ ^ ♦ φ1, ..., φn |= ψ classically iﬀ φ1 , ..., φn |= Eti → ψ i

But we can’t, because in non-free classical logic we have 1 6= 1 |= ∃x(x 6= x) while in our free modal logic we have 1 6= 1 6|=♦ ♦∃x(x 6= x). So we need an existential condition on the left side of the turnstile too.

Similar to above, we can let |=♦ be entailment in S4.2 plus existential stability axioms for all atomic sentences.

Theorem 2.5. For any sentences φ1, ..., φn, ψ, the classical entailment φ1, ..., φn |= ψ holds iﬀ

^ ♦ ♦ ♦ ^ ♦ Etk, φ1 , ..., φn |= Eti → ψ k i where the tk’s are the terms occurring in the φj’s and the ti’s are the terms occurring in ψ.

V ♦ ♦ ♦ V Proof. (⇒) Let M = hW, R, D, V i be a countermodel to k Etk, φ1 , ..., φn |= k Eti →

♦ V ♦ ♦ ψ , so there is some w ∈ W such that M, w |= k Etk, and M, w |= φ1 ∧ ... ∧ φn, and

V ♦ 0 M, w |= i Eti, but M, w 6|= ψ . Then deﬁne a classical ﬁrst-order model M = hd, Ii

31 such that [ d = D(v) v:wRv and for any predicate letter P or function symbol f,

[ [ I(P ) = V (v, P ) and I(f) = V (v, f) v:wRv v:wRv

Convergence of R plus the fact that extensions of relations and graphs of functions are growing will guarantee that M 0 is well-deﬁned. We want to show that M 0 will be a countermodel to φ1, ..., φn |= ψ. I claim that:

(*) Let φ be a non-modal formula, and suppose that every term occurring in φ denotes in some world v such that wRv. Then M 0 |= φ iﬀ for every v such that

wRv and every term occurring in φ denotes in v, M, v |= φ♦.

Since we know that all the terms occurring in the φi’s or in ψ exist at the world

♦ ♦ ♦ w, and also that φ1 , ..., φn, ¬ψ all hold at w, it will follow from Lemma 2.4 that

♦ ♦ ♦ φ1 , ..., φn, ¬ψ all hold at every world v s.t. wRv. Then it follows from (*) that

0 0 φ1, ..., φn, ¬ψ are all true in M , so M is a countermodel to φ1, ..., φn ψ, as required. It remains to prove (*). This can be done by an easy induction on the complexity of φ. The basis step is straightforward. For the induction step, I consider the cases where φ begins with ¬, ∀, or ∃ as examples. To reduce clutter, I will assume for the proof of (*) that v and u are such that wRv and wRu. First, let φ := ¬θ. M 0 |= φ iﬀ M 0 6|= θ iﬀ (by the i.h) for some v in which every

term occurring in θ denotes M, v 6|= θ♦. By Lemma 2 this will hold iﬀ M, v 6|= θ♦ for

all v such that every term occurring in θ denotes at v iﬀ M, v |= ¬θ♦ for every such v.

32 Next, let φ := ∀xθ(x). For all o ∈ d, let o be a term denoting o. Then M 0 |= φ iﬀ for all o ∈ d, M 0 |= θ(o) iﬀ for o ∈ d, for all v s.t. every term occurring in θ(o)

denotes in v, M, v |= θ♦(o) iﬀ for no v such that every term occurring in θ denotes at

v is there a world u (where vRu) and an object o ∈ D(u) such that M, u 6|= θ♦(o) iﬀ

for every v such that every term occurring in θ denotes at v, M, v |= ∀xθ♦(x). Finally, let φ := ∃xθ(x). M 0 |= φ iﬀ for some o ∈ d, M 0 |= θ(o) iﬀ for all v such

that every term occurring in θ(o) denotes at v, M, v |= θ♦(o). Since R is convergent, for any world u at which every term occurring in θ denotes there is a world v at which every term occurring in θ(o) denotes. And since R is reﬂexive, it is obvious that for every world v at which every term occurring in θ(o) denotes there is a world u at which every term occurring in θ denotes such that uRv. Thus we continue: for all v

such that every term occurring in θ(o) denotes at v, M, v |= θ♦(o) iff for every u at which every term in θ denotes, M, u |= ♦∃xθ♦(x). The other cases are all straightforward. This completes the induction and the proof of (*). (⇐) Any classical first order model can be transformed in the obvious way to an S4.2 model which has only one world. In this model the modalized quantifiers will behave just like normal quantifiers since they only range over the objects in the one world. All existential stability axioms will also hold in this model. Thus a

V ♦ ♦ ♦ countermodel establishing that φ1, ..., φn 6|= ψ will also establish k Etk, φ1 , ..., φn 6|=

V ♦ Eti → ψ . i

33 2.4 Intuitionistic Free Logic

In this section I prove the mirroring theorem using an intuitionistic modal logic that is a negative free logic. Following Linnebo and Shapiro, I use the framework of Simpson (1994) for intuitionistic modal logic. In this framework, a model of intuitionistic modal logic is a quintuple M = hW, ≤ ,R,D,V i. The idea is that ≤ is the usual accessibility relation from the Kripke semantics for intuitionistic logic, and R is the accessibility relation for the modalities

and ♦. Thus, we require that ≤ be reflexive, transitive, and anti-symmetric, and that the domain and the interpretations of predicates and function symbols be growing along both the ≤ relation as well as along the R relation. The frame conditions for an S4.2 model are slightly different from the classical case. As above, we require that R be reflexive, transitive, and anti-symmetric; the difference in the intuitionistic case is that we require R to satisfy the following property, which we can call ‘I-convergence’ of a frame F :

(IC) If wRu and wRv, then there for some s, t, vRt and u ≤ sRt.

(In AppendixA I show that I-convergence is necessary and suﬃcient for validating the G axiom). It is also necessary to impose the following requirements on how ≤ and R interact:5

(F1) If w0 ≥ wRv then there is a v0 such that w0Rv0 ≥ v.

(F2) If wRv ≤ v0 then there is a w0 such that w ≤ w0Rv0.

Putting all this together we can give our formal deﬁnition of model.

5See (Simpson, 1994, 50) for discussion of these requirements.

34 Deﬁnition 2.6. A model of intuitionistic S4.2 is a quintuple M = hW, ≤,R,D,V i such that:

• W is a non-empty set

• R is a binary relation on W which is reﬂexive, transitive, anti-symmetric, and I-convergent.

•≤ is a binary relation on W which is reﬂexive, transitive, and anti-symmetric.

• R and ≤ satisfy (F1) and (F2).

• D is a function that assigns to each element of W a non-empty set, such that if w ≤ v then D(w) ⊆ D(v).

• V is a valuation function indexed by elements of W that assigns subsets of D(w) to predicates and assigning graphs of (not necessarily total) functions on D(w) to function symbols.

The forcing relation M, w φ is deﬁned recursively using the usual clauses for non-modal vocabulary, and with the added clauses for the modal vocabulary.

Deﬁnition 2.7. Let M be a model of intuitionistic S4.2, and let w ∈ W . Deﬁne

M, w φ recursively:

• M, w P a1, ..., an iﬀ all of V (a1), ..., V (an) are deﬁned and hV (a1), ..., V (an)i ∈ V (w, P )

• M, w f1(a1, ..., an) = f2(b1, ..., bm) iﬀ for some o ∈ D(w), hV (a1), ..., V (an), oi ∈

V (w, f1) and hV (b1), ..., V (bm), oi ∈ V (w, f2)

• M, w ¬φ iﬀ for all u ≥ w, M, u 1 φ

35 • M, w φ ∧ ψ iﬀ M, w φ and M, w ψ

• M, w φ ∨ ψ iﬀ M, w φ or M, w ψ

• M, w φ → ψ iﬀ for all u ≥ w either M, u 1 φ or M, u ψ

• M, w ∀xφ(x) iﬀ for all u ≥ w for all o ∈ D(u), M, u φ(o)

• M, w ∃xφ(x) iﬀ for some o ∈ D(w), M, w φ(o)

0 0 • M, w φ iﬀ for every w ≥ w, for all v: if w Rv, then v φ

• M, w ♦φ iﬀ for some v, wRv and v φ.

Because of the additional structure of intuitionistic models (both modal and non- modal), the proof of the intuitionistic mirroring theorem will necessarily be more involved than the classical version. The core idea of the proof is similar, though. To obtain a modal intuitionistic model from a non-modal intuitionistic model we make each world bear R to itself and only itself. Conversely, to obtain a non-modal intuitionistic model from a modal model, we deﬁne the worlds of the non-modal model from unions of R-chains of worlds in the modal model. The key diﬀerence from the classical case is just that a classical model is like an intuitionistic model with only one world.

♦ Now let |=I be intuitionistic entailment and let |=I be entailment in intuitionistic S4.2 plus existential stability axioms for atomic sentences.

Theorem 2.8. For any sentences φ1, ..., φn, ψ, the entailment φ1, ..., φn |=I ψ holds iﬀ

^ ♦ ♦ ♦ ^ ♦ Etk, φ1 , ..., φn |=I Eti → ψ k i

36 where the tk’s are the terms occurring in the φj’s and the ti’s are the terms occurring in ψ.

V ♦ ♦ ♦ Proof. (⇒) Let M = hW, ≤,R,D,V i be a countermodel to k Etk, φ1 , ..., φn |=I

V ♦ V k Eti → ψ , so there is some w ∈ W such that M, w k Etk, and M, w

♦ ♦ V ♦ φ1 ∧ ... ∧ φn, and M, w i Eti, but M, w 1 ψ . Deﬁne an intuitionistic model M 0 = hW 0, ≤0,D0,V 0i as follows. For each w ∈ W ,

0 0 let wR := {u ∈ W : wRu}. Then let W = {wR : w ∈ W }. Put wR ≤ uR iﬀ w ≤ u. Let D0(w ) = S D(u), and for any predicate P or function symbol R u∈wR f, let V 0(P, w ) = S V (P, u) and V 0(f, w ) = S V (f, u). Along with the R u∈wR R u∈wR requirements on the relation R, the conditions (F1) and (F2) guarantee that M 0 will be a well-deﬁned intuitionistic model. The following claim, which can be easily proved by induction on the complexity of φ, will be useful:

0 (**) For any non-modal sentence φ and any u, v ∈ W , if uRv, then M , uR φ iﬀ 0 M , vR φ.

I now claim that for any non-modal sentence φ and w ∈ W such that every term

♦ 0 occurring in φ denotes at w: M, w φ iﬀ M , wR φ. From this claim it will follow 0 that M is a countermodel to φ1, ..., φn |=I ψ. The claim itself is easily proved by an induction on the complexity of φ. The only tricky case is when φ := ∀xψ(x).

I begin with the right-to-left direction. Suppose that M, w 1 ∀xψ♦(x). Then for some u ≥ w and some v such that uRv, M, v 1 ∀xψ♦(x). Hence for some s ≥ v

♦ 0 and some o ∈ D(s), M, s 1 ψ (o). By the i.h., M , sR 1 ψ(o). By (F2) there is a

0 0 t ≥ u ≥ w such that tRs. Note that o ∈ D (tR), so by (**), M , tR 1 ψ(o). Hence

0 M , wR 1 ∀xψ(x).

37 0 For the left-to-right direction, suppose M , wR 1 ∀xψ(x). Hence for some uR ≥

0 0 wR, and some o ∈ D (uR), M , uR 1 ψ(o). Find some v such that uRv and o ∈ D(v).

0 ♦ ♦ Then by (**), M , vR 1 ψ(o) and by the i.h. M, v 1 ψ (o). Hence M, v 1 ∀xψ (x).

Since w ≤ uRv, it follows that M, w 1 ∀xψ♦(x). This completes the proof for the case of φ := ∀xψ(x). The other cases are straightforward.

(⇐) Let M = hW, ≤,D,V i be a countermodel to φ1, ..., φn |=I ψ. From this model, define a modal intuitionistic model M 0 = hW, ≤,R,D,V i by putting uRv iff u = v, for all u, v ∈ W . Clearly R is reflexive, transitive, anti-symmetric, and I-convergent. And it is easy to check that this model also satisfies (F1) and (F2). We also need to check that the existential stability axioms for atoms are valid in M 0. This follows from the fact that true atomic facts are preserved along ≤ in an intuitionistic model. Finally, I claim that, for any non-modal sentence φ and world w, if every term

0 occurring in φ denotes at w, then M, w φ iﬀ M , w φ♦. This is proved by an easy induction on the complexity of φ. It follows that M 0 is a countermodel to

V ♦ ♦ ♦ V ♦ k Etk, φ1 , ..., φn |=I k Eti → ψ .

2.5 Conclusion

What the mirroring theorems show is that, although the potentialist regards ordinary mathematics as less-than-explicit about its inﬁnitary commitments, she is nonetheless free to accept the canon of classical or intuitionistic mathematics under a suitable and straightforward interpretation of its non-modal language. If we interpret quantiﬁers potentially—ranging over potential objects, as it were—then we can accept all the same mathematical entailments as the standard mathematical theory would have it.

38 Thus accepting the potentially inﬁnite need not commit us to any deviant mathematics, and conversely accepting standard mathematics may not after all require us to accept the existence of actual inﬁnities.6 All this depends, however, on the fact that we are dealing with convergent possibilities, since that is what licenses S4.2 as the right modal logic. In the next chapter we will consider what happens when we drop this assumption.

6Of course, this will depend to some extent on which pieces of standard mathematics we accept. While the ♦ interpretation allows us to make perfectly good sense of arithmetic without accepting actual infinities, perhaps accepting even the potential existence of infinite sets requires the existence of actual infinities.

39 Chapter 3

Branching Possibilities

The mirroring theorems of Chapter 2 provide us with an elegant way to interpret standard mathematical theories from a potentialist point of view that does not (or does not necessarily) countenance any actual infinity. Those theorems depend essentially, however, on the possibilities in question being convergent. Without this assumption, however, adequately representing the potentialist point of view will require more expressive resources than our standard modal logic has to offer. In this chapter we will be concerned with the modal logic of potentialism for branching possibilities. Both to fix intuitions and for continuity with the literature that this chapter builds on (see §3 below), it is convenient to consider the modal in question to be a temporal modal. Thus φ is interpreted to mean that φ holds at all possible future times and ♦ means that φ holds at some possible future time. What neither of these says, however, is that something will happen in the future. If we imagine time branching so that there are multiple possible ways for the future to proceed, and none of them has a privileged status as the ‘real’ future, then to say that something will happen in the future requires us to say something about each of the possible futures. Specifically, we want an operator expressing that inevitably φ, with the intended semantics that in every future branch of time there is a moment at which φ holds. I take it as relatively clear that this is something that the potentialist might want to express. Exactly what the potentialist wants to express using such an operator will depend

40 on the particular subject matter that the potentialist is theorizing. But we will see specific and important uses of the inevitability operator in Part II of this dissertation (especially concerning the bar theorem). Thus the reader who is skeptical of the usefulness of the inevitability operator is asked to be patient. The goal of this chapter is not so much to establish the utility of the inevitability operator as to establish how significantly we increase the strength of our logic by adding the inevitability operator. Specifically, in this chapter I will show that there is no recursive axiomatization of the first-order temporal logic of branching time with an inevitability operator expressing that in every branch there is a time at which such-and-such is true. Indeed,

1 the validity problem for this logic is shown to be Π1-hard. In §1 I will discuss work on potentialism by Joel David Hamkins and some of his collaborators. This work is significant both because it provides a purely mathematical analysis of the logic of potential infinity—in contrast to the philosophical analysis I have offered above—and because this work isolates branching potentialist systems that are subject to the logic S4 rather than S4.2. I will argue that, although mathematically interesting, this work fails to provide a philosophical motivation for accepting S4 as the right modal logic of any potential infinities. In §2 I introduce the distinction between potential infinities with branching and convergent possibilities; I explain why potential infinities with branching possibilities require using the logic S4 rather than S4.2; and I show that branching possiblities also require an increase in expressive power. In §3 I prove the technical results of this chapter. §4 provides a sound, though necessarily incomplete, axiomatization for the logic of branching possibilities. §5 explains the philosophical significance of the technical results proved here. §6 is a bit of a digression, commenting on the role of model theory in the dialectic surrounding potentialism.

41 3.1 Hamkins on Potentialism

The approach I outlined in the previous section attempts to determine the right modal logic for potential inﬁnity by a philosophical analysis of potentiality. An alternative approach pursued by Joel Hamkins, along with a number of collaborators, gives a purely mathematical analysis of the modal logic of potentialism. The basic idea

behind this analysis is to start with a mathematical theory T ; then we deﬁne ♦φ to be true in a model M of T when φ is true in some extension of M. Various more speciﬁc modalities can be studied by restricting the types of extensions of M that determine

whether ♦φ is true in M. For instance, Hamkins and Linnebo(2019) study the set-

theoretic modalities true in some larger Vκ, true in some larger transitive sets, and true in some forcing extension, among others. Hamkins and Woodin(2018) analyze the set-theoretic modality true in some top-extension. Hamkins(2018) studies the arithmetic modalities true in some extension, true in some end-extension, and true

in some Σn-elementary extension, among others. One particular result of Hamkins (2018) that is worth noting is that the standard model of arithmetic validates exactly S4 as the modal logic of true in some end-extension. (Recall that a model M of arithmetic is an end-extension of a model N if for every a ∈ M \ N, M |= b < a for every b ∈ N. Thus an end extension results from adding new non-standard numbers to the domain which are larger than all the old numbers.) While this work is of great interest both technically and philosophically, I think its relevance for analyzing the modal logic of potential inﬁnity is mixed. To see why, let

us begin with the case of arithmetic. On the Hamkins analysis, whether ♦φ is true in a model depends on whether φ is true in various other models that extend it. But all the models in question include the totality of natural numbers in their domain, and

42 they extend each other by adding non-standard elements. It is not clear what this has to do with the view that the natural numbers themselves are a merely potential infinity. Recall that a collection being potentially infinite means that it is always possible to find a new element of that collection. So in an account of the modal logic of potentialism, the new objects that get added to the domain should be the elements of the collection that is supposedly potentially infinite. In the case of arithmetic, this means that the possibilities should involve adding new (true) natural numbers to the domain, not non-standard numbers. Moreover, this also means that in any given model there should be (true) natural numbers that are not yet in the domain of that model. Neither of these are the case in Hamkins’ account, which seems better suited as a potentialist account of non-standard numbers. Perhaps there is a case to be made that such a view deserves to be called arithmetic potentialism; but at the very least it seems importantly different from the traditional view that the natural numbers form a potential infinity. On the other hand, the analysis of set theory in Hamkins and Linnebo(2019) seems more promising as a philosophically-motivated account of set-theoretic potentialism. In this case the structures extend each other by adding new sets. And of course, this is exactly what the set-theoretic potentialist thinks is happening. In the set-theoretic case, the closer alignment between the informal philosophical idea and the formal account makes that formalism more attractive as an analysis of the philosophical idea. The various versions of set-theoretic potentialism that Hamkins and Linnebo (2019) analyze all have S4.2 as a lower bound for their right modal logics, however. Thus, Hamkins’ work on arithmetic potentialism gives a mathematical case for accepting S4 as the right modal logic for some forms of potentialism, but the philo-

43 sophical motivation for those forms of potentialism is lacking. On the other hand, Hamkins and Linnebo’s work on set-theoretic potentialism has a more solid philosophical motivation, but leads us to accept logics at least as strong as S4.2. What we do not get is a philosophical motivation for accepting S4 as the right modal logic for some forms of potentialism. Exactly such a philosophical motivation is what I will provide in the next section.

3.2 Branching vs. Convergent Possibilities

The mirroring theorems of Chapter 2 provide an elegant way to give a potentialist interpretation of mathematical discourse. The mirroring theorems rely essentially on the presence of the G axiom, however. And the G axiom should not always be included in a modal logic of potential infinity. Recall that the idea motivating the G axiom is that constructing one object should never preclude later going on to construct another object. Or, to put the matter another way, when we are choosing to realize various possibilities that witness the potentially infinite character of some object(s), those possibilities are never incompatible. For some potential infinities, this is a reasonable assumption. For instance, this assumption is well-motivated for geometrical constructions, natural numbers, and sets. But for other potential infinities, this assumption is problematic. We have already seen one example of non-convergent possibilities in §2, namely end-extensions of models of arithmetic. I argued above that Hamkins’ analysis does not give us reason to drop the G axiom from a potentialist treatment of the natural numbers. However, the mathematical tools he uses in his analysis suggest a different motivation for dropping the G axiom.

44 Hamkins(2018) shows that S4 is the right logic for the arithmetic modality true in some end-extension by appeal to the so-called universal sequence. The universal sequence is the code e of a Turing machine that enumerates a sequence of natural numbers and that has the following properties. In every model, e enumerates a ﬁnite sequence; in the standard model, e enumerates the empty sequence; if e enumerates the sequence s in M, then in every end-extension of M, e enumerates some (not necessarily proper) extension of s; and in any model M where e enumerates a sequence s, for any sequence s0 that extends s, there is an end extension of M where e enumer-

0 ates s . Thus if e enumerates s in M, there are end-extensions M1 and M2 in which e enumerates s_h1i and s_h2i, respectively.1 Accordingly, in every end extension

_ of M1, e enumerates a sequence beginning with s h1i; and in every end-extension

_ of M2, e enumerates a sequence beginning with s h2i. So there can be no mutual

end-extension of M1 and M2. Thus the end extension relation is not convergent, and the modalities of true in some end extension do not satisfy the G axiom. This sketch of Hamkins’ argument suggests a more direct way to isolate S4 as the right modal logic for certain potential infinities. The infinities in question are infinite sequences of natural or rational numbers. Given the common definition of real numbers as Cauchy sequences (or equivalence classes thereof), the possibility of giving a potentialist treatment of infinite sequences is of obvious philosophical and mathematical interest. Consider Hilbert’s remarks on Weierstrass’ accomplishments:

[T]he discussions about the foundations of analysis did not come to an end when Weierstrass provided a foundation for the infinitesimal calculus. The reason for this is that the significance of the infinite for mathematics

1 As usual, hx1, ..., xni denotes the number which codes the sequence x1, ..., xn, and if n and m are codes of two sequences, then n_m codes the result of concatenating n and m.

45 had not yet been completely clarified. To be sure, the infinitely small and the infinitely large were eliminated from analysis, as established by Weier- strass, through a reduction of the propositions about them to [propositions about] relations between finite magnitudes. But the infinite still appears in the infinite number sequences that define the real numbers, and, further, in the notion of the real number system, which we conceive to be a an actually given totality, complete and closed. (Hilbert, 1925, 369)

Hilbert, for obvious reasons, did not consider the sort of potentialism developed here as a way to ‘clarify the significance of the infinite for mathematics.’ But there is nothing to stop us from taking that view. The viability of a potentialism that extends to include analysis, then, requires an account of infinite sequences, taken as potential infinities. To take a sequence of natural numbers to be potentially infinite is to think of it as incomplete, or never actually infinite, but as possibly exceeding any finite bound in length. Temporal language is well-suited to describe the idea: a potentially infinite sequence can be thought of as one that begins with some finite length and grows in time as more entries are successively added to the sequence. Not all potentially infinite sequences will exhibit branching behavior. For instance, a sequence generated by a recursive operation will never branch because each value is completely determined by the (finitely many) preceding values. But recursive sequences are the exception rather than the rule. Consider an arbitrary 0-1 valued potentially infinite sequence. Because the sequence is potentially infinite, there must be some argument on which it is not yet defined. The two possible values are obviously mutually exclusive. Hence, if both values are genuine possibilities, then this sequence

46 exhibits branching behavior. Thus, we will have branching possibilities unless there is some pre-determined fact of the matter about what value the sequence will take on each argument. And it is hard to see what could determine such a fact of the matter. One way to pre-determine the values of the sequence is to define those values by some function. But this assumes that the values of the function are themselves pre-determined, or somehow given in advance. The claim that values of a function are pre-determined or given in advance, independently of having any rule or procedure for determining those values, looks suspiciously like actualism about that function. And since an infinite sequence is itself a function from naturals to naturals, this assumes an actualist view of the very objects we were trying to take a potentialist view of. So a potentialist treatment of infinite sequences of natural numbers seems to require branching possibilities. Moreover, we can explicitly find branching behavior by focusing on the notion of a free choice sequence. A free choice sequence is a potentially infinite sequence of natural numbers, thought of as created by an (idealized) mathematician who freely chooses each value of the sequence one after another. Free choice sequences have mostly been studied as part of intuitionistic mathematics, though in fact they predate Brouwerian intuitionism, having first been introduced in du Bois-Reymond(1882). 2 (McCarty(2005) notes the irony of the fact that du Bois-Reymond introduced free choice sequences as part of a defense of realist platonism). Now consider the idealized mathematician making a choice for the nth value of a free choice sequence. They may be free to choose multiple values, but once they have chosen, say, 0 for the nth value of the sequence they are no longer able to choose 1 for

2I should note that (Linnebo and Shapiro, 2019, note 16) acknowledge that S4.2 would not be appropriate for a theory of free choice sequences. So I am not objecting to their account, but supplementing it in a way they should be comfortable with.

47 the nth value of this sequence. Since the values of the sequence are freely chosen, both 0 and 1 are genuine possibilities. And clearly making one choice precludes making another choice. The possible ways this potentially inﬁnite sequence may be extended are branching rather than convergent. Thus we have a potential inﬁnity for which S4 rather than S4.2 is the right logic. Since the mirroring theorem essentially relies on the G axiom, we can no longer rely on

the ♦ translation to achieve a satisfactory potentialist interpretation of mathematical

discourse. In fact, even the motivation for the ♦ translation looks questionable in the context of branching possibilities. In the context of convergent possibilities, the

background space of possibilities does not change based on what one does. So ♦∃x behaves just like an ordinary existential quantiﬁer ranging over the background space of possibilities. (Indeed, the mirroring theorem turns essentially on this fact). But this is no longer the case when possibilities can branch. If the objects that you

In fact, the problem is more acute than the ♦ translation being unsatisfactory. The

problem is that the usual modal operators and ♦ are not sufficiently expressive. Consider again a potentially infinite sequence of natural numbers. One thing that the potentialist should be able to say is that there will be a billionth entry in this sequence. The idea that potentially infinite sequences grow without bound and each entry eventually gets defined is an essential part of the conception of a potentially infinite sequence. But given that the possibilities for such a sequence are branch-

ing, this cannot be expressed in terms of and ♦. For instance, if α : N → N

48 is the potentially inﬁnite sequence in question, ∃x(α(1, 000, 000, 000) = x) says that at every stage in the construction of α there exists a billionth value of α;

♦∃x(α(1, 000, 000, 000) = x) says that there could be a stage at which α has a billionth value; and ♦∃x(α(1, 000, 000, 000) = x) says that it is never ruled out that α has a billionth value. By contrast, what we want to say is that on every possible future branch of time, there is a stage at which α has a billionth value. In other words, we want to say that it is inevitable that α eventually have a billionth value. To say this we need to introduce another modal operator beyond and ♦, which we may write Iφ with the meaning that it is inevitable that φ eventually be true. (A formal semantics will be given in the next section). That this is a genuinely new operator which cannot be expressed in terms of and ♦ will be established as Corollary 3.7. This expressive deficit is unique to the case of branching possibilities, since when we have convergent possibilities and a growing domain, ♦∃x will range over the background space of possibilities and will be equivalent to I∃x (assuming the stability axioms). Another place the inevitability operator is needed is to express the condition of Cauchy convergence. Recall that a sequence α, which we may take here to be rational- valued, is Cauchy if for every rational number p > 0 there is some N such that for all n, m > N, |α(n) − α(m)| < p. Let’s consider how to formulate this condition when α is potentially infinite. We want to say that necessarily, for any p > 0, there is or will be an N such that, necessarily, for any n, m > N, if α(n) and α(m) have been defined, then |α(n) − α(m)| < p. If the sequence has a known rate of convergence, then we may be able to express N as a function of p; but this need not be the case. Moreover, the witness for N may depend on how α evolves in the future. So to express the condition that α is Cauchy, we need to say that, for every p > 0, no matter how the

49 future develops, there will come some N such that for any n, m > N, if α(n) and α(m) have been defined, then |α(n) − α(m)| < p. In other words, we need to say that for every p > 0 there will inevitably be some such N. Thus in the context of branching possibilities, the potentialist needs to extend her expressive resources. There are two significant facts about this extension. The first is that the logic that results from adding I to S4 is not axiomatizable. Thus the logical commitments of the potentialist cannot be codified in a complete set of deductive practices. The second fact is that the validity problem of this logic is very hard. In

1 fact, it is Π1-hard. In other words, if the potentialist could determine whether an arbitrary formula was valid in her logic, then she could serve as an oracle for any set of

1 3 natural numbers that can be deﬁned by a Π1 formula in the language of arithmetic. So the purely logical commitments of the potentialist incur signiﬁcant mathematical commitments. These facts are proved in the next section.

3.3 An Incompleteness Theorem

What I am calling the inevitability operator originates in the context of temporal logic, speciﬁcally the temporal logic of branching time. As with much of modern temporal logic, the study of branching time goes back to Prior.4 The intuitive idea is that if the future is open then there should be multiple possible ways the future could develop, multiple future branches of time, as it were. The inevitability operator is intended to capture the expression “at some point in the future...” in the context of branching time. If there are multiple future branches,

3See Sacks(1990). 4See Prior(1967). See also Reynolds(2002) for a more recent comparison of the so-called Ock- hamist and Piercian approaches to branching time.

50 then “at some point in the future, φ” should require that on every future branch

there is a future time at which φ is true. This contrasts with the usual ♦ operator, since ♦φ requires merely that φ be true at some future time on some branch. The inevitability operator was introduced by McCall(1979), who called it the strong future tense. Subsequently, Burgess(1980) proved the completeness and decidability of a propositional logic for branching time which included this operator.5 While ﬁrst- order temporal logics have not received much attention in philosophy, they have been studied extensively by theoretical computer scientists. The results of this section are very much of a piece with theorems that are well-known in computer science.6 For instance, Hodkinson et al.(2002) and Montagna et al.(2002) prove incompleteness results for ﬁrst-order logics of branching time, similar to the results below. The proofs presented below are novel and slightly more general than those of Hod- kinson et al.(2002) and Montagna et al.(2002) in the range of frames they apply to. Those articles assumed that the underlying frame is a tree and that the structure of time is discrete; I allow the accessibility relation to be any partial order, and the structure of time can be dense or even continuous.7 This includes the possibility that

time has a linear structure, and if time is linear then Iφ is equivalent to ♦φ. Thus a 5More recently, Doder et al.(2010) have proved that the first-order logic of branching time with this operator is complete for a non-standard class of models. (They also invoke an infinitary proof system for this completeness result, though the infinitary rule of their system governs an operator which I will not consider here.) Roughly, their models stand to my standard models as Henkin models of second-order logic stand to standard (or ‘full’) models of second-order logic. Somewhat more precisely, the truth of ‘inevitably φ’ in my models depends on what happens in all branches; the models of Doder, Ognjanović,and Markovićinclude a set Σ of branches, and the truth of ‘inevitably φ’ in their models depends only on what happens in the branches in Σ. 6In computer science, the logic of branching is known as Computational Tree Logic (CTL); there is also a slightly more expressive extension CTL∗. My operator I can be expressed in these systems by the operator combination AF. See (Gabbay et al., 1994, Ch. 4) and (Gabbay et al., 2000, Ch. 3) for an overview of first-order temporal logic and logics of branching time as well as for further references. 7On the other hand, the results of Hodkinson et al.(2002) are more informative in that they show even the two-variable fragments of their logics to be unaxiomatizable.

51 further corollary of Theorem 3.5 is that the ﬁrst-order modal logic of linearly ordered

8 frames is incomplete even with only the standard operators and ♦. Neverthe- less, these are relatively minor technical differences and it is unsurprising that the incompleteness results still apply. The proofs are worth including, though, both for their intrinsic interest and so the present work can be self-contained given that these results are not as well known in philosophy. Moreover, the main idea behind the proof of Theorem 3.5 will be relevant to the philosophical points developed in §4. Let us now turn to formally defining the potentialist’s logic. I will call this logic S4+. The logical vocabulary of S4+ will be that of first-order logic plus the modal operators I, , and ♦.

Deﬁnition 3.1. A model for S4+ is an ordered quadruple M = hW, R, D, V i, where:

• W is a non-empty set (intuitively, the set of moments of time or, as I will usually say, worlds).

• R is a reﬂexive, transitive, anti-symmetric relation on W .

• D is a non-empty set of objects (intuitively, the domain of individuals).

• V is a function that assigns to each individual constant a member of D, assigns to each n-ary relation and world a subset of Dn, and assigns to each n-ary function symbol and world the graph of an n-ary function on D.

We regard individual constants as 0-place function symbols. Note that, under this deﬁnition of model, the domain is constant from world to world. This assumption

8 Montagna et al.(2002) also show how to adapt their proof to the case of linear time. This, of course, still assumes that time is discrete.

52 is dispensable for the results that follow, though. Since the Barcan Formula char- acterizes the constant-domain models, if there were an axiomatization of S4+ with growing domains, adding the Barcan Formula would give us an axiomatization of S4+ with constant domains. But this would contradict the main theorem below. Note also that although the idea of branching time suggests that hW, Ri should be a tree, this is allowed but not required by this definition of a model.9 Aside from I, the definition of truth at a world will be familiar. For I, however, we will need one more definition:

Deﬁnition 3.2. In a model M, a chain above w is a set X of worlds linearly ordered by R, such that for all u ∈ X, wRu. A chain X above w is maximal if there is no proper superset Y ⊃ X that is also a chain above w.

Now we can deﬁne truth at a world. For simplicity, I will again assume a substitutional interpretation of quantiﬁers; if o ∈ D, let o be an individual constant such that V (o) = o.

Deﬁnition 3.3. Let M be an S4+ model, and let w ∈ W . Deﬁne M, w |= φ recursively:

• M, w |= P a1, ..., an iﬀ hV (a1), ..., V (an)i ∈ V (w, P )

• M, w |= f1(a1, ..., an) = f2(b1, ..., bm) iﬀ for some o ∈ D, hV (a1), ..., V (an), oi ∈

V (w, f1) and hV (b1), ..., V (bm), oi ∈ V (w, f2)

• M, w |= ¬φ iﬀ M, w 6|= φ

9It is worth noting that the following arguments would still go through if we required hW, Ri to be a tree. The same is not true of other restrictions that could be placed on the underlying frame. For instance, if we required either that R be an equivalence relation on W , then Iφ would be equivalent to ♦φ and the logic would simply be ﬁrst-order S5.

53 • M, w |= φ ∧ ψ iﬀ M, w |= φ and M, w |= ψ

• M, w |= φ ∨ ψ iﬀ M, w |= φ or M, w |= ψ

• M, w |= ∀xφ(x) iﬀ for all o ∈ D, M, w |= φ(o)

• M, w |= ∃xφ(x) iﬀ for some o ∈ D, M, w |= φ(o)

• M, w |= φ iﬀ for all u ∈ W such that wRu, M, u |= φ

• M, w |= ♦φ iﬀ for some u ∈ W such that wRu, M, u |= φ

• M, w |= Iφ iﬀ for all maximal chains X above w there is a u ∈ X such that M, u |= φ.

The reason that X must be a maximal chain in the clause for Iφ is twofold. The ﬁrst reason is to ensure that X ‘goes far enough’. To see what this means, suppose X may be any chain, not just a maximal chain. Now consider a model M where

W = {r ∈ R : r ≥ 0}, and suppose P is an atom such that M, r |= P iﬀ r ≥ 2. Intuitively, IP should be true at time 0: just wait until time 2, and P will be true. But the interval [0, 2) is a (non-maximal) chain above 0 such that P is false at every member of the chain. So if Iφ were true iﬀ φ is true somewhere in every chain, then IP would be false at time 0. Hence we must consider only the maximal chains when evaluating Iφ at a world. The second reason that X must be a maximal chain in the clause for I is to ensure that no moment in time gets skipped or overlooked. To see what I mean by

this, consider a model M with W = N, and suppose M, n |= P iﬀ n is odd. Then the set X of even numbers is a chain, and for no member n ∈ X does it hold that M, n |= P . Thus X is a chain none of whose members makes P true; so without the

54 maximality condition in the clause for I, IP would not be true in M. Intuitively, however, IP should be true in M, since at every moment in time, P either is already true or will be true at the very next moment. The chain X misses this because it omits moments of time that matter for evaluating IP . Requiring X to be maximal avoids this problem.

1 We can now establish that the set of validities of S4+ is Π1-hard and hence not recursively axiomatizable. The strategy of the proof is to ﬁnd a set S of natural numbers that is not recursively enumerable, and then show that the decision problem for this set is Turing-reducible to the validity problem of S4+. This means, in other words, that there is a Turing machine T which maps natural numbers to logical formulas, such that any number n is in S just in case the corresponding logical formula is a validity of the logic. Now, if the set of validities was recursively enumerable, this would mean that there is a Turing machine T 0 which, when given a logical formula, eventually halts if and only if that formula is valid in S4+. Accordingly, S would then also be recursively enumerable, since we could take a number n, run the Turing machine T to ﬁnd the corresponding logical formula, and then run the T 0. Then T 0 would eventually halt if and only if n is in S. And this would contradict the fact that S is not recursively enumerable. Thus, the set of validities cannot be recursively enumerable. This in turn will imply that there is no recursive axiomatization that is complete for validity in S4+. Now for the details. Let P be a new one-place predicate and let ≺ be any two-place relation. Make the following abbreviations:

W 1 ∀x∀y[(x ≺ y → x ≺ y) ∧ (x ⊀ y → x ⊀ y)]

W 2 ∃!xP x ∧ ∀x[P x → x ∈ ﬁeld(≺)]

55 W 3 ∀x∀y[P (x) ∧ (y ≺ x → ♦P y)]

W 4 ∀x[(P x ∧ ∃y(y ≺ x)) → I∃y ≺ xP y]

W 5 I∃x[P x ∧ ∀y(y ⊀ x)]

Lemma 3.4. S4+ |= (W 1 ∧ W 2 ∧ W 3 ∧ W 4) → W 5 iﬀ ≺ is well-founded.

Proof. (⇒) Suppose ≺ is not well-founded, so we have an inﬁnite series x1 x2 ....

Deﬁne a model M as follows: Let the set of worlds be {n : n ∈ N} and put nRm iﬀ

n ≤ m. Let the domain of objects be {xi : i ∈ N}. Assign the true interpretation of

≺ {xi : i ∈ N} to ≺ in every world. And for each world n, let {xn} be the extension of P . Then it is easy to check that M, 0 6|= (W 1 ∧ W 2 ∧ W 3 ∧ W 4) → W 5. (⇐) Assume that ≺ is well founded. Let M be an arbitrary model and w an arbitrary world in M such that M, w |= W 1 ∧ W 2 ∧ W 3 ∧ W 4. It will suﬃce to show that M, w |= W 5. Suppose not. Then there is some maximal chain X above w such

that for all u ∈ X, M, u 6|= ∃x[P x ∧ ∀y(y ⊀ x)]. Hence M, u |= ∀x[P x → ∃y(y ≺ x)]. By W 2, we know that there is some x such that P x, so then there must be some y ≺ x. So by W 4 there is some world v ∈ X, where uRv, and some y ≺ x such that M, v |= P y. But since v ∈ X, we also know that (relabeling variables for clarity)

M, v 6|= ∃y[P x ∧ ∀z(z ⊀ y)]. Thus by the same line of reasoning we can ﬁnd a z ≺ y, and so on. So ≺ is not well-founded, contradicting our assumption.

To proceed to the main theorem, we will need a few facts about recursive ordinals. (The material referred to here can be found in (Sacks, 1990, Ch. 1).) An ordinal α is said to be recursive if there is a recursive function e such that {hx, yi : {e}(hx, yi) = 0} is a well-ordering of the natural numbers with order-type α. The question of whether

1 e is the code of a recursive ordinal is Π1-complete.

56 Theorem 3.5. Assume that we have a language that includes the language of arithmetic, and a unary predicate P . Let V Q be the conjunction of axioms of Robinson arithmetic Q, and deﬁne ≺ as follows (note that e is a parameter):

x ≺ y := {e}(hx, yi) = 0

Then e codes a recursive ordinal just in case:

^ S4+ |= Q ∧ ∀x∀y(x = y ∨ x ≺ y ∨ y ≺ x) ∧ [(W 1 ∧ W 2 ∧ W 3 ∧ W 4) → W 5]

Proof. The conjunction V Q is included so that all recursive functions are repre- sentable. Since Robinson arithmetic is finitely axiomatized, the conjunction is finite, and we still have a well-formed formula. The second conjunct, ∀x∀y(x = y ∨ x ≺ y ∨ y ≺ x) is satisfied iff e (codes a function that) defines a linear order. By Lemma 3.4, the third conjunct is satisfied iff the order defined by (the function coded by) e is well-founded. A well-founded linear order is a well-order, as required.

1 Thus, the validity problem for S4+ is Π1-hard.

Corollary 3.6. S4+ is not recursively axiomatizable.

Proof. By Theorem 3.5, if S4+ were recursively axiomatizable, then the problem of whether e codes a recursive well-ordering is reducible to the problem of whether a

0 certain formula is provable. But this latter problem is Σ1, in contradiction to the

1 Π1-completeness of the recursive ordinals.

57 Corollary 3.7. I cannot be deﬁned in terms of or ♦ by the standard logical connectives.

Proof. Since S4 (i.e. S4+ without the operator I) is axiomatizable, if I were deﬁn- able from or ♦, then S4+ would be axiomatizable.

3.4 Axioms for Inevitability

Here is a Hilbert-style proof system for the logic S4+. The main theorem of the previous section entails that this system is incomplete, though it is easy to check that it is sound. What I have labeled M1 is of course the familiar K axiom, M3 is the T axiom, and M6 is the usual 4 axiom.

M0 Axioms for classical quantiﬁcational logic

M1 (φ → ψ) → (φ → ψ)

M2 (φ → ψ) → (Iφ → Iψ)

M3 φ → φ

M4 φ → Iφ

M5 φ → ¬I¬φ

M6 φ → φ

M7 IIφ → Iφ

M8 Iφ → Iφ

M9 ∀σφ ↔ ∀σφ

58 M10 I∀σφ → ∀σIφ

M11 ∃σIφ → I∃σφ

The rules are modus ponens, substitution, and necessitation. The usual completeness theorems for S4, S4.2, and other similar modal logics allow us to reason either model theoretically or deductively, as is more convenient. When, in Part II, we use S4+ as our logic, we will have to pay attention to the deductive structure of our arguments to ensure that theorems are indeed provable and not merely valid. It is reassuring to note, however, that we can still safely invoke model theoretic reasoning as long as we limit ourselves to I-free formulas.

Proposition 3.8. If φ has no occurrences of I, then:

S4+ |= φ ⇔ S4 |= φ ⇔ S4 ` φ

Proof. The first equivalence follows from the facts that (i) any S4+ model is also an S4 model, and (ii) the definition of M, w |= φ is the same in S4+ and S4 for formulas that have no occurrences of I. Thus φ holds in every S4+ model iff φ holds in every S4 model. The second equivalence is the standard completeness theorem for S4.

It also follows that S4+ is conservative over S4.

3.5 The Philosophical Upshot

These technical results have several interesting philosophical implications. I will mention three.

59 First, the diﬃculty of the validity problem for S4+ reveals the logical commitments of the potentialist to be signiﬁcant and substantive. As shown above, for any

1 Π1 sentence φ in the language of arithmetic, there is a formula that is an S4+-validity just in case φ is true. This is partly interesting as a matter of bookkeeping. Whether the complexity of S4+ tells against its status as logic is outside the scope of this work, but is of a piece with debates about, for instance, the status of second-order logic in light of its mathematical commitments.10 Whether these commitments tell against the potentialist position more generally is a matter of what other commitments the potentialist has. I am not arguing here that these commitments are necessarily problematic for the potentialist, but they are interesting to note and form part of the backdrop against which potentialism should be evaluated. Another part of why these commitments are interesting is because they are somewhat surprising. The distinction between actual infinity and potential infinity is a metaphysical one, and actual infinity is seemingly the more demanding metaphysical notion. One might have expected the metaphysically weaker commitment to a merely potential infinity to correspond to a weaker logical/mathematical commitment. But this turns out not to be the case. The second philosophical conclusion is that the result of adding I gives the potentialist increased expressive power over the classical mathematician with a first-order non-modal language. As Lemma 3.4 shows, S4+ is able to express that a given order relation is well-founded. As is well known, this is not expressible in first-order logic. The language of the potentialist thus enables them to draw important distinctions that elude the non-modal first-order theorist. This again mirrors debates about

10See Jané(1993), Väänänen(2001), Koellner(2010), Shapiro(2012), and Brauer(2018) for some of the threads in this debate.

60 second-order logic: the putative disadvantage of carrying substantial mathematical commitments is weighed against the putative advantage of increased expressive power. The third philosophical conclusion is that there is little hope of providing anything like the mirroring theorems of Chapter 1 for potential infinities with branching possibilities. The philosophical upshot of the mirroring theorem was that we could systematically reinterpret standard mathematical discourse in a potentialist-friendly way that did not require us to substantively tinker with the mathematics. But the incompleteness theorem above suggests that there cannot be any effective translation T of standard mathematical discourse into the language of S4+ which is both sound and faithful, i.e. which satisfies the following equivalence (where the unsubscripted |= is first-order classical logical consequence):

T T ∆ |= φ ⇔ ∆ |=S4+ φ

The reason, of course, is that classical logical consequence is axiomatizable, by the completeness theorem. So S4+ consequence cannot be reduced to classical consequence. To conclusively establish that such a translation is impossible, we would have to show that for any ostensible translation T , the fragment of S4+ that is in the image of T is not axiomatizable. I do not have a sufficiently general characterization of all possible translations T , and so I am not in a position to definitively prove that there can be no sound and faithful translation. But the proof strategy behind Theorem 3.5 is very flexible, and incompleteness results in modal logic tend to be quite robust. So it is quite likely that for any translation T , the fragment of S4+ in the range of that translation is also unaxiomatizable. If this conjecture is correct, then any putative

61 potentialist translation of non-modal language into a modal language cannot be both sound and faithful. Without such a translation in hand, the potentialist would not be able to give an account of inﬁnite sequences of natural or rational numbers, such as Cauchy sequences, that preserved all the standard mathematical results about such objects. Thus, insofar as one wants to retain such standard results, the actualist seems to have an advantage.

3.6 A Remark on Models and Potentialism

Throughout this chapter and the previous one I have freely indulged in talk of possible worlds and availed myself of Kripke models. While possible worlds and Kripke models provide an illuminating heuristic for thinking about the modal logic of potential infinity, it is worth reflecting on how this use of model theory fits into the dialectic between the actualist and potentialist. Kripke models can play different roles for the potentialist and the actualist. The actualist of course accepts the existence of actual infinities and may even find the idea of potential infinity obscure.11 What the Kripke models provide, then, is a way for the actualist to make sense of the notion of potential infinity and the potentialist’s reasoning about infinity. From the perspective of the actualist, modal logic and its attendant Kripke models provide a way of understanding or explicating potentialism.

11Recall the remarks of (Niebergall, 2014, 256-7), quoted in Ch. 1: “Personally, I simply have no ordinary understanding of [the phrase ‘potentially infinite’], and I do not find much help in the existing literature. ... The reason is that those philosophers who are interested in the theme of the potentially infinite are usually drawn to it because they regard it as desirable to avoid assumptions of infinity (i.e., of the actual infinity), yet do not want to be restricted to a mere finitist position. An assumption of merely the potentially infinite seems to be a way out of this quandary: it seems to allow you to have your cake and eat it too.”

62 For the potentialist, however, the use of Kripke models is problematic. Recall that we thought of the worlds in these models as the stages in the construction of the potentially infinite domain. It would be rather awkward to hold simultaneously that some domain was merely potentially infinite while the collection of stages in the construction of that domain formed an actual infinity. So the potentialist should not accept that there is an actual infinity of stages in the construction of a potentially infinite domain. The stages should be potentially infinite just like the domain. On the other hand, it is also natural to think that any infinite set is an actual infinity, since the members of a set form a ‘completed’ collection. And since Kripke models include a set of all worlds, the use of Kripke models carries a commitment to actual infinity.12 Thus the potentialist cannot take the use of Kripke models at face value. For the potentialist, an infinite Kripke model will only potentially exist. (This is an instance of the familiar fact that notions encoded in an object theory often reappear in the metatheory.)13 Despite this, Kripke models are useful to the potentialist in allaying actualist objections to their point of view. In essence, the potentialist can say to the actualist: “I don’t think these models actually exist. But by your lights they do, so here is a way for you to understand what I’m up to.”14 This difference between how the actualist and potentialist understand the model theory has implications for assessing potentialism. Hamkins(2018) has argued that an infinite collection is only truly a potential infinity if it exhibits branching behavior. Infinite collections with convergent possibilities are, in Hamkins’ view, implicitly

12At least in the cases that interest us, where there are inﬁnitely many worlds. 13This suggests an interesting task for the potentialist: using a potentialist metatheory, can one describe a class of potentially inﬁnite models that provide an adequate semantics for potentialist discourse? 14Thus Kripke models play a similar role in the potentialist’s argument as representation theorems do in the nominalist project of Field(1980). Thanks to Chris Pincock for suggesting the analogy.

63 actualist. “The reason,” he explains, “is that if the universe fragments of ones potentialist system are mutually coherent with one another, [i.e., if the possible extensions are always convergent] ... then there is a unique limit model to which the system is converging.” Hamkins makes two points here. First, he notes that the potentialist

can ‘refer to’ what happens in the limit model via the ♦ translation, so that “it is as though the limit model actually exists, for all the purposes of speaking about what is true or false there.” Moreover, because the limit model is unique and wholly determined by the potentialist’s universe fragments, the actualist’s limit universe can be said to “supervene on the potentialist ontology,” so that the “limit model has an implicit existence” (Hamkins, 2018, 33). All this is exactly right from the actualist point of view. But to even put the matter in terms of a limit model is to beg the question against the potentialist. From the potentialist point of view, the possible extensions of a potentially inﬁnite system do not converge to a unique limit because they do not converge at all. The potentialist can regard ‘the limit model’ as a way of speaking about what is ultimately possible. Since the potentialist’s ontology includes the given range of possibilities, Hamkins is right in a sense that the limit universe supervenes on the potentialist’s ontology. But from the potentialist point of view, while we can always approach this widest range of possibilities, we can never actually reach it. For the potentialist, then, the ‘limit universe’ has no more existence than the horizon.

And while it is true that the ♦ translation provides a sense in which the actualist and potentialist agree about what is true, this fact cuts both ways. The actualist can interpret this fact as showing that the potentialist is really talking about actual inﬁnities but doesn’t want to admit it; the potentialist, by contrast, can interpret this fact as showing that actual inﬁnities are a mere fa¸conde parler which the actualist has

64 taken too literally. Ultimately, the fact that a metaphysical disagreement does not lead to mathematical disagreement does not show that the metaphysical disagreement is not genuine.

3.7 Conclusion

Recognizing the modal character of the concept of potential infinity allows us to draw on the tools of modal logic in giving an account of how to reason about potential infinity and of the commitments of potentialism. We saw previously that there is an important choice point in determining the right modal logic for potentialism about a given collection. This choice is a matter of whether the possible ways of expanding that collection are ever mutually exclusive. If not, then we have convergent possibilities, and as we saw in Chapter 2, S4.2 is an appropriate modal logic and we can avail ourselves of the mirroring theorem to interpret mathematical discourse on behalf of the potentialist. On the other hand, if the possible ways of expanding the collection are ever mutually exclusive, then we have branching possibilities. In this case, I have argued in the present chapter, the modal logic should be S4 rather than S4.2; but standard S4 is not sufficiently expressive to allow the potentialist to say everything that they want to say. To account for this shortcoming, we can add the inevitability operator and use the extended logic that I have called S4+. The result, however, is not axiomatizable. This reveals the significant mathematical commitments of potentialism and quite likely precludes a sound and faithful potentialist interpretation of standard non-modal mathematical discourse.

65 Part II

A Modal Theory of Free Choice Sequences

66 Chapter 4

Free Choice Sequences

In developing intuitionistic mathematics, L.E.J. Brouwer introduced the notion of a free choice sequence. Roughly, these are potentially inﬁnite sequences of natural numbers whose values are freely chosen sequentially, one by one. Free choice sequences are a central piece in the development of the intuitionistic theory of the continuum, being key to Brouwer’s continuity theorem, Brouwerian counterexamples to the law of excluded middle, and other characteristically intuitionistic theorems.1 Free choice sequences are also among the more controversial aspects of Brouwerian intuitionism, and it is not hard to see why. The very conception of freely choosing the values of a sequence one after the other introduces agent-centric and temporal dimensions to mathematics that are foreign to traditional ways of thinking about mathematics. The fact that free choice sequences also lead to results that contradict classical mathematics makes them doubly suspect. In this respect the use of choice sequences in intuitionistic analysis diﬀers from intuitionistic arithmetic. The intuitionistic theory of Heyting Arithmetic (HA) is a proper subtheory of classical Peano Arithmetic (PA), so the classical mathematician can regard HA as simply being the constructive fragment of PA.2

1See, e.g. van Atten(2018) for an assortment of paradigmatic examples. 2Another route to making sense of HA from the perspective of PA is to use G¨odel’smodal translation of intuitionistic logic into S4. One can thus obtain a (faithful) epistemic interpretation of HA in PA plus a modal operator. See Shapiro(1985), Goodman(1974).

67 Free choice sequences are thus often seen as, at best, the awkward stepchild left by Brouwerian intuitionism.3 For instance, in an often-quoted passage, Bishop writes:

[Brouwer] seems to have [had] a nagging suspicion that unless he personally intervened to prevent it the continuum would turn out to be discrete. He therefore introduced the method of free-choice sequences for constructing the continuum, as a consequence of which the continuum cannot be discrete because it is not well enough deﬁned. This makes mathematics so bizarre it becomes unpalatable to mathematicians, and foredooms the whole of Brouwer’s program. (Bishop, 1967, 6)

Feferman also comments (albeit somewhat less pessimistically):

Brouwer introduced ... a novel conception, that of free choice sequences (f.c.s), ... of which one would have only ﬁnite partial information at any stage. Then with the real numbers viewed as convergent f.c.s. of rationals,

a function f from R to R can be determined using only a ﬁnite amount of such information at any given argument. Brouwer used this line of

reasoning to conclude that any function from R to R must be continuous, in direct contradiction to the classical existence of discontinuous functions. With this step Brouwer struck oﬀ into increasingly alien territory, and he found few to follow him even among those sympathetic to the constructive position. (Feferman, 1998, 47)

3At least by those outside the intuitionist school. Within intuitionist camps, choice sequences have received a fair amount of attention; though even within such camps their status is subject to some debate. For instance, see (Troelstra, 1977, Appendix C), Troelstra(1983), and (Dummett, 2000, §7.5) for the justiﬁcation of theories of choice sequences, and see Myhill(1967), Troelstra(1968), (Troelstra, 1977, Appendix B), and van Atten and van Dalen(2002) for discussion of controversial continuity principles for choice sequences.

68 And Tait writes:

I am ... rejecting the Brouwerian conception that the subject has indi- viduated an inﬁnite choice sequence by the act of beginning to choose its successive members. For this idea depends upon a subjectivist stance: the sequence in question is the one that I am choosing; it is always unﬁn- ished but becomes more determinate in time, as I make more and more choices. ... From an objectivist point of view, time does not enter into mathematics: its truths are time independent. (Tait, 2005, 16)

As suggested above, we can discern at least three diﬀerent objections to free choice sequences: they introduce a temporal aspect to mathematics, they introduce a subjective or agential aspect to mathematics, and they lead to results that contradict classical mathematics. Together, these objections can foster a suspicion that the concept of choice sequences as mathematical objects is deeply bizarre, if not incom- prehensible. I aim to dispel this suspicion. I will be developing a modal theory of free choice sequences in a classical background theory. As a result, each of the three objections to free choice sequences will be shown to be either erroneous or ultimately unproblematic. The objection that free choice sequences introduce a temporal dimension into mathematics will be defused. My account does take the temporal nature of free choice sequences seriously and gives it an explicit treatment using the resources of modal logic. The result is a perfectly sensible mathematical theory of intratemporal objects, and the truths of this theory are as objective and time-independent as those in any other area of mathematics—just as Tait would have it.

69 The objection that free choice sequences introduce a subjective or agential aspect to mathematics is shown to be simply erroneous. Nowhere does the notion of an agent or a subject enter into my theory. The notion of a mathematical agent helps motivate certain choices of axioms, but this is entirely at the pre-formal level. Finally, the objection that free choice sequences lead to results that contradict classical mathematics is also shown to be erroneous, since my theory explicitly includes standard second-order arithmetic alongside free choice sequences, and all with classical logic in the background. The modal theory of free choice sequences does suf- ﬁce to prove modal versions of some of the intuitionists’ anti-classical results. These include the unprovability of a real number being determinately rational or irrational, the existence of a bounded monotone sequence of rationals that does not necessarily converge, and a weak version of the result that there exist no discontinuous functions on the reals. In the modal setting, however, these theorems concern the intratemporal free choice sequences, and hence are modal in character, rather than concerning the familiar classical real numbers.

4.1 The goal

The goal of Part II of this dissertation is to analyze choice sequences in modal terms in a classical background theory. There are two things that I hope to accomplish with this project: ﬁrst, to show that the notion of free choice sequence is compre- hensible from the classical point of view, and, second, to show how, having done so, portions of intuitionistic analysis—or rather, modal analogues thereof—are available to the classical point of view. To the extent that this second goal is accomplished, classical and intuitionistic mathematics need not be seen as competing or alternative

70 approaches to mathematics, and intuitionistic mathematics—in particular, portions of intuitionistic analysis—can be seen to be legitimate from the classical perspective. This project can be seen in part as following up on a recent proposal from Kripke (2019). There, Kripke “outline[s] how a concept of free choice sequence could be combined with an acceptance of classical mathematics” (3). The essential idea is to imagine a classical mathematician facing a potential infinity of points in time at which they can freely choose values for a growing sequence of natural numbers. Kripke’s paper is largely programmatic, though; he does not present any explicit theory of temporal free choice sequences that is meant to extend classical analysis and only briefly discusses which intuitionistic principles might carry over to the setting he proposes. My aim in this paper is to address exactly such questions, moving beyond informal conceptions or intuitive motivations to give an explicit and properly mathematical theory of free choice sequences. Moschovakis(2017) has also introduced a system inspired by Kripke’s proposal. Her approach is to use a multi-sorted theory that contains intuitionistic analysis, a negative translation of classical analysis, and an axiom asserting that for any choice sequence it is not impossible that there is a determinate (lawlike) sequence that agrees everywhere with it. A modified realizability argument shows the theory to be consistent.4 Perhaps the most important difference between Moschovakis’ system and the approach below is that I take the temporal aspect of choice sequences to be an integral part of the concept; accordingly, the temporal aspect figures explicitly in my theory in the form of modal operators. As Moschovakis(2017) observes in concluding

4The idea that classical and intuitionistic analysis are in some sense compatible is also found in Moschavakis’ earlier work, e.g. Moschovakis(1996); cf. also Moschovakis(2016). In these pieces Moschovakis studied a system that extended Kleene’s system FIM of intuitionistic analysis and whose lawlike portion coincides with classical analysis.

71 her article, her theory “gives no further insight into the stage-by-stage activity of a creating subject. All we can claim is that from the perspective (unattainable by the creating subject) of the end of time, Kripke’s idea is classically feasible.” In the following chapters, by contrast, I will give the step-by-step activity of the idealized mathematician a central role. I will proceed by informally sketching the concept of a free choice sequence in §2. In §3 I will introduce the modal framework and show how various formal and quasi- formal conceptions of free choice sequences can be represented in this framework, making plausible the claim that modal logic can be used to capture the general notion of a free choice sequence. In the next two chapters I then pursue a speciﬁc implementation of this idea, developing a formal theory of free choice sequences. In Chapter 5 I present a theory of so-called lawless sequences, and in Chapter 6 I generalize the modal theory from Chapter 5 to include non-lawless choice sequences.

4.2 Free choice sequences

The basic idea of a choice sequence can be seen as arising from two components of Brouwer’s thought. The ﬁrst component was his metaphysical views: Brouwer, of course, thought of mathematical objects as purely mental constructions with no objective or mind-independent existence. Accordingly, one could not truly assert the existence of some mathematical object without a method for constructing that object mentally. The second component of Brouwer’s thought was his acceptance of the arithmetized account of the continuum that had become widespread in classical mathematics

72 since Dedekind and Cantor.5 The arithmetized view of the continuum required defining real numbers in terms of infinite sets or sequences of rational numbers (àla Dedekind cuts, Cauchy sequences, or infinite decimal expansions). Brouwer’s constructivism required that any such sets or sequences must be given as possible mental constructions. (For simplicity I will simply talk of sequences rather than sequences or sets). If there is a law that can be followed to construct the sequence, such a sequence passes constructivist muster, since the law provides a method for constructing the sequence. But the fullness of the continuum is not exhausted by the Cauchy sequences that can be given by a law. One wants to countenance any arbitrary Cauchy sequence. And the idea of an arbitrary Cauchy sequence emerges in intuitionism as a free choice sequence: an infinitely proceeding series of choices of rationals not bound by any law, but determined freely by an agent. Since mathematical agents are limited, we must regard these sequences as necessarily unfinished. But the intuitionist regards objects that could, at least in principle, be constructed by a mathematician as legitimate objects of study. And since there is no in-principle finite limit to how long a sequence a mathematician could create, we can regard a choice sequence as indefinitely proceeding or potentially infinite, though not as a completed or actually infinite sequence.

5Before 1914 Brouwer had adopted a non-punctiform, geometric view of the continuum as given to intuition as a unified whole. After 1914, however, he espoused an arithmetized account of the continuum. See Troelstra(1982) for an account of the history of Brouwer’s thought in this area. Troelstra identifies Brouwer(1914) as the first appearance in print of choice sequences as acceptable intuitionstic objects. Some readers have thought that this shift indicated a rejection of the geometric conception, e.g Placek(1999), Troelstra and van Dalen(1988). van Atten(2007), however, argues that although Brouwer adopted the arithmetized account, he never rejected the geometric conception. On this reading, the geometric conception does not figure directly in Brouwer’s later writings because the arithmetic conception (developed with choice sequences) suffices for the mathematical development of analysis (cf. Heyting(1974)). But, van Atten argues, the philosophical views that led Brouwer to accept the primitive geometric intuition of the continuum remain in his later writings (van Atten, 2007, 34).

73 Because of the idealization involved, it is common to describe choice sequences as potentially inﬁnite sequences of numbers created by an idealized mathematician successively picking each member of the sequence. They are free to pick any number they like; but they are also free to impose constraints on their future choices. Thus we countenance choice sequences that are, after some stage, bound by some laws as well as those that are free from all constraints. As Brouwer(1952) described the matter:

[Intuitionism] recognizes the possibility of generating new mathematical

entities: ﬁrstly in the form of inﬁnitely proceeding sequences p1, p2, ..., whose terms are chosen more or less freely from mathematical entities previously acquired; in such a way that the freedom of choice existing

perhaps for the ﬁrst element p1 may be subjected to a lasting restriction

at some following pv, and again and again to sharper lasting restrictions

or even abolition at further subsequent pv’s, while all these restricting

interventions, as well as the choices of the pv’s themselves, at any stage may be made to depend on possible future mathematical experiences of the creating subject. (p. 140)

These ideas express the basic concept of a free choice sequence which I will be trying to capture in a modal framework. At the same time, in giving the modal account I do not take myself to necessarily be bound by any particular statement or idea of Brouwer’s. Given that his own ideas about choice sequences changed a number of times over his lifetime, this is not a feasible or even desirable goal.6 Similarly, the variety of explicit accounts of choice sequences that developed in later

6See (Troelstra, 1977, Appendix A) and Troelstra(1982) for an overview of some of these shifts in Brouwer’s thinking.

74 intuitionistic literature led Troelstra to claim that “there are a great many notions of choice sequence which have to be regarded as distinct primitive notions ” (Troelstra, 1983, 225, emphasis original). This should not be taken to suggest that there is no core intuitive notion of a free choice sequence—in this section I have sketched just such a notion. Troelstra’s point, rather, is that in ﬂeshing out a fully detailed account of choice sequences there are a number of decision points where multiple ways of proceeding are compatible with the core idea of a free choice sequence. Although this situation precludes a simple appeal to a generally accepted deﬁnition of a choice sequence, we can still take it as an ideal for the modal account of choice sequence to cohere with the conception(s) of choice sequences found in Brouwer, Heyting, Troelstra, and other intuitionists. The important thing is that the modal account should allow us to capture the role that choice sequences play in intuitionistic arguments. I will make good on this by showing how a modal theory of choice sequences allows us to prove modal analogues of important intuitionistic results.

4.3 The modal approach

In this section I will outline the modal approach to choice sequences and show how they allow us to express various versions of choice sequences that have been proposed in the literature. This section will be a mostly informal outline before giving an explicit theory of choice sequences in the following chapters. The guiding idea is that choice sequences are intra-temporal mathematical objects, and we capture their temporal character using modal operators. The main modal operator will be , meaning “at all times henceforth ...”, as well as its dual ♦, “at some later time ...” By ﬁat, we can take the later than relation to be reﬂexive, so that

75 henceforth includes the present moment. Obviously, later than is transitive, and we can take it that the ﬂow of time is not cyclical, so that later than is anti-symmetric. Since we cannot assume that the ﬂow of time has any other structure, the appropriate modal logic to use will be S4.

4.3.1 Notational preliminaries

We will use a sort of variables for choice sequences α, β, γ, ...; intuitively these will be partial functions on N (though in the formal theories of Chapters 5 and 6 we will use a trick to dodge the technicalities of a free logic that partial functions would entail). Individual variables x, y, z, ... will be taken to range over natural numbers. Numerical terms will always be taken to denote. There will also be a sort of variables X, Y, Z, ... ranging over sets of natural numbers as in classical analysis. These sets are extensional objects whose membership does not change over time. We can assume in the background all the usual coding apparatus for finite sequences of natural numbers as found in, for instance, Hájekand Pudlák(1993). This provides us with a formula Seq(x) which is true of exactly those numbers x which code finite sequences; such numbers x are known as sequence numbers. It is sometimes also convenient to write x ∈ Seq rather than Seq(x). We will write hn1, ..., nki

th for the (code of) the ﬁnite sequence with i member ni. There are also length and projection functions for sequences. We will use lh for the length function and (·)i for the projection function. So if x = hn1, ..., nki, then lh(x) = k, and for all i ≤ k,(x)i = ni. There are two important pieces of notation that are unique to the literature on choice sequences (although widespread within that literature). First, α(x) is used

76 to denote the initial segment of α of length x. Thus, α(x) = y iﬀ y ∈ Seq and for

all z < x, α(z) = (y)z. Second, it is common to use α ∈ x as an abbreviation for α(lh(x)) = x. What this means is that x is a sequence number encoding an initial segment of α. Since Greek letters are being used for choice-sequence variables, we will use A, B, ... as metavariables ranging over formulas.

4.3.2 Conceptions of choice sequences

As noted above, there are a variety of accounts of choice sequences which, although sharing an intuitive core, differ in the precise details attributed to choice sequences. In this section I show how modal resources suffice to capture a number of the most influential conceptions of choice sequences. Part of the core conception of a choice sequence is that they are sequences of numbers that grow with time. Thus for any conception of choice sequence we will impose the following requirements:

• ∀x(∃yα(x) = y → ∀z < x∃yα(z) = y)

• ∀x♦∃yα(x) = y

•∀ x∀y(α(x) = y → α(x) = y)

• ∃x∀yα(x) 6= y

The first requirement says that if α is defined on x then it is defined on every number less than x as well. The second says that for any number x, it is possible for α to be defined on x. The third says that once a value for α(x) is chosen, it cannot

77 be changed. The fourth says that α is never deﬁned everywhere, it always remains uncompleted.7 Now I turn to showing how we can use this framework to represent various semi- formal conceptions of free choice sequences that have been proposed in the literature, making it plausible that the modal framework does have the potential to unify classical and intuitionistic analysis.

Lawless Sequences One point of divergence between choice sequences is whether they are subject to any restrictions. For instance, the idealized mathematician might restrict herself to choosing even numbers for a given sequence. Lawless sequences are sequences that have no such restrictions. Troelstra suggests the image of tossing a die to make each choice. This is represented here as:

∀x(¬∃yα(x) = y → ∀y♦α(x) = y)

There is also the possibility that a ﬁnite number of choices have been made determinately before starting to toss the die:

Seq(m) ∧ ∃x(α(x) = m ∧ ∀y > x∀z♦α(y) = z)

Recall α(n) is the initial sequence of the ﬁrst n values of α.

7The statement of the fourth postulate here is based on the assumption that the natural numbers are a completed totality, rather than a potential inﬁnity that grows with time. If one were inclined to a more thoroughgoing potentialism that included potentialism about natural numbers as well as potentialism about free choice sequences, then the fourth postulate might be replaced with ♦∃x∀yα(x) 6= y.

78 Lawlike Sequences These are sequences given by some law. The mathematical importance of lawlikeness, for the intuitionist, is that lawlike sequences are entirely determinate. Each value is preordained. Since we have our stock of classical, determinate functions available, we can say that a choice sequence is lawlike when its values necessarily coincide with those of a given determinate function. Letting f : N → N be a classical function variable (using the usual abbreviation of a function variable for a relational variable that is total and many-one), a sequence α is lawlike if it satisﬁes:

∀x∀y(∃zα(x) = z → (α(x) = y ↔ f(x) = y))

Note that we are not committed to saying that for every classical function there is a lawlike choice sequence that coincides with it. The formula above does not assert the existence of any lawlike functions, but merely gives us a way of asserting that the sequence α is lawlike. Which functions f are able to serve as laws for constructing sequences can be left as an open question for now. If we can say that every function satisfying A(f) can be given by a law, then we can say that for every such f there is an α as above.

Constrained sequences These are choice sequences that are, from the beginning, subject to some constraints. Intuitionistically, these are choice sequences each of whose initial segments is a member of some initially given spread.8 There are two diﬀerent ways to express this, which may or may not be diﬀerent depending on what

8The term ‘constrained sequence’ is my own; in the literature people talk about sequences that are subject to a spread law or that are lawless within a spread. The notion of a constrained sequence corresponds more closely to a sequence that is subject to a spread law, since the sequence may or may not be lawless within the spread.

79 comprehension principles we have in our classical theory. The ﬁrst version is:

∀x∀y(α(x) = y → A(x, y))

The second version uses a set variable in place of the formula:

∀x∀y(α(x) = y → hx, yi ∈ X)

Choosing constraints As well as giving a constraint from the beginning that the sequence will satisfy, the subject may at any stage choose to impose new constraints that the rest of the sequence will satisfy. The idea here is that we will have an indexed family of sets {Xn} and the subject can choose to constrain all future choices to one of those sets. Instead of adding third-order variables, this family of sets will be coded as a single set X = {hn, xi : x ∈ Xn}. Admittedly, whether this is really an adequate formalization of the intuitionistic idea will depend on what choices of constraints we take to be permissible options for the subject, and what comprehension principles we have in the background. Having noted those caveats, we can express that α is a sequence for which we choose constraints as follows:

∀x∀y[α(x) = y →

0 0 0 0 0 [∃n∃m(y = hn, mi) ∧ ∀x > x∀y (α(x ) = y → h(y)1, (y )0i ∈ X)]]

The idea here is that the subject is choosing ordered pairs of numbers. The ﬁrst coordinate is the ‘real’ sequence of numbers, and the second coordinate is the code of the set that constrains all future choices. To allow that a constraint need not

80 th be imposed, we can stipulate that X0 = N, so choosing the 0 set amounts to not imposing a constraint yet. van Atten has also discussed the possibility of imposing provisional constraints which may be lifted subsequently. To capture this, we will have the subject again choose a pair of numbers, but only require that the constraint govern the very next choice:

∀x∀y(α(x) = y → (∃n, m(y = hn, mi ∧ h(α(x − 1))1, yi ∈ X)))

As long as the subject wishes to keep a given constraint in place, they can keep choosing ordered pairs with the same second coordinate.

Projections of sequences From whichever account of choice sequences one starts, a natural extension is to consider the result of creating a new sequence by applying some mapping to an already existing sequence. The resulting sequence is called a projection of the original sequence, in analogy to the geometrical notion of project- ing a ﬁgure by applying a mapping. The intuitive picture that motivates studying projections of choice sequences is that of the idealized mathematician creating a new sequence whose values are chosen by reference to the values of a sequence that they have already begun. For instance, if the idealized mathematician has begun a choice sequence α, then they could decide to create a new sequence β such that β(x) = 1 when α(x) is even and β(x) = 0 when α(x) is odd.

81 This idea can be captured modally as follows, where β is the sequence being formed by projection of the α’s according to the mapping f:

∀x(β(x) = y → (α1(x) = x1 ∧ ... ∧ αk(x) = xk ∧ f(x1, ..., xk) = y))

This concludes our brief survey of how to express various conceptions of choice sequences in the modal framework. Clearly the modal approach has the resources to capture many of the central ideas about free choice sequences, and the prospects are good for developing a theory of free choice sequences in a modal extension of classical mathematics. The remainder of this dissertation aims to make good on those prospects. Of necessity, I will not develop a theory that includes all of the conceptions outlined here. I will begin in the next chapter by developing a theory of lawless sequences. Then the ﬁnal chapter will expand the theory of lawless sequences by allowing sequences to be formed by projection. As we will see, this suﬃces to develop some interesting and important pieces of intuitionistic analysis.

82 Chapter 5

Lawless Sequences

In this chapter I introduce a modal theory of lawless sequences. I begin in §1 by outlining the intuitionistic theory of lawless sequences, called LS. Then in §2 I

develop the modal theory of free choice sequences, called MCLS. A Kripke model shows the theory to be consistent. Moreover, MCLS proves modal-logical analogues of the axioms of LS, which supports the claim that MCLS adequately captures the intuitionistic concept of a lawless sequence.

5.1 The Intuitionistic Theory of Lawless Sequences

In this section I will provide a brief overview of the intuitionistic theory of lawless sequences. I will both present the precise theory of lawless sequences and discuss its informal motivation. In the next section I give an explicit modal theory of lawless sequences. The intuitionistic theory of lawless sequences is a nice place to begin because it is well-understood and relatively simple, and also because it provides a starting point for deﬁning more complex theories of choice sequences. To reduce clutter, in this section only I will use n, m as variables ranging over sequence numbers, so that ∀n and ∃n should be taken as abbreviating ∀n ∈ Seq and ∃n ∈ Seq.

83 There are four axioms of this theory. The first axiom says that for every possible initial segment, there is a lawless sequence with that initial segment. The second says that (extensional) identity is decidable for lawless sequences. The third says that any property A which holds of a lawless sequence α depends only on some finite initial segment of α. Thus, the third axiom justifies a continuity principle for mappings from lawless sequences to numbers. The fourth axiom is also a continuity principle, saying that whenever a lawlike sequence can be chosen from a lawless sequence, then there is a uniform, lawlike way of choosing each such lawlike sequence on the basis of the neighborhood in which each lawless sequence is found. I will begin by formally presenting the theory of lawless sequences before elaborat- ing on its intuitive justification. My exposition here largely follows that of Troelstra (1977). Formally, the first three axioms are:

LS1 ∀n∃α(α ∈ n)

LS2 α = β ∨ α 6= β, i.e. ∀xα(x) = β(x) ∨ ¬∀xα(x) = β(x)

V V LS3 A(α, β1, ..., βl) ∧ i α 6= βi → ∃n(α ∈ n ∧ ∀γ ∈ n( i γ 6= βi → A(γ, β1, ..., βl)))

The fourth axiom is more technical, and requires the notion of a neighborhood function. If F is a tree (or, in intuitionistic terms, a spread) with the usual topology,1 a neighborhood function on F encodes a continuous functional F → N. Intuitively, letting KF be the class of neighborhood functions on F , a particular function e ∈ KF takes initial segments of a sequence ξ ∈ F , and e(ξ(x)) = 0 if the initial segment ξ(x) is not long enough to determine the value of the function that e encodes and

1If a tree is a set of sequences of natural numbers (or their codes), then the topology is generated by the sets which contain a given sequence and all the sequences which extend it. Pictorially, these sets include a given node in the tree and all the nodes above it.

84 e(ξ(x)) = y + 1 if the function that e codes takes value y on any argument that agrees with ξ on the ﬁrst x elements. The class of neighborhood functions on lawless sequences can be deﬁned as the class of e satisfying:

∀α∃x(e(α(x)) 6= 0) ∧ ∀n, m(e(n) 6= 0 → e(n_m) = e(n))

For brevity, deﬁne e(α) = x :↔ ∃y(e(α(y)) = x + 1). Finally, let νp(α1, ..., αp) :=

λx.νphα1(x), ..., αp(x)i be a pairing function on sequences. Let a, b, ... range over

lawlike sequences, and (b)x := λy.b(x, y). Now we can state the fourth axiom of lawless sequences, LS4:

^ ∀α1, ..., αp( αi 6= αj → ∃aA(α1, ...αp, a)) → ∃e∃b∀α1, ..., αpA(α1, ...αp, (b)e(ν(~α))) i,j

We can now consider the justification of these axioms. LS1 states that for any finite sequence of numbers, there is a lawless sequence with that finite sequence as an initial segment. Troelstra describes this as corresponding to the idea that we can pick any initial segment of a sequence before we let it go on its own. (As an alternative motivation, we might think that if LS1 were not the case, then the sequences would not be lawless, since a certain initial segment was forbidden. But note that this line of reasoning is a classical reductio, so the intuitionist will not find it persuasive.) We can also think of LS1 as a density principle or maximality principle. LS2 states that extensional identity is decidable. It is justified by reference to intensional identity: two segments are intensionally identical, written α ≡ β, if they are given as the very same procedure for picking numbers. Intuitively, intensional identity should be decidable. And since at any stage of picking, one only knows a

85 finite initial segment of a sequence, the only way one could know two sequences to be always coextensive is if they were given as the very same sequence of choices. Hence α = β → α ≡ β.2 It is evident that if two sequences are intensionally identical, they must be extensionally identical as well, so α ≡ β → α = β. So the decidability of intensional identify entails the decidability of extensional identity, per LS2. LS3 is justified by the idea that anything that can be asserted about a lawless sequence is asserted at some finite stage, and hence is asserted on the basis of only a finite segment of the sequence. Thus the assertion should also hold of any sequence with that same initial finite segment. This is often called the axiom of open data. It is also easy to see that this axiom justifies the principle of ∀α∃x-continuity (also called weak continuity for numbers):

∀α∃xA(α, x) → ∀α∃x∃y∀β ∈ α(y)A(β, x)

To obtain this schema from LS3, for each α and x satisfying A(α, x) we ﬁnd an n as in LS3 (i.e. ∀γ ∈ nA(γ, x)). Then trivially there is a y such that α(y) = n, so we have ∀α∀x[A(α, x) → ∃y∀β ∈ α(y)A(β, x)]

Then some easy quantificational logic gives us the principle of ∀α∃x-continuity. LS4 is similar in justification to LS3. If for any lawless sequences you can find a lawlike sequence satisfying A, then this must be possible on the basis of some initial data from the lawless sequences. So for some neighborhood there is a lawlike

2 Note that this is an intuitionistic argument. It asks what it would take to prove α = β and concludes that one would have to have proved α ≡ β. Moreover, it does not rely on the ‘unﬁnished’ character of choice sequences, for even if we think of lawless sequences as already completed by the ideal mathematician, our inability to survey the entire sequence (or predict its course via a law) would preclude us from asserting α = β unless we had a proof of α ≡ β.

86 way of finding the lawlike sequence from the value of the neighborhood function on those lawless sequences, taken together (via pairing) as a single sequence. Troelstra’s gloss on this axiom is that “with respect to operations of types NN → N, p-tuples of independent lawless sequences behave like single lawless sequences” (Troelstra, 1977, 28). As noted above, the theory of lawless sequences includes a notion of intensional identity for choice sequences, which is invoked in the justification of LS2. We could likewise take a notion of intensional identity as primitive in our modal account of choice sequences, but I will suggest that we do not need to. To see why, let us consider extensional identity. Extensional identity between α and β has to mean not just that α and β are coextensive at some time, but that they are coextensive at every time, i.e. ∀x(α(x) = β(x)). Now suppose that two sequences are extensionally identical. Then at no moment can it ever be the case that one is defined on x while the other is undefined. So choosing an element for one sequence is choosing an element for the other sequence as well, i.e. both sequences are given as the very same process of choosing numbers. This is why we do not need a separate notion of intensional identity.

5.2 The Modal Theory of Lawless Sequences

Our language will be that of second-order arithmetic, plus an additional constant f to be explained shortly:

{0, f, s, +, ×, <}

87 The logical vocabulary is:

{∀, ∃, ∧, ∨, →, ¬, =, ∈, , ♦, I}

There are three sorts of variables: x, y, z, ... ranging over natural numbers, X, Y, Z, ... ranging over sets of natural numbers, and α, β, γ, ... ranging over choice sequences. The objects in the range of first-order variables x, y, z, ... are the familiar natural numbers. I assume that the natural numbers form a completed totality rather than themselves being a mere potential infinity. Thus the natural numbers all exist at each moment in time rather than coming to exist at some point in time. Sets of natural numbers X, Y, Z, ... are the extensional objects familiar from classical second-order arithmetic, and they will neither gain nor lose members with time. Since choice sequences are supposed to be uncompleted objects, they should be partial functions of natural numbers. This would require our logic to be free. We do not want a simple negative free logic, though, because we want ∀x(α(x) = α(x)) to be true. Since α will not be total in any world, under a negative free logic, α(x) = α(x) would be false if α were not defined on x.

To avoid the technical complications of a free logic, we introduce the constant f to denote a dummy object that is the ‘referent’ of undefined terms. We can let f be its own denotation, and will refer to it as the null object. Obviously the null object is a convenient technical fiction, and no metaphysical significance should be attached to it. Grammatically, f can occur anywhere an individual constant can occur, but it is not in the range of quantifiers over individual variables x, y, z.... We stipulate the following axioms governing f:

f1 ∀x(x 6= f)

88 f2 ∀x(x ≮ f ∧ f ≮ x)

f3 f = f

f4 f ≮ f

f5 s(f) = f

f6 ∀x((x + f) = (f + x) = f)

f7 ∀x((x × f) = (f × x) = f)

f8 ∀X(f ∈/ X)

f9 ∀α(α(f) = f)

f10 ∀x(α(x) = f ↔ ∀yα(x) 6= y)

We can separate the axioms of the theory into three categories: logical axioms, arithmetic axioms, and sequence axioms. The logical axioms will be those of S4+, including the Barcan formula and the converse Barcan formula for all types of variables. For arithmetic axioms we include necessitations of universal closures of the following:

A10 6= s(x)

A2 s(x) = s(y) → x = y

A3 x + 0 = x

A4 x + s(y) = s(x + y)

A5 x × 0 = 0

89 A6 x × s(y) = x × y + x

A7 x ≮ 0

A8 x < s(y) ↔ (x < y ∨ x = y)

A9( x ∈ X → x ∈ X) ∧ (x∈ / X → x∈ / X)

IND ∀X[(0 ∈ X ∧ ∀x(x ∈ X → s(x) ∈ X)) → ∀x(x ∈ X)]

We also include every instance of the comprehension scheme, where A does not include

3 any instances of f or any choice variables (free or bound):

CA ∃X∀x(x ∈ X ↔ A(x))

The reason for not allowing choice variables in the comprehension formula is because we want sets X to be extensional objects and for membership in X to be positively and negatively stable. If we allowed choice variables in the comprehension formula, these conditions would be jeopardized. For instance, if we could form the set X of arguments on which α is deﬁned, membership in X would not be negatively stable. For sequence axioms we include the following:

S1 ∀x(∃yα(x) = y → ∀z < x∃yα(z) = y)

S2 ∀xI∃yα(x) = y

S3 (∀x∀y(α(x) = y → α(x) = y))

S4 ¬♦∀x∃yα(x) = y 3One could of course study various sub-theories based on weaker comprehension axioms, as is done in standard second-order arithmetic, cf. Simpson(1999). While this would likely be interesting, it is not something I will pursue in this work.

90 S5 ∀α∀x(¬∃yα(x) = y → (♦(∃yα(x) = y ∧ ¬∃yα(x + 1) = y) ∧ ∀y♦α(x) = y)

S6 ∀n ∈ SeqI∃α(α ∈ n)

S7 ∀α∀Y [∀x∃y(hx, yi ∈ Y ∧ ∀x∀y(α(x) = y → hx, yi ∈ Y ) → ¬I∃x∃y(α(x) = y ∧ hx, yi ∈/ Y ]

V V S8 ∀α1...∀αk∀x1...∀xk[ i αi(xi) = f∧ i,j ¬∀xαi(x) = αj(x) → ∀z1...∀zk♦(α1(x1) =

z1 ∧ ... ∧ αk(xk) = zx)]

The first four axioms ensure that the choice sequences are growing with time, and there is no end of time. The fifth axiom says two things. First, it says that all sequences are individually lawless in the sense that if a sequence is not yet defined on a given argument, then there are no constraints on the values that it can take at that argument. Second, it says that the values of sequences can be chosen one at a time. The sixth axiom says that for every possible initial segment of a choice sequence, there will inevitably eventually be a choice sequence with that initial segment. There are two motivating ideas behind this axiom. The first is that it should be possible that there is a lawless sequence with any given initial segment. If the sequences are lawless, then no initial segment should be ruled out from the beginning. The second idea is that we want our universe of choice sequences to be maximally inclusive: anything that can happen should happen. The idealized mathematician should do everything that they can do. In this sense, S6 combines an insight about what is possible for lawless sequences with a maximality principle or density principle about how many lawless sequences eventually get created in the course of time. The seventh axiom says that, when Y is a total relation, if the graph of α is so far contained within Y , there is no guarantee that the graph of α will eventually leave

91 Y . The motivation for this axiom is similar to the first part of the motivation for S6: if there were a guarantee that α would eventually disagree with Y , then any path through Y would be outlawed or off limits, contradicting the idea that the sequences in question are lawless. To put it another way, we can think of sets of ordered pairs as drawing a line in the space of potential histories of choice sequences. Then S7 says that no such line can isolate or characterize the potential histories available to a given lawless sequence. The eighth axiom says that any k distinct lawless sequences which are respectively undefined on x1, ..., xk can go on to take any k values z1, ..., zk on those arguments.

(S8 is really a schema, including an instance for each k ≥ 2). Since the zi’s need not bear any relation to each other, this axiom captures that idea that there are no necessary connections between distinct lawless sequences. Just as there are no laws governing the evolution of a single lawless sequences, there should be no laws governing the evolution of k-tuples of lawless sequences. Appendix B sketches a model for the theory without S8 which reveals how the theory without S8 allows for a sort of global coordination among the lawless sequences.

Let MCLS be the theory that consists of axioms f1-f10, A1-A9, IND, CA, and S1-S8, because it is a modal theory of lawless choice sequences.

5.2.1 A Kripke Model for MCLS

The intuitive picture behind this theory is very similar to the motivating gloss for the theory LS that we saw in the previous section: The arithmetical facts are held ﬁxed throughout all time, and an idealized mathematician successively chooses values

92 for potentially inﬁnite sequences of natural numbers. At each moment, the mathematician can extend a sequence that has already been begun, or they can start a new sequence. These choices introduce diﬀerent possible branches in time. When the mathematician chooses, say, 15 as the next value for a sequence α, there are other possible branches where they chose 14, 17, 892, and so on. But after choosing 15, those other branches are no longer accessible, because entries in a sequence cannot be revised once they have been chosen. In what follows I describe a Kripke model

for the theory MCLS, vouchsaﬁng the consistency of the theory. Comfortingly, the model adheres closely enough to the intuitive picture just described that it gives us some warrant for trusting intuitions derived from that picture.4

The set W of worlds will be the set N

4Which is of course not to say that those intuitions should be substitutes for actual proofs.

93 Figure 5.1: Sketch of a Kripke model for MCLS

h0i h1i

. . h0, 0i h0, 1i .

. . h0, 0ih0i h0, 0ih1i .

. . . h0, 0, 0ih1i h0, 0, 1ih1i

h0, 0, 0ih1, 0i h0, 0, 0ih1, 1i h0, 0, 1ih1, 0i h0, 0, 1ih1, 1i

. . . . h0, 0, 0ih1, 0ih0i h0, 0, 0ih1, 0ih1i . . h0, 0, 1ih1, 1ih0i h0, 0, 1ih1, 1ih1i

......

1, 2, ..., k, then the sequence of length k gets extended in the next world n by concatenating it with (n)lh(n). Thus we now have sequences of lengths 1, 2, ..., k − 1, k + 1. In the next world m, we extend the sequence of length k − 1 by concatenating it with

(m)lh(m). So we now have sequences of length 1, 2, ..., k − 2, k, k + 1. We proceed in this way until we have sequences of length 2, 3, ..., k + 1, at which point we add a new sequence of length 1 so that we have sequences of length 1, 2, ..., k + 1. To help illustrate how this will give us an assignment with choice sequences that are independent and, Figure 5.1 traces out the resulting sets of sequences on a small portion of our frame of possible worlds. Even this small fragment of the tree, however, will give a sense of how anything that can happen will happen with the choice sequences under this assignment. The reader is invited to ﬁll in the rest of the frame. To turn this intuitive picture into an explicit assignment for choice sequence variables, assume that we have an enumeration of choice sequence variables αi. Now deﬁne an assignment

94 for choice variables as follows: let Sm be the set of sequence numbers that we have

assigned to the world m by the process above. Assign α1 the empty sequence at the

root node, and then proceed inductively as follows. Suppose αi has been assigned a

ﬁnite sequence sn at a world n and world m is an immediate successor of world n; if sn ∈ Sm, then assign sn to αi at m; if sn 6∈ Sm, then there is a unique sequence

_ sm ∈ Sm that extends sn by one value (namely, sm := sn h(m)lh(m)i), and assign αi

the sequence sm at m. If there is a sequence of length 1 in Sm that was not in Sn, then

assign this sequence to the first choice sequence variable αi which was not assigned a finite sequence at n. These finite sequences that we have assigned to choice sequence variables at worlds are to be the initial defined segment of those choice sequences at those worlds. When α has been assigned a finite sequence at some world in this way, set the rest of the

graph of α at that world to be constantly f; and if α has not been assigned a ﬁnite sequence at a world, then set its entire graph to be constantly f at that world.

5.2.2 Translations into MCLS

Assuming that we are content using modal resources to describe the ever-continuing

but never-completed nature of choice sequences, MCLS has a strong claim to capturing the guiding informal idea of a lawless sequence. In particular, S5 says that if a value for α(x) has not been chosen yet, then it can be chosen to be anything, there are no restrictions on this choice—the characteristic feature of lawless sequences. Of course, neither LS nor any other theory of intuitionistic analysis includes modal operators. So to develop the theory of choice sequences in our modal framework, and to vindicate the claim that this provides a uniﬁed framework for developing classical

95 and intuitionistic analysis, we would like a way of translating formulas of the intuitionistic language into the modal language. A natural way to try to do this is to use the G¨odel-Tarski modal translation. We will use P, Q, ... to range over atoms and A, B, ... as metavariables ranging over formulas:

A A†

P P A ∧ B A† ∧ B† A ∨ B A† ∨ B†

† † A → B (A → B ) † ¬A ¬A † ∀σA ∀σA These clauses of the translation are well-justified. Unfortunately, there is a difficulty with translating the existential quantifier. The usual translation is:

A A†

∃σA ∃σA†

Alternatively, however, we could deﬁne the translation ? which is like † except for the existential quantiﬁer:

A A?

? ∃σA ♦∃σA The motivation for this alternative translation of the existential quantifier is this: intuitionistically, to say that some σ satisfies A amounts to saying that for some particular σ one has a method for establishing that it satisfies A. Since choice sequences are temporal objects, this method may involve waiting some period of time, though. For instance, even if the 7th element of α has not been chosen yet, one can assert

96 intuitionistically ∃y(α(6) = y).5 The method for ﬁnding a particular y satisfying α(6) = y is to simply wait until the 7th element has been chosen. So even if we do not yet have a σ satisfying A, we can assert ∃σA as long as we have a guarantee that we will after some ﬁnite period of time have a witness for σ. Making the modal explicit, this amounts to saying that no matter how the future develops, there will eventually come a particular σ satisfying A. Hence we might translate (∃σA)? as at all future

? times, it is possible that, or in symbols: ♦∃σ(A ). Both the † and ? translations get some things right and some things wrong. For instance, since ∀x∃y(α(x) = y) is a theorem of the intuitionistic theory of lawless sequences, the modal translation of it should be a theorem of MCLS. And

? † (∀x∃y(α(x) = y)) is a theorem of MCLS; by contrast, (∀x∃y(α(x) = y)) is not a

? theorem of MCLS. On the other hand, under the translation ∃xα(x) = 0 would translate to ♦∃xα(x) = 0, which is a theorem of MCLS, following from S4 and S5. But ∃xα(x) = 0 is not a theorem of LS, and for good reason. Since there is no restriction on any of the choices in a lawless sequence, we cannot assert that every lawless sequence has a zero. It might not. In this case, the † translation gets matters

† right: (∃xα(x) = 0) is not a theorem of MCLS. The reason that both the † and ? translations get some things right and some thing wrong is that neither is calibrated quite right. The † translation of ∃σA is too strong: ∃σ(A†) requires that we actually have in hand a witness for σ. But the potentially inﬁnite nature of free choice sequences requires us to reason about the potential existence of objects. Provided we have a guarantee that the potential existence of some σ will eventually be actualized, we can assert ∃σA. In interpreting existential quantiﬁers actualistically, † is too strong.

5Since choice sequences start on the argument 0, α(6) will be the 7th value of α.

97 ? ? Conversely, the translation of ∃σA is too weak: ♦∃(A ) requires only that it always be possible that there be a witness for σ. In other words, it requires that the existence of such a witness is never ruled out. But this is consistent with there never materializing a witness at any moment in history. For instance, if I am choosing values indeﬁnitely, then even if I have chosen only 0’s up to a given moment, it is always possible that I choose a 1 for the next value. And this is true even if at no moment in time do I ever actually choose a 1. What we want is not a mere guarantee that the existence of some σ will never become impossible; we want a guarantee that the existence of some σ will eventually become actual. What we really want for a translation of the intuitionistic existential quantiﬁer into our modal setting is a way of expressing that inevitably, there will eventually be some x such that... . Since, as we saw in Chapter 3, this cannot be expressed in terms of the usual operators and ♦, we can add a new operator I, with the semantics that Iφ is true at a world w just in case every maximal R-chain of worlds above w includes a world u such that φ is true at u. (For a formal semantics, refer back to §3.3.)6 Now we could consider a translation T that agrees with † and ? for logical constants other than ∃, which it translates:

A AT

∃σA I∃σAT

6Under the assumption that the underlying frame of a model is a tree, this semantic clause is equivalent to saying IA is true at w just in case there is a set of worlds X that bars w and A is true at each member of X. Thus, I∃xA would be true at w just in case there a set of worlds X that bars w and for each u ∈ X there is an object o that satisﬁes A at u. This is the same as the clause for satisfaction of existential formulas in the Beth semantics for intuitionistic logic. Incidentally, Kripke (2019) suggested that Beth semantics would be the appropriate tool for developing a theory of choice sequences for essentially the reasons adduced here (though as applied to disjunctions rather than existential formulas). Indeed, van Dalen(1978) used Beth trees to give a model of choice sequences. I am not aware, however, of any use of Beth trees to give a semantics for classical modal logic, hence they will not ﬁgure in my approach below.

98 This would provide a nice way of expressing the intuitionistic existence claims in our modal theory. However, there are two difficulties with appealing to this T translation. The first difficulty can be dealt with, but the second difficulty is more serious. The first difficulty is that the modal logic S4+ that results from adding I to S4 is not axiomatizable. (Again, see §3.3.) Recall that the goal is to provide a set of postulates that, as far is possible, captures in the classical modal setting the assumptions and reasoning of the intuitionist. The fact that a candidate modal logic is not axiomatizable is some reason for pessimism that a set of postulates based on that logic can adequately and faithfully capture intuitionistic reasoning. On the other hand, if a large enough fragment could be axiomatized to adequately capture the relevant intuitionistic arguments, then that would suffice for my purposes even though the logic is incomplete. By way of analogy, the fact that second-order logic is not axiomatizable does not prevent us from axiomatizing a large enough fragment to express many interesting mathematical arguments. Furthermore, intuitionistic logic itself is incomplete from the perspective of an intuitionistic metatheory.7 So the fact that a logic is incomplete is only evidence that it does not suffice to codify intuitionistic reasoning insofar as intuitionistic logic itself does not suffice to codify intuitionistic reasoning. Thus the incompleteness of S4+ does not necessarily preclude its use in a faithful regimentation of intuitionistic analysis. (This is an interesting contrast to the classical case. Recall that in Chapter 3 I argued that the incompleteness of S4+

7 McCarty(2008) has a general incompleteness result for intuitionistic logic (as well as superin- tuitionistic logics); see also references therein for previous results. I should note, however, that the status of intuitionistic logic as complete or incomplete is somewhat controversial. The completeness of intuitionistic logic would entail the intuitionistically problematic Markov’s Principle, see (Dum- mett, 2000, 172-4). On the other hand, de Swart(1976) and Veldman(1976) prove completeness results for intuitionistic logic with an intuitionistic metalogic but a slightly modiﬁed notion of model. (Dummett, 2000, 193-200) raises doubts about whether these alternate models faithfully capture the intuitionist’s intended interpretation of the logical vocabulary.

99 made a faithful interpretation of classical analysis problematic). The second diﬃculty, however, is more serious. This concerns not the inclusion of the operator I as such, but its use in translating the existential quantiﬁer. I argued above that I∃σ is the right translation of ∃σ to capture the potential nature of existence in our modal-temporal setting. However, there are problems with the way it interacts with other connectives. For instance, the following is intuitionistically valid: ∃x(Ax ∨ Bx) → (∃xAx ∨ ∃xBx). Under the T translation, this becomes

T T T T [I∃x(A x ∨ B x) → (I∃xA x ∨ I∃xB x)], which is not valid in the modal logic. Thus, although the T translation gets existentials right, it is not in general a sound translation because of the way its translation of existentials interacts with other connectives. Since there seems to be no satisfactory uniform translation scheme available, our strategy going forward will be piecemeal and somewhat informal. I will show how

MCLS can prove modal analogues of important pieces of the theory of lawless sequences, without being bound by a particular way of finding those modal analogues. Although I will not be bound by a particular translation of non-modal formulas into modal language, the translation † will serve as a general guide. Since it is only with the clause for ∃ that our three translations disagree, † is a good guide for translating the other logical vocabulary. Moreover, since the problem with the † translation is that it is too strong, taking it as our guide will never leave us with less than we were after. Because of the difficulties with the T translation, I will be judicious in using I, appealing to it only when clearly appropriate. It is worth including, however, because it will be essential for expressing the content of the treatment of the bar theorem, as well as the definition of real number generators in the next chapter.

100 5.2.3 Basic Properties of MCLS

We can begin by noting that MCLS proves modal versions of LS1 and LS2, along with a few other basic properties.

Lemma 5.1. MCLS ` ∀n♦∃α(α ∈ n)

Proof. By S6 and axiom M4.

? Lemma 5.2. MCLS ` LS1 .

Proof. By axiom S6 and the CBF we have ∀n♦∃α(α ∈ n), which by necessitation ? gives us ∀n♦∃α(α ∈ n), i.e. LS1 .

† Lemma 5.3. MCLS ` LS2 .

† Proof. Recall that α = β is an abbreviation for ∀x(α(x) = β(x)), so LS2 is ∀x(α(x) = β(x)) ∨ ¬∀x(α(x) = β(x)). Classical logic gives us ∀x(α(x) = β(x)) ∨ ¬∀x(α(x) = β(x)). First we want to show ∀x(α(x) = β(x)) → ∀x(α(x) = β(x)). By the modal 4 axiom we have ∀x(α(x) = β(x)) → ∀x(α(x) = β(x)), and then the converse Barcan Formula gives the desired result.

Next we want to show ¬∀x(α(x) = β(x)) → ¬∀x(α(x) = β(x)). Given the equivalence in S4 of φ and φ and the presence of the BF and CBF, it will suﬃce to show ¬∀x(α(x) = β(x)) → ¬∀x(α(x) = β(x)). Assume ¬∀x(α(x) = β(x)). This is equivalent to ♦∃x(α(x) 6= β(x). So α and β disagree on some x, and at least one of α(x) and β(x) must denote.

If α(x) and β(x) both denote, then by axiom S3 we have α(x) 6= β(x), which entails ♦∃x(α(x) 6= β(x), i.e. ¬∀x(α(x) = β(x)).

101 On the other hand, suppose that α(x) denotes and β(x) does not denote. Then by axiom S5 it is possible for β(x) to denote a value that disagrees with α(x). So

♦♦∃x(α(x) 6= f ∧ β(x) 6= f ∧ α(x) 6= β(x)); then by the modal 4 axiom ♦∃x(α(x) 6= f ∧ β(x) 6= f ∧ α(x) 6= β(x)). Now, as before, from S3 this diﬀerence between α(x) and β must persist, so we have ♦∃x(α(x) 6= β(x), i.e. ¬∀x(α(x) = β(x)).

As plausibility considerations for continuity principles or principles of open data such as LS3, intuitionistic writers often argue that anything that can be proved about a choice sequence has to be proved at some ﬁnite stage, and moreover that any proof given at a ﬁnite stage will never cease to be legitimate proof. The next lemma captures the second of these claims, showing that when (the translation of) an intuitionistic formula holds, it will continue to hold in the future.

? ? † † Lemma 5.4. MCLS ` A ↔ A . Also, MCLS ` A ↔ A .

Proof. We prove both claims at once. (⇐) Immediate from modal logic.

? (⇒) Induction on the complexity of A. If A is atomic, then A := A, and the claim follows from the modal 4 axiom. For the induction step, the cases of →, ¬, ∀, are all also immediate from the modal

? ? logic, since A will be of the form B. Likewise for ∃ under the translation. For ∃ under the † translation, (∃σA)† is of the from ∃σA†, which by i.h. is equivalent to

† † ∃σA , which entails ∃σA . ? ? For the case where A := B ∨ C, we have by the i.h. B → B , whence it follows ? ? ? ? ? ? ? ? B → (B ∨ C ). Similarly we have C → C , whence C → (B ∨ C ). Hence ? ? ? ? by dilemma (B ∨ C ) → (B ∨ C ).

102 ? ? ? ? For the case where A := B ∧ C, we have by the i.h. B → B and C → C , ? ? ? ? ? ? ? ? whence (B ∧ C ) → (B ∧ C ) and thus (B ∧ C ) → (B ∧ C ).

As explained above, we will not in general be relying on either the † or ? translation, but we will rely on the general idea behind this lemma, namely that, for the intuitionist, once a sentence is assertable8 it will remain assertable in the future. Thus, ﬁnding modal analogues of intuitionistic claims will generally involve judicious

addition of ’s to the intuitionistic formula. As in the previous chapters, a formula that, once true, will continue to be true into the future can be called stable. More precisely, let us say that a formula A is positively stable if it satisﬁes:

~ ~ ~ ∀~α∀X∀~x[A(~α, X, ~x) → A(~α, X, ~x)]

And a formula is negatively stable if it satisﬁes:

~ ~ ~ ∀~α∀X∀~x[¬A(~α, X, ~x) → ¬A(~α, X, ~x)]

A formula that is both positively and negatively stable will be called stable sim- pliciter.

Lemma 5.5. If A contains no occurrences of , ♦, I, or f, and no choice sequence variables, then A is stable.

Proof. Note that A will be a formula in the language of second-order arithmetic. We

0 proceed by induction on the quantiﬁcational complexity of A. MCLS is ∆0-complete, 8Or is true, or is known, or has a construction or proof, depending on your preferred gloss on the intuitionistic approach.

103 0 since it extends Robinson arithmetic. So if A is ∆0 and true, then MCLS ` A and

0 hence MCLS ` A and thus MCLS ` A → A. On the other hand, if A is ∆0 and

false, then MCLS ` ¬A and hence MCLS ` ¬A and thus MCLS ` ¬A → ¬A. For the induction step, ﬁrst assume that A is positively stable. So we have ∀x(A →

A). By standard quantiﬁcational logic we have ∃xA → ∃xA, from which we can derive ∃xA → ∃xA. Likewise for ∃X and ∃α. Similarly, if A is positively stable, then from ∀x(A → A) we get, by standard quantiﬁcational logic,∀xA → ∀xA, from which we can derive ∀xA → ∀xA. Like- wise for ∀X and ∀α. So ∃σA and ∀σA are positively stable.

Now assume A is negatively stable. So we have ∀x(¬A → ¬A). By standard quantiﬁcational logic we have ∀x¬A → ∀x¬A, from which we can derive ¬∃xA → ¬∃xA. Likewise for ∃X and ∃α. Again, if A is negatively stable, we have ∀x(¬A → ¬A). By standard quantiﬁ- cational logic we have ∃x¬A → ∃x¬A, from which we can derive ¬∀xA → ¬∀xA. Likewise for ∀X and ∀α. So ∃σA and ∀σA are negatively stable. This completes the induction.

Appendix C collects a few more results on stability which, although moderately interesting, will not be needed in what follows.

Theorem 5.6. MCLS ` ∀xα(x) = β(x) → (A(α) ↔ A(β))

Proof. Induction on the complexity of A. If A is atomic, then it has one of the

following forms: t1(α) < t2(α), t1(α) = t2(α), t(α) ∈ X. Each case is straightforward, since <, =, ∈ are all extensional. For the induction step, the cases where the main logical operator in A is either a truth-functional connective or a quantiﬁer are all straightforward. Consider the case

104 where A is of the form B. By the i.h and necessitation we have:

` (∀xα(x) = β(x) → (B(α) ↔ B(β)))

From this it follows that:

` ∀xα(x) = β(x) → (B(α) ↔ B(β)))

And thus, as required:

` ∀xα(x) = β(x) → (B(α) ↔ B(β)))

The argument for ♦B proceeds in the same way up to the last step, where one appeals 0 0 to (B ↔ B ) → (♦B ↔ ♦B ).

5.2.4 Continuity Principles

Having established these basic properties of MCLS, we now want to show that the continuity principles embodied in LS3 and LS4 hold in this theory. Let n = isα abbreviate the formula Seq(n) ∧ α ∈ n ∧ ∀m(Seq(m) ∧ lh(n) < lh(m) → α∈ / m) which says that n is the initial deﬁned segment of α.

Lemma 5.7. If A(α1, ..., αk) contains no modal operators, no occurrences of f, and no lawless variables other than α1, ..., αk (free or bound), then there is an arithmetical formula A0(n) such that:

!! ^ 0 MCLS ` A(α1, ..., αk) ↔ ∃m1...∃mk mi = isαi ∧ A (m1, ..., mk) i

105 Proof. I will define a method for finding an A0 as required in two steps. First, from a formula A(α1, ..., αk) we can find a formula A1(α1, ..., αk−1, isαk) with one less choice sequence variable, but possibly including occurrences of f. Iterating this procedure will give us a formula Ak(isα1, ..., isαk) with no choice sequence variables, but possibly including f. Then we can systematically remove occurrences of f to obtain the formula A0 as required. The intuitive idea of the proof is nicely illustrated by considering an atom P (α(x)) with only one occurrence of α. Then the first translation is (x ≤ lh(n)∧P ((n)x))∨(x > lh(n) ∧ P f). When α is defined on the argument x, the value α(x) can be replaced with (n)x (where n is the initial defined segment of α) and otherwise the value of

α is undefined and can be replaced with f. This is simplest case, though. In full generality we need to account for the possibilities that A is not an atom and that there are multiple occurrences of α, and some occurrences may be nested within one another. The possibility that A is not an atom is easily dealt with. The translation we define next will apply to any quantifier-free formula. Then, since we can assume that any formula is in prenex normal form, the translation can simply ‘push through’ the initial quantifiers. To address the possibility that there are multiple occurrences of α is more involved, but conceptually straightforward. Let αi(t1), ..., αi(ts) be all the occurrences of αi and its arguments in a quantifier-free formula P . We will simply consider all the possibilities of which αi(t1), ..., αi(ts) are defined and which are not, and then disjoin those possibilities. I will introduce some ad hoc notation for this proof only: let 2s be the set of 0,1-

i valued sequences of length s. Let tl be the result of replacing every every occurrence

106 s χ of βi(tj) in tl with (mi)tj . Finally, for χ ∈ 2 , let P be the result of replacing every

occurrence of tj in P with f for exactly those j such that (χ)j = 0.

Now let P0 be the maximal quantiﬁer-free subformula of A, and deﬁne Pi recursively to be:

     _ ^ i ^ i χ  tj < lh(mi) ∧  tj ≮ lh(mi) ∧ Pi−1 s χ∈2 j:(χ)j =1 j:(χ)j =0

Thus Pk will be a formula with all choice sequence variables αi replaced by numbers mi; it is easy to verify that if then P0 and Pk will be equivalent. So if A is ∀~x∃~y...∀~zP0, and if for all i, mi = isαi, then A will be equivalent to ∀~x∃~y...∀~zPk.

Pk may still have occurrences of f in it, however, and those must be removed to get 0 the ﬁnal formula A . This is simple: for any atom containing f, it is decidable from the syntax whether that atom is true or false. (This is easily veriﬁed by induction on the number of function symbols, appealing to the axioms governing f). Thus, we can 0 obtain an arithmetical P from Pk by replacing each atom containing f with 0 = 0 when the atom is true and with 0 = 1 when false. Thus we have our arithmetical formula A0 := ∀~x∃~y...∀~zP 0.

Lemma 5.8. Let A and A0 be as in Lemma 5.7. Then:

^ MCLS ` [ βi 6= βj → [A(β1, ..., βk) ↔ i

^ 0 ∃n1...∃nk[ ni = isβi ∧ A (n1, ..., nk) i

^ 0 _ _ ∧ ∀m1...∀mk( Seq(mi) → A (n1 m1, ..., nk mk))]]] i

107 Proof. We reason model-theoretically. Since there are no occurrences of I in A, this is V kosher in virtue of Proposition 3.8. Assume w is an arbitrary world at which i

(⇒) Assume w A(β1, ..., βk), and let n1, ..., nk be initial segments of β1, ..., βk

deﬁned at w, respectively. Let m1, ..., mk be arbitrary numbers such that Seq(mi).

V _ By S7, there is a world u such that wRu and u i ni mi = isβi. Furthermore, 0 _ _ u A(β1, ..., βk), so by Lemma 5.7 u A (n1 m1, ..., nk mk). Since this formula is 0 _ _ arithmetic, it is stable and thus must hold at w as well, i.e. w A (n1 m1, ..., nk mk).

Since this holds for every m1, ..., mk such that Seq(m1), ..., Seq(mk), we have that w V 0 V 0 _ _ ∃n1...∃nk[ i ni = isβi∧A (n1, ..., nk)∧∀m1...∀mk( i Seq(mi) → A (n1 m1, ..., nk mk))]. V 0 V (⇐) Suppose w ∃n1...∃nk[ i ni = isβi∧A (n1, ..., nk)∧∀m1...∀mk( i Seq(mi) → 0 _ _ A (n1 m1, ..., nk mk))]. We want to show w A(β1, ..., βn). Let wRu. Then 0 V 0 _ _ u A (n1, ..., nk) ∧ ∀m1...∀mk( i Seq(mi) → A (n1 m1, ..., nk mk)), because this for- _ _ mula is arithmetic and hence stable. In particular, if n1 m1, ..., nk mk are the initial

0 _ _ segments of β1, ..., βk deﬁned at u, then u A (n1 m1, ..., nk mk). Thus by Lemma

5.7, u A(β1, ..., βk); and hence w A(β1, ..., βk).

~ Theorem 5.9. Suppose A(α, β) contains no modal operators, no occurrences of f, only the displayed free lawless variables, and no bound lawless variables. Then:

~ ^ MCLS `[A(α, β) ∧ ¬∀xα(x) = βi(x)] → i ^ ~ ∃x∀γ[α(x) = γ(x) ∧ ¬∀xγ(x) = βi(x) → A(γ, β)] i

~ V Proof. Let w A(α, β) ∧ i ¬∀xα(x) = βi(x). Let n0 be the initial segment of α

108 defined at w and let ni be the initial segment of βi defined at w. We want to show ~ that ∀γ(α(lh(n0)) = γ(lh(n0)) → A(γ, β)). Let γ be an arbitrary choice sequence V such that w α(lh(n)) = γ(lh(n))∧ i ¬∀xγ(x) = βi(x). And let u be an arbitrary _ world such that wRu, and let s be such that n0 s is the initial segment of γ defined ~ V at u. Now since w A(α, β), Lemma 5.8 entails that w ∀m0...∀mk( i Seq(mi) → 0 _ _ V 0 _ _ _ A (n0 m0, ..., nk mk)), and hence w ∀m1...∀mk( i Seq(mi) → A (n0 s, n1 m1..., nk mk)). V 0 _ _ _ Since this formula is stable, u ∀m1...∀mk( i Seq(mi) → A (n0 s, n1 m1..., nk mk)), ~ so by Lemma 5.7, u A(γ, β).

This theorem is an analogue of LS3. It shows that for arithmetical formulas A with lawless parameters, if A holds necessarily for α, then it does so on the basis of a ﬁnite amount of information about α, and hence will also hold necessarily of any lawless variable that agrees with α on a given initial segment. Formally, this captures the intuition that whatever properties can be established to persistently hold of a lawless sequence must be based on only an initial segment of the sequence. (That the theorem captures this intuition is particularly evident from the proofs of Lemmas 5.7 and 5.8.) Thus we have a version of the open data principle.

Next, we want to show how MCLS can prove a modal analogue of LS4. When e is a classical function variable (regarded as an abbreviation from a classical set variable X), let e ∈ Kp be an abbreviation for the displayed formula below, which expresses that e is a p-adic neighborhood function. Where neighborhood functions, as introduced above, encoded continuous functionals NN → N, p-adic neighborhood functionals encode p-adic functionals NN×...×NN → N which are continuous in each of their arguments. The deﬁnition for the p-adic case is entirely analogous to that from before. The only diﬀerence is that where we earlier required neighborhood functions to

109 be non-zero only on sequence numbers, now we require p-adic neighborhood functions to be non-zero only on sequence numbers each of whose entries is the code of a p-tuple of numbers.9

∀x[e(x) 6= 0 → (Seq(x) ∧ ∀y ≤ lh(x)(Seq((x)y) ∧ lh((x)y) = p))]

∧ ∀x, y[e(x) 6= 0 → [Seq(y) ∧ ∀z < lh(y)(Seq((y)z) ∧ lh((y)z) = p)

→ e(x_y) = e(x))]]

As above, let e(α) = x be an abbreviation for ∃y(e(α(y)) = x + 1); when used as a term, e(α) can be understood as a deﬁnite description denoting the unique y such that e(α) = y + 1, and this description can be eliminated `ala Russell. And

when n is a sequence number each of whose entries is a p-tuple, let ni abbreviate

h((n)1)i, ..., ((n)lh(n))ii. Informally, if we think of n as a code of p distinct sequences

th of length lh(n), then ni is the i such sequence.

Theorem 5.10. Suppose A(α1, ..., αk) contains no modal operators, no occurrences

of f, and only the displayed lawless variables. Then:

^ MCLS ` ∀α1...∀αk[ ¬∀xαi(x) = αj(x) → ∃fA(α1, ..., αk, f)] → i,j

k ∃e ∈ K ∀n[e(n) 6= 0 → ∃f∀α1 ∈ n1...∀αk ∈ nkA(α1, ..., αk, f)]

Proof. Let α1, ..., αk be arbitrary distinct choice sequences, and suppose we have

∃fA(α1, ..., αk, f). Fix a witness f. Then by a k-fold application of Theorem 5.9, 9In Troelstra(1977) there is no special treatment of p-adic neighborhood functions because he assumes that the pairing function—and accordingly the encoding of p-tuples—is surjective. Thus for him, every natural number is the code of a p-tuple, and hence every sequence number is a sequence number each of whose entries is the code of a p-tuple.

110 V there exist n1, ..., nk such that ∀α1 ∈ n1...∀αk ∈ nk( i,j ¬∀xαi(x) = αj(x) → 0 A(α1, ..., αk, f)). By Lemma 5.8, we can ﬁnd a purely arithmetical formula A such that the following two formulas are equivalent:

^ ∀α1...∀αk[ ¬∀xαi(x) = αj(x) ∧ α1 ∈ n1 ∧ ... ∧ αk ∈ nk → A(~α, f)] (5.1) i,j

^ 0 _ _ ∀m1...∀mk[ Seq(mi) → A (n1 m1, ..., nk mk, f)] (5.2) i

Each ni is a sequence number, but they may not be of the same length. We can,

however, assume without loss of generality that lh(n1) ≤ ... ≤ lh(nk). Now we deﬁne

a neighborhood function e as follows. For all n1, ..., nk, if there is an f that satisﬁes

equation 5.2, then if n is a sequence of length k such that (n)k = nk and for all i < k,

ni is an initial segment of (n)i, then e(n) = 1. Otherwise, e(n) = 0. Then by the equivalence between equation 5.1 and equation 5.2, if e(n) 6= 0, we have that:

^ ∃f∀α1...∀αk[ ¬∀xαi(x) = αj(x) → ∀α1 ∈ n1...∀αk ∈ nkA(~α, f)] i,j

To see that this indeed gives us a modal analogue of LS4, observe that we can infer:

Corollary 5.11. Suppose that A contains no modal operators, no occurrences of f, and only the displayed lawless variables. Then for g a function variable that has arity

111 one greater than the arity of f:

^ MCLS ` ∀α1...∀αk[ ¬∀xαi(x) = αj(x) → ∃fA(~α, f)] i,j

k → ∃e ∈ K ∃g∀α1...∀αkA(α1, ..., αk, g(e(hα1, ..., αki), ...))

Proof. We will define a function g from the neighborhood function e we defined in the proof of Theorem 5.10. The idea is to parameterize the functions f by n in such a way that from ∀n∃fn... we can infer ∃g∀n...g(n).... This will require a few steps. First, we will define a different neighborhood function e0. Like e, we will let e0(n_m) = e0(n) whenever this value is non-zero. The only difference is that when e(n) = 0 and e(n_hsi) = 1—so that e is non-zero on n_hsi, but is zero on all initial segments of n_hsi —we will define e0(n_hsi) = n_hsi. Let x ≺ y abbreviate the claim that y codes a finite sequence of numbers and x codes an initial segment of y (and x y iff x ≺ y ∨ x = y). Then we can explicitly define e0:

e0(x) = y :↔ e(x) = 1 ∧ y x ∧ e(y) = 1 ∧ ∀z(z ≺ y → e(z) = 0)

Now we will deﬁne an order ¡ on functions of a given arity. Intuitively, ¡ will order functions by lexicographically by their graphs. Explicitly:

f1 ¡ f2 :↔f1(0, ..., 0) < f2(0, ..., 0)∨

∃x1, ..., xm[f1(x1, ..., xm) < f2(x1, ..., xm)∧

∀y1, ..., ym(hy1, ..., ymi < hx1, ..., xmi → f1(y1, ..., ym) = f2(y1, ..., ym))]

112 00 V 0 _ _ For brevity, let A (~n,f) abbreviate ∀m1...∀mk( i Seq(mi) → A (n1 m1, ..., nk mk, f). V 0 00 00 ~0 Clearly, if i ni ≺ ni and A (~n,f), then A (n , f). Now we can deﬁne our function g, as desired:

g(n, ~x) = y :↔ ∀z(n z → e(z) = n)∧

00 0 00 0 ∃f[A (n1, ..., nk, f) ∧ ∀f (A (n1, ..., nk, f) → f ¡ f ) ∧ f(~x) = y]

5.2.5 The Bar Theorem

One of the characteristic pieces of intuitionistic analysis is the so-called bar theorem, and its main corollary, the fan theorem. The bar theorem asserts the validity of a form of induction, and the fan theorem is a constructive version of K¨onig’sLemma. The central role that these principles have in intuitionistic analysis is due to their role in proving that every function on the reals is continuous and every function on a closed interval is uniformly continuous, one of the more striking anti-classical results of intuitionistic analysis. Let us begin by considering the bar theorem. We will see the fan theorem in Chapter 6. Intuitionistically, a node n in a spread S is said to be barred by P by if for every α ∈ S such that α ∈ n, ∃xP α(x). Informally, a bar above n is a set of nodes above n such that every branch above n includes a node in that set: it is impossible to travel through the tree above n without passing through the bar. If P bars the root node of the spread, then we can simply say that P is a bar. In the general case, we can take S to be the so-called universal spread (i.e. the full tree ω<ω). Then bar induction is

113 the following schema:

[∀α∃xP α(x) ∧ ∀x(P x → Qx) ∧ ∀x(∀yQx_hyi → Qx)] → Qhi

As Kleene showed, this schema is intuitionistically problematic valid unless the bar is decidable.10 The schema of decidable bar induction is:

[∀x(P x ∨ ¬P x) ∧ ∀α∃xP α(x) ∧ ∀x(P x → Qx) ∧ ∀x(∀yQx_hyi → Qx)] → Qhi

Alternatively, one can add as an assumption that the bar is monotone, resulting in monotone bar induction:

[∀x∀y(P x → P x_hyi) ∧ ∀α∃xP α(x) ∧ ∀x(P x → Qx) ∧ ∀x(∀yQx_hyi → Qx)] → Qhi

Intuitionistically, monotone bar induction entails decidable bar induction.11 The bar theorem is the (ambiguous) assertion that bar induction in one of the above forms is valid.12 As an informal gloss in our temporal terms, we can describe a bar as a set X of sequence numbers such that, for any free choice sequence α that the idealized mathematician deﬁnes, no matter which choices they make along the way, they will eventually have deﬁned an initial segment of α that is in X. Then the bar theorem says that if any sequence number in a bar has a property Q, and whenever every successor of a sequence number has Q that sequence number itself also has Q, then

10(Kleene and Vesley, 1965, Lemma 9.8). 11This result, although not diﬃcult, is not trivial; see (Dummett, 2000, 63) for a proof. 12The name ‘bar theorem’ is something of an historical accident. Brouwer claimed to have a proof of the theorem, though this proof is widely seen as inadequate today, cf. (Troelstra, 1977, 24-25) for discussion.

114 the empty sequence has Q. Somewhat more abstractly, the bar theorem says that the if the root node of a tree is barred, then the portion of the tree which lies below the bar is well-founded.

It is at this point in the development of MCLS that we need the inevitability operator. Without it, there is no way to express the claim that every choice sequence will eventually hit the bar. Taking advantage of the inevitability operator, the modal analogue of the principle of bar induction (BI) can be formulated as follows. (Note that we do not need to impose any decidibility condition on the bar X because in our theory sets are classical objects, distinguished extensionally and with membership satisfying the excluded middle).

_ [∀αI∃x(α(x) ∈ X) ∧ ∀x(x ∈ X → x ∈ Y ) ∧ ∀x ∈ Seq(∀y(x hyi ∈ Y ) → x ∈ Y )] → hi ∈ Y

Theorem 5.12. MCLS ` ∃α(α(0) = f) → BI

Proof. Assume we have an α which is not yet deﬁned on 0 and that Y includes X and is downwards hereditary:

∀x(x ∈ X → x ∈ Y ) ∧ ∀x(x ∈ X → x ∈ Y ) ∧ ∀x ∈ Seq(∀y(x_hyi ∈ Y ) → x ∈ Y )

We will proceed by contraposition: assuming hi ∈/ Y , we will prove that X is not a

bar, i.e. ♦∃α¬I∃x(α(x) ∈ X). Since hi ∈/ Y , it also follows that ∃x(hxi ∈/ Y ) and then ∃y(hx, yi ∈/ Y ), and so on. Continuing in this way, we will deﬁne an inﬁnite path that lies outside of Y . More

115 exactly, deﬁne Z as follows:

z ∈ Z ↔ ∃x∃y(z = hx, yi ∧ ∃w ∈ Seq((w)x = y ∧ w∈ / Y ))

By the induction axiom, it follows that ∀x∃yhx, yi ∈ Z.

Now since we have α(0) = f, we know ∀x∀y(α(x) = y → hx, yi ∈ Z, or there is no such α. Axiom S7 gives us that ¬I∃x∃y(α(x) = y ∧ hx, yi ∈/ Z). Note that

∀x[α(x) ∈ X → ∀z < x∀y(α(z) = y → hz, yi ∈/ Z]. It follows that Iα(x) ∈ X → I∀z < x∀y(α(z) = y → hz, yi ∈/ Z. Thus we have that ¬I∃xα(x) ∈ X. So X is not a bar.

The assumption that there exists an α which is not yet defined is important in this theorem.13 If time were not well-founded, then it is possible that the idealized mathematician had already created an initial segment of every choice sequence and by a cosmic coincidence they had all veered away from the branch Z that avoids the bar X. For instance, suppose that X is the set of all sequence numbers that contain a non-zero entry, and suppose that for every finite sequence there is a choice sequence beginning with that finite sequence, but that every choice sequence that begins with a string of zeros also has a value 1 on some argument.

5.3 Conclusion

In this chapter I introduced the informal concept of a lawless sequence. After presenting the intuitionistic theory LS of lawless sequences, I proposed a modal theory

13In fact it can be weakened slightly to the assumption that there is an α whose initial deﬁned segment is contained in Z. Some assumption in this neighborhood is necessary, though.

116 of lawless sequences MCLS. The axioms of this theory have a good claim to capturing the intuitive concept of lawlessness—certainly they are all individually justiﬁed on

the basis of this concept. Moreover, the fact that MCLS proves modal analogues of the axioms of the intuitionistic theory LS provides further support for this claim. Despite their conceptual and metamathematical interest, however, lawless sequences are not particularly interesting mathematically. To develop any interesting mathematics one needs a universe that includes sequences which obey some constraints, or, in other words, which are not lawless. In the next chapter we will explore a theory that countenances non-lawless sequences. And as we will see, it suﬃces for some interesting mathematical results.

To close this chapter, I will note two important open questions concerning MCLS. The ﬁrst is whether Theorems 5.9 and 5.10 still hold when A is allowed to include bound choice variables. Those theorems, recall, were the modal analogues of LS3 and LS4. In LS3 and LS4, however, the main formula was allowed to have bound choice variables. Thus the fact that A was restricted to have no bound choice variables in Theorems 5.9 and 5.10 represents a shortcoming in the extent to which MCLS captures all the content of LS. I conjecture that this restriction can be lifted, but the question remains open. The second cluster of open questions concerns the metamathematical status of

MCLS. What is its proof-theoretic strength? Can it be interpreted in an existing non-modal theory? As defined above, MCLS includes the full second-order comprehension scheme. If we only allowed restricted forms of comprehension (e.g. recursive comprehension or predicative comprehension), how would that affect the deductive strength of MCLS? Would there be any interesting differences between the models of

MCLS with these weaker forms of comprehension? Interesting questions abound.

117 Chapter 6

Non-Lawless Sequences and Real Numbers

In the previous chapter I introduced a modal theory of lawless sequences. The fact that that theory was able to capture the central parts of the intuitionistic theory of lawless sequences goes some way to establishing my claim that the modal framework allows us to incorporate elements of intuitionistic mathematics into a fundamentally classical setting. But while lawless sequences have some interest, they are insuﬃcient for developing the most interesting portions of intuitionistic mathematics, especially intuitionistic analysis. To capture elements of intuitionistic analysis we have to expand our universe to include choice sequences that are not lawless. The basic idea is that the ﬁrst four axioms which characterize sequences as growing with time are kept as is, while the other axioms will be relativized to lawless sequences. Comprehension principles can then be added to this theory to allow projections of lawless sequences. To accomplish all this, we will add a new non-logical symbol L to the language. L will be a one-place predicate of choice sequence variables with the intended interpretation that Lα means α is lawless. It will be convenient to use the familiar abbreviations ∀α ∈ L and ∃α ∈ L for ∀α(Lα → ...) and ∃α(Lα ∧ ...), respectively.

The new theory will consist of all the same arithmetical axioms and f axioms, and the following sequence axioms:

S1 ∀x(∃yα(x) = y → ∀z < x∃yα(z) = y)

118 S2 ∀xI∃yα(x) = y

S3 (∀x∀y(α(x) = y → α(x) = y))

S4 ¬♦∀x∃yα(x) = y

S5r ∀α ∈ L∀x(¬∃yα(x) = y → (♦(∃yα(x) = y ∧ ¬∃yα(x + 1) = y) ∧ ∀y♦α(x) = y)

S6r ∀nI∃α ∈ L(α ∈ n)

S7r ∀α ∈ L∀Y [∀x∃y(hx, yi ∈ Y ∧∀x∀y(α(x) = y → hx, yi ∈ Y ) → ¬I∃x∃y(α(x) = y ∧ hx, yi ∈/ Y ]

V V S8r ∀α1 ∈ L...∀αk ∈ L∀x1...∀xk[ i αi(xi) = f ∧ i,j ¬∀xαi(x) = αj(x) →

∀z1...∀zk♦(α1(x1) = z1 ∧ ... ∧ αk(xk) = zx)]

S9( Lα → Lα) ∧ (¬Lα → ¬Lα)

Axioms S1-S4 are the same as in MCLS; S5r-S8r are the same as S5-S8 in MCLS, but with choice sequence variables explicitly relativized to L (hence the use of ‘r’ for relativized). S9 says that the property of being lawless is stable. We will also add a new axiom that allows us to show that the graph of a choice sequence has a given property by induction on the argument of the choice sequence. This will be useful in establishing certain results about real number generators below.

S-IND A(α(0)) ∧ ∀x[(α(x) 6= f → A(α(x))) → (α(x + 1) 6= f → A(α(x + 1)))] → ∀x(α(x) 6= f → A(α(x)))

This set of axioms forms a base theory to which we can add comprehension schema that allow us to consider more inclusive universes of choice sequences. We can largely

119 remain neutral about which formulas A can legitimately ﬁgure in a choice sequence comprehension schema. But for a given class of formulas Ξ, we can introduce either of the following schemas, where A is required to be in Ξ:

ΞSC ∀x∃yA(x, y) → ∃α∀x∀y(α(x) = y → A(x, y))

Ξ!SC ∀x∃!yA(x, y) → ∃α∀x∀y(α(x) = y → A(x, y))

ΞISC ∀xI∃yA(x, y) → ∃α∀x∀y(α(x) = y → A(x, y))

(SC stands for ‘sequence comprehension’). At the very least, formulas which de- ﬁne recursive functions should count as acceptable presentations of lawlike sequences.

0 1 So ∆1!SC would be a natural minimal theory to study. The focus below will pri- marily be on the theory which we may call MC, which results from adding to the sequence axioms the instances of ISC where A has no second-order variables and no choice sequence quantiﬁers (but may contain choice sequence parameters), and

0 whose ﬁrst-order quantiﬁcational complexity is at most ∆1. The reason for disal- lowing second-order parameters in A is based on taking a narrow view of what the idealized mathematician is capable of. If we allowed second-order parameters to occur in A, then for every classical function, there would be a choice sequence agreeing with it. To imagine the idealized mathematician creating such sequences would seemingly involve her having access to an oracle for every classical function. This, however, does not seem to be implicit in the idea of an idealized mathematician creating sequences in time. The most conservative idealizations involve a mathematician who is not bounded in resources of time, attention, memory, or computation; but this does not

1 0 Of particular interest would be the relation between ∆1!SC and the theory FIM of Kleene and Vesley(1965), since—to a ﬁrst approximation and modulo the bar theorem—Kleene and Vesley’s theory is obtained by adding a comprehension principle for choice sequence variables to a theory of recursive functions.

120 require her to have an oracle for every classical function. On the other hand, a more dramatic idealization would imagine not a mere extension of a limited mathematician like you or I, but instead would consider the creative activity of an omniscient godlike agent that can survey the platonic realm of mathematical truths. While certainly dramatic, this conception of the idealized mathematician does not seem incoherent. And under this conception, it would be perfectly reasonable to allow set variables to

occur in the sequence comprehension axioms. Let MC1 be the theory which is like MC except that it allows free set variables to occur in the sequence comprehension axiom. One of the important uses of sequence comprehension will be to define new choice sequences by projections of existing sequences. For instance, if we define a new sequence by taking each value as twice the corresponding value of a lawless sequence, then we will get a sequence that has only even values, but otherwise appears lawless. We can use SC to define such projections as follows: if A(y, z) defines a mapping on the natural numbers, then ∃y(α(x) = y ∧ A(y, z)) will define a projection of the sequence α by applying the mapping A to its values. Letting β be the new sequence so defined, it follows that A(y, z) ∧ ♦α(x) = y → ♦β(x) = z and A(y, z) ∧ α(x) = y → (β(x) 6= f → β(x) = z). The fact that we can define new sequences by projection from existing sequences entails by a fairly standard diagonal argument that we cannot enumerate the choice sequences.

Theorem 6.1. Assume MC + ΞSC. Then there is no formula F (n, α) that deﬁnes an enumeration of the choice sequences and is such that ∃α[F (x, α) ∧ ♦α(x) = y + 1] is in Ξ.

121 Proof. Suppose that there is such an F . By S2,

∀x∃y∃α[F (x, α) ∧ ♦α(x) = y + 1]

So by ΞSC we have:

∃β∀x∀y[β(x) = y → ∃α[F (x, α) ∧ ♦α(x) = y + 1]

Since F deﬁnes an enumeration of the choice sequences, there is some z such that

F (z, β) ∧ ∀α(F (z, α) → ∀xα(x) = β(x)). Then we can infer:

∀y[β(z) = y → ♦β(z) = y + 1]

But this contradicts S3.

By an easy adaptation of this proof one can also show that there is no formula that provably defines an enumeration of the non-lawless sequences (and which bears the same relation to Ξ)—the only modification is that one also has to show that β is not lawless. This argument for this proof is adapted from Troelstra’s paradox, which afflicts some versions of the theory of the creative subject, as developed by Kreisel(1967). 2 In outline, Troelstra’s argument goes as follows: consider the activity of the idealized mathematician (the ‘creative subject’). The potential actions of this agent, we may assume, have order type of the natural numbers, and each stage in their activity, the

2Troelstra ﬁrst presented it in (Troelstra, 1969, 105-6); see also van Atten(2016) and van Atten (2018) for discussion.

122 creating subject establishes certain propositions.3 By inspection the creative subject can determine whether a given proposition they have established is of the form “α is a sequence whose values are ﬁxed by a law”. Thus the creative subject can enumerate

the lawlike sequences that they create. Let βn be such an enumeration. Then deﬁne the sequence γ(x) := βx(x) + 1. Since the values of γ are ﬁxed by a law, γ must occur somewhere in the enumeration βn. But this is obviously impossible. By formalizing this argument, Troelstra(1969) showed Kreisel’s theory of the creative subject to be inconsistent. The fact that I used a similar argument to show that there is no enumeration of choice sequences in MC naturally raises the question of whether we could show MC to be inconsistent by providing just such an enumeration of choice sequences. This fear, however, can be allayed by providing a model for MC.

6.1 A Kripke Model for MC

In this subsection I will show how the Kripke model for MCLS from §5.2.1 can be extended to deﬁne a model of MC. The construction will assign a set of choice sequences to each world in the frame in a series of stages. Recall that the worlds in the underlying frame are themselves ﬁnite sequences of natural numbers, so we can refer to lh(w).

0 Let An(x, y) be a an enumeration of the ∆1 formulas with (at least) two free numerical variables, no bound choice sequence variables, and no set parameters. The construction will proceed in a series of stages, deﬁned inductively. At stage 0, lawless

3Kreisel’s theory of the creative subject was intended as a regimentation of Brouwerian intuitionism, so the epistemic act of ‘establishing’ a proposition and the metaphysical act of creating mathematical objects should be regarded as truly one and the same act.

123 sequences are assigned to the Kripke frame as in §5.2.1. For stage n > 0, we add new choice sequences to the model according to the following steps (using µ for G¨odel’s minimization operator):

• Step 1: If no world w veriﬁes ∀xI∃yAn(x, y)∧∀x∀y(An(x, y) → An(x, y)), then stage n is the same as stage n − 1.

• Step 2: If An has no choice sequence parameters, and a world w veriﬁes

∀xI∃yAn(x, y) ∧ ∀x∀y(An(x, y) → An(x, y)), then add a choice sequence

αAn to the domain of w. Deﬁne the graph of αAn at w as follows: let m be the largest number such that a lawless sequence is deﬁned on m and such that

∀x ≤ m∃yAn(x, y). For x ≤ m, put αAn (x) = µyAn(x, y), unless this conﬂicts

with the value assigned to αAn (x) at a world uRw, in which case αAn (x) is

4 assigned the same value at w as at u; and for x > m, put αAn (x) = f. If

there are any worlds w at which αAn (x) has not been assigned any graph, put

αAn (x) = f for all x.

• Step 3: An has choice sequence parameters ~α, and a world w, with choice

sequences ~γ that were deﬁned at a stage < n, veriﬁes ∀xI∃yAn(~γ, x, y) ∧

∀x∀y(An(~γ, x, y) → An(~γ, x, y)), then add a choice sequence αAn,~γ to the

domain of w. Define the graph of αAn,~γ at w as follows: let m be the largest number such that a lawless sequence is defined on m, each γ is defined on m,

and ∀x ≤ m∃yAn(~γ, x, y). For x ≤ m, put αAn,~γ(x) = µyA(~γ, x, y), unless

this conﬂicts with the value assigned to αAn,~γ(x) at a world uRw, in which

case αAn,~γ(x) is assigned the same value at w as at u; and for x > m, put

4This assumes that the assignment of choice sequences to worlds proceeds inductively on the worlds. Since our underlying frame is well-founded, this is unproblematic.

124 αAn,~γ(x) = f. If there are any worlds w at which αAn,~γ(x) has not been assigned

any graph, put αAn,~γ(x) = f for all x.

• Step 4: For each s < n: if As has choice sequence parameters, and if there are some ~γ each of which were deﬁned at either an earlier stage or an earlier step of stage n, and at least one of which was deﬁned in the two previous steps, and

if w veriﬁes that ∀xI∃yAs(~γ, x, y) ∧ ∀x∀y(As(~γ, x, y) → As(~γ, x, y)), then

form αAs,~γ as in Step 3.

• Step 5: If An has choice sequence parameters, and if there are some ~γ that were defined at either an earlier stage or an earlier step of stage n, and at least one of which was defined in one of the two previous steps, and if w verifies that

∀xI∃yAn(~γ, x, y) ∧ ∀x∀y(An(~γ, x, y) → An(~γ, x, y)), then form αAn,~γ as in Step 3.

• Steps 2n + 6, 2n + 7: Iterate Steps 4 and 5 ω-many times.

Iterating this process through all ω stages will provide us with a model for MC.

Theorem 6.2. MC is consistent.

Proof. It is easy to check that each of the choice sequences in the model constructed above satisfy axioms S1-S4, and since we started with a model for MCLS, the lawless sequences will also satisfy S5r-S9. The sequence induction schema S-IND will also hold since the domain of natural numbers is just the standard natural numbers. If an instance of ISC were not satisﬁed, then there would have to be a formula An and existent choice sequences ~γ such that at some world w it is true that

∀xI∃yAn(~γ, x, y)∧∀x∀y(An(~γ, x, y) → An(~γ, x, y)) but ¬∃α∀x∀y(α(x) = y →

125 An(~γ, x, y)). But an induction on the stage and step at which the γ’s were deﬁned easily shows this to be impossible.

Note that the model construction actually does not depend on the restriction to

0 ∆1 formulas A. Essentially the same process can be used to yield the more general result that allows A to have arbitrary complexity, as long as it does not contain bound choice sequence variables.

6.2 Kripke’s Schema

Formal theories of choice sequences often include Kripke’s Schema, which is stan- dardly formulated:5

∃α[∀x(α(x) = 0 ∨ α(x) = 1)

∧ ∀x(α(x) = 0) ↔ ¬A

∧ ∃x(α(x) = 1) → A]

The motivation for this is to capture Brouwer’s so-called Creating Subject arguments. In these arguments, Brouwer describes a free choice sequence whose values are determined by the mathematical activities of a creating subject. For instance, we might imagine that the creating subject is performing various mathematical constructions, and as long as those constructions fail to establish A, the subject chooses the value 0 for the next entry of the choice sequence. And if the subject ever eﬀects a mathematical construction that veriﬁes A, then they will choose the value 1 for the sequence.

5For instance, Myhill(1968), Hull(1969), Moschovakis(2017). See also van Dalen(1999), who discusses Kripke’s Schema without giving a formal theory of choice sequences, and van Atten(2018), who identiﬁes implicit appeals to Kripke’s Schema in various arguments of Brouwer.

126 A straightforward way to try to modalize Kripke’s Schema would be:

∃α[∀x∀y(α(x) = y → y = 0 ∨ y = 1) ∧ ∀x∀y(α(x) = y → y = 0) ↔ ¬A

∧ ∃xα(x) = 1 → A]

This schema is only satisﬁable by formulas A that are positively stable, though. For let A satisfy the schema and suppose A. By the second conjunct of the schema,

α(x) 6= 0 for some x, so by the ﬁrst conjunct of α(x) = 1. By S3, then, α(x) = 1, so by the third conjunct A. One way to account for this result is to require that Kripke’s Schema only be applied to positively stable formulas. Alternatively, we can replace the second conjunct with the weaker condition: ¬A → ∀x(∃yα(x) = y → y = 0). This is in turn implied by the other two con- juncts, and hence we arrive at another version of modalized Kripke’s Schema:

∃α[∀x(∃yα(x) = y → y = 0 ∨ y = 1) ∧ ∃xα(x) = 1 → A]

Let ΞKS denote this second modalized version of Kripke’s Schema, restricted to the instances where A is in Ξ. KS will denote the unrestricted modalized Kripke’s Schema. It is easy to see that accepting lawlike comprehension for a class of formulas will entail Kripke’s Schema for that same class of formulas:

Theorem 6.3. Provided Ξ includes numerical identities and is closed under truth- functional operations and necessitation, Ξ!SC entails ΞKS.

127 Proof. Let A be in Ξ. Then (A ∧ y = 1) ∨ (¬A ∧ y = 0) is also in Ξ, thus by Ξ!SC we have

∃α(∀x∀yα(x) = y → ((A ∧ y = 1) ∨ (¬A ∧ y = 0)))

As is easy to see, it also follows that ∀x(α(x) = 0 ∨ α(x) = 1) and that ∃x(α(x) = 1) → A as required.

Ξ!SC and ΞKS are not equivalent, however, since MC + KS is consistent with

0 all sequences being either lawless or necessarily 0, 1-valued, while even MC + ∆1!SC is not.

6.3 Bar Induction and the Fan Theorem

In MCLS we saw that the bar theorem follows from the assumption that there are still new choice sequences that have yet to be defined. In MC we can remove this qualification and prove the bar theorem directy. Moreover, in MC we can countenance choice sequences that are subject to some constraints in their possible futures. Thus, we can consider not only bar induction as it applies to the full tree of Baire space, but also the principle of bar induction as relativized to a given tree, or, in the intuitionistic parlance, relativized to a given spread. If F is a set of sequence numbers closed under initial segments, then F is a spread (abbreviated sp(F )) if ∀x ∈ F ∃y(x_hyi ∈ F ). A spread in which each node has only finitely many successors is a fan. A choice sequence α is in a spread (abbreviated

α ∈ F ) if ∀x(∃y(α(x) = y) → α(x) ∈ F ).

128 Now let BIR be the following relativized version of bar induction:

[sp(F ) ∧ ∀α ∈ F I∃x(α(x) ∈ X) ∧ ∀x(x ∈ X → x ∈ Y )∧ ∀x ∈ Seq(∀y(x_hyi ∈ Y ) → x ∈ Y )] → hi ∈ Y

Theorem 6.4. 1. MC ` BI

2. MC1 ` BIR

Proof. Claim 1 will follow from claim 2. Claim 2 can be proved in a way largely similar to the original proof of Theorem 5.12. If hi ∈/ Y , then we could define a path through F that does not inevitably intersect X. To show that this is impossible, we need to show that necessarily there is a choice sequence in F that does not inevitably diverge from that path. And we do know that necessarily there is a sequence α that does not inevitably diverge from that path: let λ be a lawless sequence which is not yet defined on n, and define α(x) := λ(x + n). Then α will not inevitably intersect X, since it does not inevitably diverge from the path that avoids X. But α need not in general be in F . So we will define a projection β of α as follows: If α(x) ∈ F , then β(x) = α(x); if α(x) ∈/ F , then β(x) is the least y such

_ that β(x − 1) y ∈ F . (The occurrence of F here is what requires MC1 rather than just MC). Visually, if F is pictured as a tree growing upward on the page, then β follows α as long as α stays in F , but if α ever leaves F , then β just keeps going left within F . Since α does not inevitably diverge from F \ X, β also does not inevitably diverge from F \ X.

129 Thus from the assumption that hi ∈/ Y we can construct a β such that β ∈ F but

¬I∃xβ(x) ∈ X, and this is the contrapositive of BIR. Note that if F is recursive, then the second claim can be proved in MC rather than

0 MC1 by replacing occurences of F with its deﬁning formula, which will be ∆1.

We can now also prove the fan theorem. Informally, the fan theorem says that if we have a bar in a fan, then there is a uniform ﬁnite bound on how any sequence in the fan can proceed before it hits the bar. This is essentially the contrapositive of K¨onig’slemma.6

Theorem 6.5. MC1 ` F an(F )∧∀α ∈ F I∃x(α(x) ∈ X) → ∃y∀α ∈ F I∃z < yα(z) ∈ X.

Moreover, if F is recursive, then MC suﬃces in place of MC1.

Proof. We will use BIR. Deﬁne the set Y by:

n ∈ Y :↔ ∃y∀α ∈ F (α ∈ n → I∃z < yα(z) ∈ X)

To show that this set is well-defined we have to show that the comprehension formula is equivalent to one with no choice sequence variables. But the formula ∃y∀α ∈ F (α ∈ n → I∃z < yα(z) ∈ X) is easily seen to be equivalent to ∃y∃z < y∃m ∈ Seq(lh(m) + lh(n) = z ∧ n_m ∈ X). Clearly if hi ∈ Y , then ∃y∀α ∈ F I∃z < yα(z) ∈ X. It is also clear that ∀x(x ∈ X → x ∈ Y ). Thus it suffices to show ∀w(n_hwi ∈ F → n_hwi ∈ Y ) → n ∈ Y . Suppose ∀w(n_hwi ∈ F → n_hwi ∈ Y ). Then since F is a fan, there are finitely

6For the intuitionist the difference between König’slemma and its contrapositive is important, since contraposition is not constructively valid. Indeed, while the fan theorem is intuitionistically acceptable, König’slemma is not. Cf. (Dummett, 2000, 49-53).

130 _ many w1, ..., wk such n hwii ∈ F . Thus by the deﬁnition of Y there are ﬁnitely many

_ y1, ..., yk such that ∀α ∈ F (α ∈ n hwii → I∃z < yiα(z) ∈ X). Thus, if y is the maximum of these yi’s, then ∀α ∈ F (α ∈ n → I∃z < yα(z) ∈ X), i.e. n ∈ Y , as required.

6.4 Real numbers

Real numbers, as conceived by the intuitionist, are given to us by real number generators, sequences that, in the limit, pin down a single point on the continuum. For instance, Cauchy sequences can serve as real number generators, as can series of convergent nested open intervals of rationals. Real numbers themselves can then be taken to be equivalence classes of co-convergent real number generators. This would officially make real numbers second-order objects, and functions on reals third-order operations. However, to avoid this level-raising, functions on real number generators can serve as representatives of, or means of apprehending, functions on real numbers themselves.7 I will follow Heyting(1971) in much of the material for this section. In particular, I will define real number generators as Cauchy sequences of rational numbers. For this purpose I will take as given some standard way of defining the rational numbers and their operations from the natural numbers.8 Moreover, for simplicity I will assume that the coding of rational numbers is onto the natural numbers. I will take variables

7In this respect my treatment accords with that of (Bishop and Bridges, 1985, 65), who write: “Notice that a real number is a regular sequence of rational numbers, not an equivalence class of regular sequences of rational numbers. To deﬁne a real number to be such an equivalence class would be either pointless or incorrect: pointless (but correct) if the equivalence class is required to be speciﬁed by giving some particular regular sequence of rational numbers that belongs to it, and incorrect otherwise.” 8For one such approach, see Feferman(1964).

131 p, q, ... to be variables ranging over rationals. The addition and multiplication signs will thus be used ambiguously to denote the functions on naturals and rationals. Context will serve to disambiguate. We can now define real number generators and the usual field operations on them. As well as the standard operations familiar from classical analysis, we will define the apartness # relation that figures in constructive analysis.

Deﬁnition 6.6. We can deﬁne a number of notions concerning real numbers.

1. α is a real number generator (abbreviated Rα) when:

∀p > 0I∃x∀y (α(x + y) 6= f → |α(x) − α(x + y)| < p)

2. Deﬁne α = β if Rα, Rβ, and:

∀p > 0I∃x∀y (α(x + y) 6= f ∧ β(x + y) 6= f → |α(x + y) − β(x + y)| < p)

3. Deﬁne α#β if Rα, Rβ, and:

∃p > 0I∃x∀y (α(x + y) 6= f ∧ β(x + y) 6= f → |α(x + y) − β(x + y)| ≥ p)

4. Deﬁne α < β if Rα and Rβ, and:

∃p > 0I∃x∀y (α(x + y) 6= f ∧ β(x + y) 6= f → β(x + y) − α(x + y) > p)

5. Deﬁne γ = α + β if ∀x(γ(x) 6= f → γ(x) = α(x) + β(x))

132 6. Deﬁne γ = α − β if ∀x(γ(x) 6= f → γ(x) = α(x) − β(x))

7. Deﬁne γ = α × β if ∀x(γ(x) 6= f → γ(x) = α(x) × β(x))

It is worth noting that +, −, × are only well-defined if the theory includes instances of SC where A has choice sequence parameters. For a rational p we can let p be the choice sequence with constant value p. (Note that this is a different use of the overline than that used to denote the initial segment of a choice sequence; in that latter usage the line extends only over the sequence variable.) Having flagged these assumptions, we can observe that these definitions are well behaved.

Theorem 6.7. 1. If Rα and Rβ, then R(α + β), R(α − β), and R(α × β).

2. If Rα ∧ α#0, there is a α−1 such that Rα−1 ∧ α × α−1 = 1.

3. α = α; if α = β then β = α; and if α = β and β = γ, then α = γ.

4. α + β = β + α.

5. α × β = β × α.

6. If α = β and Rγ, then:

(a) α + γ = β + γ,

(b) α − γ = β − γ and γ − α = γ − β,

(d) If α#0, then α−1 = β−1.

Proof. The proofs are all straightforward. We can prove claims 1 and 2 as illustra- tions.

133 1. Suppose γ = α + β. We need to show that ∀pI∃x∀y(γ(x + y) 6= f → |γ(x) − γ(x + y)| < p). By the definition of +, if γ(x + y) 6= f, then γ(x + y) = α(x + y)+β(x+y). Let p > 0 be given, and consider an x1 such that ∀y(α(x1 +y) 6= f → p |α(x1)−α(x1 +y)| < 2 ). Now since Rβ, we also have that I∃x2∀y(β(x2 +y) 6= f → p |β(x2 + y) − β(x2)| < 2 . Put x = max(x1, x2). Then ∀y(α(x + y) 6= f ∧ β(x + y) 6= f → |(α(x) + β(x)) − (α(x + y) + β(x + y))| < p. Since inevitably such an x exists, we have our result.9 The arguments for γ = α − β and γ = α × β are similar. 2. Assume Rα and α#0. Define α−1 by setting α−1(x) := (α(x))−1 when α(x) 6= 0, and α−1(x) := 0 when α(x) = 0. Provided Rα−1, it is obvious by definition that

−1 −1 −1 α × α = 1. So we want to show that ∀pI∃x∀y(α (x + y) 6= f → |α (x + y) − −1 α (x)| < p). Let p > 0. Since Rα, ∀p0I∃x1∀y(α(x+y) 6= f → |α(x)−α(x+y)| < p0. Since α#0, there exists a q such that I∃x2∀y(α(x+y) 6= f → |α(x+y)−0| ≥ q). 2 −1 Put p0 = pq , and let x = max(x1, x2). Then necessarily for all y, if α (x + y) 6= f then some routine algebra shows that

−1 −1 α(x) − α(x + y) po α (x) − α (x + y) = < = p α(x)α(x + y) q2

Since inevitably there is such an x, we have our desired result.

It is worth commenting on the choice of deﬁnition of real number generator. In

particular, why use the initial quantiﬁer ∀pI∃x rather than ∀p∃x? First, note that the latter expression is strictly stronger, so the chosen deﬁnition allows us to

9Let me indicate brieﬂy how this informal argument can be rendered fully formal: we start by arguing that necessarily, if there is an x1 as in the main text, then necessarily if there is an x2 as in the text then there is an x as in the text. Then by applying the modal axioms M1 and M2 we infer that if inevitably there is an x1, then if inevitably there is an x2, then inevitably there is an x. Since we know that inevitably there is an x1 and necessarily it is inevitable that there is an x2, we can conclude that inevitably there is an x, as required.

134 countenance more choice sequences as real number generators than the alternative definition would. But the main reason for doing so is that we may not have an effective bound on the convergence rate of a choice sequence; but as long as there is a guarantee that the choice sequence will converge, then it constitutes a perfectly reasonable way of picking out a point on the continuum. In intuitionistic and constructive mathematics there are also more stringent conceptions of real number generators. The definition above requires that the sequence eventually converge, but no particular rate for convergence is prescribed at the outset, and indeed the rate of convergence may not be known. If, however, we were to read the quantifier combination ∀p∃x constructively, that would require that for a given p we have a method for finding an x such that p is the radius of convergence at x. This reading is, thus, what one would expect from a constructive mathematician. Some authors make this reading explicit, requiring in the definition that a real number generator have a known rate of convergence, i.e. that given a real number generator

1 10 α and some k one can determine when α will have a radius of convergence k . A stronger requirement is the rate of convergence be uniformly given in advance, so that for any k there is an n such that any real number generator α will have

1 11 a radius of convergence k by the argument n. We will follow Heyting(1971) in not requiring this of all real number generators, but defining a special class of real number generators that meet this condition, called canonical real number generators. Parts 2-4 of Definition 6.6 will apply without problem to canonical real number generators. The definitions of addition, subtraction, and multiplication will need to be tweaked, however, to ensure that the canonical real number generators are closed

10Cf. (Troelstra and van Dalen, 1988, ch. 5). 11Bishop(1967) requires this of all real number generators.

135 under addition, subtraction, and multiplication.

Deﬁnition 6.8. 1. α is a canonical real number generator (abbreviated R0α) if:

n 1 ∀x α(x) 6= → ∃n ∈ α(x) = ∧ |α(x) − α(x + 1)| ≤ f Z 2x 2x+1

0 0 2. γ := α +R0 β if R α, R β, and necessarily, if γ(x) 6= f then:

  n α(x + 1) + β(x + 1) if this is of the form 2x γ(x) =  1 α(x + 1) + β(x + 1) − 2x+1 otherwise

0 0 3. γ := α −R0 β if R α, R β, and necessarily, if γ(x) 6= f then:

  n α(x + 1) − β(x + 1) if this is of the form 2x γ(x) =  1 α(x + 1) − β(x + 1) − 2x+1 otherwise

0 0 4. γ := α ×R0 β if R α, R β, and necessarily for all x, letting k be the least number

k−2 N such that |α(0)| + |β(0)| + 2 < 2 , we have that γ(x) is whichever value 2x minimizes |α(x + k)β(x + k) − γ(x)|. If there are two such N, pick the larger one.

A word on notation: recall that p denotes the real number generator with constant value p. (Recall also that p, q, ... are being used as variables ranging over rationals).

n In the context of canonical real number generators we will use 2k to denote a canonical n real number generator which has constant value 2k for all arguments x > k. Similarly, n for a rational p not of the form 2k , the canonical real number generator p can be easily deﬁned from a recursive function, and hence is entirely lawlike.

136 We now conﬁrm that these deﬁnitions have the desired properties. Two facts will

0 be useful in showing that R (α ×R0 β).

Lemma 6.9. Suppose R0α and R0β.

1 1 1. ∀x∀y ≥ x α(x) − 2x < α(y) < α(x) + 2x .

1 2. ∀x |α(k + x + 1)β(k + x + 1) − α(k + x)β(k + x)| < 2x+2 .

Proof. 1. Let x be arbitrary. By S-IND it follows that for all y ≥ x, α(y) is bounded by X 1 α(x) ± 2i x

1 1 And this bound entails α(x) − 2x < α(y) < α(x) + 2x . 0 0 1 2. Since R α and R β, we know that |α(k + x + 1) − α(k + x)| ≤ 2k+x+1 , and similarly for β. Thus we have,

1 1 |α(k +x+1)β(k +x+1)−α(k +x)β(k +x)| ≤ |α(k +x)+β(k +x)|+ 2x+k+1 22(x+k+1)

By claim 1, |α(k + x) + β(k + x)| < |α(0)| + |β(0)| + 2. By the deﬁnition of k we know that |α(0)| + |β(0)| + 2 < 2k−2, which gives:

1 1 2k−2 1 1 |α(k + x) + β(k + x)| + < + < 2x+k+1 22(x+k+1) 2x+k+1 22(x+k+1) 2x+2

This gives us our desired bound:

1 |α(k + x + 1)β(k + x + 1) − α(k + x)β(k + x)| < 2x+2

137 Theorem 6.10. Suppose R0α and R0β.

0 0 0 1. R (α +R0 β), R (α −R0 β), and R (α ×R0 β).

−1 0 −1 −1 2. If α#0, there is an α such that R α ∧ α ×R0 α = 1.

3. α +R0 β = β +R0 α.

4. α ×R0 β = β ×R0 α.

5. If α = β, and R0γ, then:

(a) α +R0 γ = β +R0 γ,

(b) α −R0 γ = β −R0 γ and γ −R0 α = γ −R0 β,

(d) If α#0, then α−1 = β−1.

Proof. 1. The cases of addition and subtraction are straightforward. For multiplica-

n tion, it is clear by construction that γ(x) will always be of the from 2x . To see that 1 |γ(x) − γ(x + 1)| ≤ 2x+1 , let x be arbitrary. By Lemma 6.9 (2), |α(x + k + 1)β(x + k + −x−2 N 1 1)−α(x+k)β(x+k)| < 2 . So if γ(x) = 2x is the multiple of 2x which is closest to 1 α(x+k)β(x+k), then the multiple of 2x+1 which is closest to α(x+k +1)β(x+k +1) 2N−1 2N 2N+1 must be one of 2x+1 , 2x+1 , or 2x+1 . By deﬁnition, γ(x+1) is deﬁned to be the nearest 1 N such value. And obviously each of these values are within 2x+1 of 2x , as required.

2. Without loss of generality, assume α > 0, and in particular that I∃x0∀y > xα(y) > p. That such a p and x0 exist follows from the assumption that α#0, and moreover x can be found eﬀectively from p in light of Lemma 6.9 (1). The basic

−1 1 0 idea will be to deﬁne α from the sequence α(x) , similar to how we deﬁned ×R : if

138 1 th we go far enough out in the sequence α(x) , the diﬀerences between the (x + k) and (x + k + 1)th values will be close enough to each other that if we deﬁne α−1(x) by

1 n −1 −1 approximating α(x+k) by the nearest value 2x , |α (x + 1) − α (x)| will never exceed 1 2x+1 .

In more detail, note that, for x ≥ x0:

x x+1 x 1 1 2 2 2 1 1 − ≤ − = = < α(x) α(x + 1) N 2N + 1 N(1 + 2N) α(x)α(x + 1)2x+1 p22x+1

1 1 −1 n Now, we can ﬁnd a k such that p22x+k ≤ 2x . Then deﬁne α (x) to be the value 2x 1 which is nearest to α(x+k+2) (and if there are two such values, pick the larger one). It is straightforward to check that α−1 is as required by the theorem. Claims 3, 4, and 5 are all straightforward.

Thus real number generators provide us with a means of apprehending the intuitionistic continuum—or at least an ersatz version of it—from the modal-temporal perspective of MC. Canonical real number generators give us a particularly well- structured way of apprehending the continuum. Canonical real number generators also have the philosophical beneﬁt of providing a constructive (indeed, uniform) bound on the rate of convergence. While MC cannot, I conjecture, prove that the canonical real number generators exhaust the continuum, a slightly stronger theory can; and hence MC cannot prove that the canonical real number generators do not exhaust the continuum.

0 Lemma 6.11. Assume MC + Π1ISC. Then:

1. If R0α then Rα.

2. If Rα then there is a β such that R0β and α = β.

139 Proof. Claim 1 is obvious.

0 For claim 2, we will deﬁne β from α using Π1ISC as follows: we have that

−x−2 ∀xI∃y∀z |α(y + z) − α(y)| < 2

N N+1 N N+1 If 2x ≤ α(y) ≤ 2x , then let β(x) be whichever of 2x and 2x is closer to α(x). By construction R0β, and it is easy to see that α = β as well.

We can call the domain obtained from our real number generators the temporal- potential continuum. Already this modest account of our ersatz intuitionistic continuum suﬃces to prove some anti-classical results. The next theorem is in the vein of so-called weak Brouwerian counterexamples. The basic idea behind a weak Brouwe- rian counterexample to a given principle is to show that if that principle were provable then we could prove some other problematic principle, such as the law of excluded middle. Hence the original principle must be unprovable. Exactly what makes a problematic principle problematic will depend on the speciﬁc example chosen, but what they all have in common is that this problematic principle itself is taken to be unprovable. The following theorem is in this vein, showing that MC does not prove that every real number generator is a determinately rational or irrational.

Theorem 6.12. MC 0 ∀α ∈ R(∃qα = q ∨ ¬∃qα = q)

Proof. Let λ be a lawless sequence with initial deﬁned segment of length n, and let

hrii be a Cauchy sequence converging to some irrational number. Then deﬁne a new real number generator β as follows: if λ(n + 1) ≤ 10, then ∀m > nβ(m) = 1 and if

λ(n + 1) > 10, then ∀m > nβ(m) = rm. Clearly in one range of possibilities β will be equal to a rational number q (viz., 1) and in the other range of possibilities β will

140 not be equal to any rational number (since β will in fact be equal to an irrational

number). Thus we have a countermodel to ∀α ∈ R(∃qα = q ∨ ¬∃qα = q).

It is worth commenting on the precise statement that I glossed as saying that not every real number generator is a determinately rational or irrational. To say that a given real number generator is rational (i.e., identical to a rational number) or not rational (i.e., not identical to any rational number) is an instance of the excluded middle. And since the background logic of MC is classical, this result is trivially provable in MC. But the mere statement that a real number generator is either rational or not rational does not faithfully capture what the constructivist means when he says that a given real number is either rational or irrational. For the constructivist, to say that a given real number is (ir)rational means that it is determinately (ir)rational: the matter has been settled, and the real number in question is demonstrably (ir)rational. In particular, when the constructivist says that a number is irrational, this result should continue to hold in the future—it should be stable. But in MC, not being rational (i.e. not being identical to any rational number) is not stable. Hence to capture the idea of a real number generator being determinately rational or determinately irrational, we use the notion of necessarily being identical to some rational number or necessarily not being identical to any rational number. As another anti-classical result, we can observe that < is not a linear order on the real number generators. While the previous theorem followed the spirit of a weak counterexample, showing merely that our theory MC does not prove that every real number generator is either determinately identical to some rational or determinately distinct from every rational, this theorem actually shows that MC proves < not to be a linear order.

141 Theorem 6.13. MC ` ¬∀α ∈ R∀β ∈ R (α < β ∨ α = β ∨ β < α)

Proof. Put α = 0. Let λ be a lawless sequence whose maximum value thus far deﬁned

−1 x is k. Deﬁne β so that if λ(y) < 2k for all y ≤ x, β(x) = 2 ; otherwise, if y < x is −1 y the least number such that λ(y) ≥ 2k, then β(x) = 2 . Now it is not inevitable that there is no such y, so it is not inevitable that β approaches α arbitrarily closely. Thus, β 6= α. It is also not inevitable that if there were such y it would have to be even, so it is not inevitable that β will eventually

be some given value greater than α. Thus, α ≮ β. And similarly, it is not inevitable that if there were such a y it would have to be odd, so β ≮ α.

One more example of an anti-classical result from intuitionistic analysis that we can capture in the modal setting is that monotone and bounded sequences need not be convergent. To the contrary, MC proves the existence of a counterexample.

Theorem 6.14. MC ` ∃α[∀x(α(x) 6= f → α(x) ≤ α(x − 1)) ∧ ∀x(α(x) ≥ 0) ∧ ¬Rα]

Proof. Let λ ∈ L and deﬁne α as follows: if fn is the least computable function (under

1 a ﬁxed ordering) that agrees with λ x, put α(x) = n . Clearly α is bounded and monotone, as required. It remains to show ¬Rα. Let n be the least number such that fn agrees with the initial deﬁned segment of λ. As long as fn and λ agree up to x, it is possible that they will disagree on x + 1. Moreover, since λ(x + 1) can be anything, the least m such that fm and λ agree up to x + 1 can be arbitrarily large. Thus, as

1 long as λ and fn agree, |α(x) − α(x + 1)| can be any value < n . So if λ and fn agree, 1 then there will not be any x such that ∀y(α(x + y) 6= f → |α(x) − α(x + y)| < 2n ).

Since it is not inevitable that λ and fn disagree, it is not inevitable that there will be such an x.

142 6.5 Continuity

One of the distinctive theorems of intuitionism is that every function is continuous. In this subsection, we will see that a weak version of this fact holds in our temporal theory as well. (Toward the end of the next section we will establish another continuity result which, although weaker than the claim that every function is continuous, is in some respects stronger than the result proved in this section). Theorem 6.15 below is a partial continuity result, which can be glossed as saying that no functions whose values are given as canonical real number generators have

deﬁnable discontinuities. Somewhat more exactly, if F (α, γ) deﬁnes a function R → R, where the arguments are given as real number generators and the values are given as canonical real number generators, then it is not the case that there is a sequence

α ∈ R and a definable set of sequences βi ∈ R that approach α arbitrarily closely and witness a discontinuity at α. (The expression ‘defines a function’ is a little misleading, since F need not actually be a definition in our formal language. It could, for instance, be a new predicate added to the language. I use the expression ‘define a function’, however, since F is syntactically taken to be a relation between choice sequences rather than a function symbol.) Before stating the theorem precisely, I should explain what it means for a set of

sequences βi to be deﬁnable. First, let us say that a sequence β is deﬁnable if there

12 is a set X such that ∀x∃!yhx, yi ∈ X and ∀x∀y(β(x) = y → hx, yi ∈ X). Recall 13 that a family of sets Xi can be coded as a single set X = {hi, xi : x ∈ Xi}. Then a set of sequences βi is deﬁnable if there is a family of sets Xi such that any sequence γ

12A more liberal notion of deﬁnability would relax the condition ∀x∃!yhx, yi ∈ X to ∀x∃yhx, yi ∈ X. For this theorem, however, we need the stronger notion. 13Of course, when we regard families of sets as coded in this way, it is not necessarily true that for every plurality of sets there is a family of sets including exactly the members of that plurality.

143 is one of the βi’s just in case it is deﬁnable by one of the Xi’s. More precisely, the set

of βi’s is deﬁnable by a family X just in case each member of the family X deﬁnes a choice sequence and the set of βi’s consists of exactly the sequences γ satisfying:

∃i∀x∀y(γ(x) = y → hi, hx, yii ∈ X)

Note that the concept of a deﬁnable set of sequences is a meta-theoretic notion. In our formal theory there are no such objects as sets of sequences. If X is a set, then we can use β ∈ X to abbreviate the claim that β is deﬁnable by X, i.e.:

∀x∃!yhx, yi ∈ X ∧ ∀x∀y(β(x) = y → hx, yi ∈ X)

th And if X is a family of sets, we can use Xi to denote the i element of the family X.

So β ∈ Xi is an abbreviation for

∀x∃!yhi, hx, yii ∈ X ∧ ∀x∀y(β(x) = y → hi, hx, yii ∈ X)

Theorem 6.15. Assume MC1. Suppose that the formula F (α, γ) deﬁnes a function on real numbers where the values of F are given as canonical real number generators, i.e.:

0 ∀α ∈ R∃γ ∈ R F (α, γ) ∧ ∀α∀γ[(F (α, γ) → F (α, γ)) ∧ ∀β ∈ R∀γ0 ∈ R0(α = β ∧ F (α, γ) ∧ F (β, γ0) → γ = γ0)]

144 Then there is no family X deﬁning sequences βi which witness a discontinuity at α, i.e.:

−k ¬∃α ∈ R∃n∃X∀k∀β ∈ Xk[β ∈ R ∧ |α − β| < 2 ∧

−n ∃ζ(F (β, ζ) ∧ ∀γ F (α, γ) → |γ − ζ| ≮ 2 )]

Proof. Suppose not, so that there are such α, n, and X. Let λ be a lawless sequence. Since λ is only defined on a finite segment of arguments, it must agree with some function f on those arguments. Now define a choice sequence ξ so that if λ and f agree up to m, then ξ(m) = α(m). On the other hand, if λ and f disagree on some argument ≤ m, then let k be the least argument λ and f disagree on; if k is even, set

ξ(m) = y for hm, yi ∈ X k , and if k is odd, set ξ(m) = α(m). 2

Let βk be the witness β which is deﬁnable by Xk. Likewise, let ζk be the witness for ∃ζF (βk, ζ). Also, ﬁx γ such that F (α, γ). Note that since γ and ζk are canonical

−n real number generators and |γ − ζk| ≮ 2 , by Lemma 6.9 (1) we know that |γ(n +

−n−2 3) − ζk(n + 3)| > 2 . Let us assume for this paragraph that λ and f do agree on the initial deﬁned segment of λ. Now by deﬁnition, ξ and α will disagree only if λ and f disagree, so ξ and α do not disagree on any argument. But since it is possible that λ and f disagree on an odd argument, it is possible that ξ = α. Fix η such that F (ξ, η). If

−n−2 |η(n + 3) − γ(n + 3)| > 2 , then η#γ, and hence α 6= ξ. But since ♦α = ξ, we have |η(n + 3) − γ(n + 3)| ≯ 2−n−2. On the other hand, for any k greater than the length of the initial deﬁned segment of λ, it is possible that ξ = βk. In this case, it

145 −n−2 ♦|η(n + 3) − γ(n + 3)| > 2 . This is only possible if either η or γ is not defined up to n + 3. Thus we have shown that if λ and f agree on the initial defined segment of λ, then either η or γ is not defined up to n + 3. But by axiom S2, it is inevitable that both η and γ are eventually both defined on n + 3. Thus it is inevitable that λ and f eventually disagree. But since λ is lawless, this contradicts axiom S7r.

There are two main restrictions that make Theorem 6.15 weaker than the intuitionistic result that every function is continuous. The ﬁrst restriction is that the values of the function are represented as canonical real number generators. In light of Lemma 6.11, this restriction can be lifted if we strengthen our comprehension principles somewhat:

0 Corollary 6.16. Assume MC1 + Π1ISC. Suppose that the formula F (α, γ) deﬁnes a function on real numbers in the sense of Theorem 6.15, except that the values of the function are given as (not necessarily canonical) real number generators. Then

there is no family X deﬁning sequences βi which witness a discontinuity at α, in the sense of Theorem 6.15.

Proof. By Lemma 6.11, the formula ∃γ ∈ RF (α, γ) ∧ γ0 ∈ R0 ∧ γ = γ0 also deﬁnes a function, but the values of this function are given as canonical real number generators. Hence the result follows by Theorem 6.15.

Unfortunately, this corollary is only as philosophically signiﬁcant as the motivation

0 for Π1ISC, and the philosophical motivation for this principle is rather lacking. On

the other hand, it does allow us to observe that in MC (not MC1!), if F (α, γ) provably deﬁnes a function on real number generators, then there is no provably deﬁnable discontinuity.

146 Corollary 6.17. Assume F (α, γ) provably deﬁnes a function on real numbers, i.e.:

MC ` ∀α ∈ R∃γ ∈ RF (α, γ) ∧ ∀α∀γ(F (α, γ) → F (α, γ))

∧ ∀β ∈ R(α = β ∧ F (α, γ1) ∧ F (β, γ2) → γ1 = γ2)]

Then MC does not prove that there is a real number generator α and a family X

deﬁning sequences βi which witness a discontinuity at α, i.e.:

−k MC 0 ∃α ∈ R∃n∃X∀k∀β ∈ Xk[β ∈ R ∧ |α − β| < 2 ∧

−n ∃ζ(F (β, ζ) ∧ ∀γ F (α, γ) → |γ − ζ| ≮ 2 )]

0 Proof. Since MC1 + Π1ISC extends MC, if MC did prove that there were such

0 a discontinuity, then so would MC + Π1ISC. But this would contradict Corollary 6.16.

The second restriction on Theorem 6.15 is that there be a deﬁnable family of choice sequences which witness the discontinuity. Although this is a signiﬁcant restriction, the theorem still rules out important functions that are classically perfectly well

deﬁned. For instance, the step function f : R → R deﬁned as

  0 x ≤ 0 f(x) =  1 x > 0

is a perfectly sensible function from the classical point of view. But there is no analogous function deﬁnable on real number generators. The reason for this is that

1 the discontinuity occurs at 0 and the constant sequences n are a deﬁnable family

147 of choice sequences approaching the discontinuity 0. Obviously the strategy behind this example can be generalized. There are no functions which are discontinuous at a point α which is deﬁnable by some X and which is the limit point of an open neighborhood on which the function takes a value at least ε away from its value at α. Another example of a classically unproblematic discontinuous function is the characteristic function of the rationals χQ, deﬁned as

  1 x ∈ Q χQ(x) =  0 x∈ / Q

But again, Theorem 6.15 precludes there being such a function deﬁned on real number generators. Let f be some recursive function deﬁning π, and put α(x) := f(x). Then

f(n) the family βn(x) = n is a deﬁnable family witnessing the discontinuity at α.

6.6 Decomposing the Continuum

One of the important further corollaries of the intuitionistic continuity theorem is that there is no decomposition of the continuum: if A and B are two sets such that

A∪B = R and A∩B = ∅, then one of A and B is equal to R and the other is equal to ∅. It is easy to see how to reach this conclusion from the continuity theorem: consider the characteristic function of A; it must be continuous, and since it only takes values

0 and 1, this entails that it is constant. So A is either all of R or is the empty set, and of course then B must be either the empty set or all of R (respectively). Without the full continuity theorem, we cannot replicate this line of reasoning in MC. In this section we will instead consider directly whether the temporal-potential

148 continuum can be decomposed. We will focus on the temporal-potential continuum as given by canonical real number generators—the canonical continuum, we may call it. Technically, this has the advantage that canonical real number generators have more structure than their motley cousins, the real number generators. Philosophically it also has the advantage that, as explained in §6.4, canonical real number generators keep us closer to the constructive spirit of the intuitionistic theorems that we are trying to capture in the modal/temporal framework. We will end up only with a partial result on the decomposition of the continuum. Theorem 6.24 shows that if sets A and B decompose the canonical continuum, then A and B are both topologically open; as a result, Corollary 6.27 shows that either there is no decomposition of the continuum or there must be ‘gaps’ in the continuum. First, however, we should precisify the notion of a set of real numbers. Since we will not be adding higher-order variables to our language, we will be directly concerned not with sets of real numbers but with formulas A(α) that deﬁne a set of real number generators that respect the identity of real number generators:14

∀α(A(α) → α ∈ R) ∧ ∀α∀γ(A(α) ∧ α = γ → A(γ))

Moreover, sets (or species, in intuitionistic jargon) are intuitionistically defined by some property the members of that set all have in common. As (Heyting, 1971, 37) puts it, “A species is a property which mathematical entities can be supposed to possess,” and once we have defined a species S, any mathematical object “which satisfies the condition S, is a member of the species S.” Since defining properties are

14As remarked earlier in connection with functions, to say that A ‘deﬁnes a set’ is somewhat misleading, since A need not be a formula that explicitly deﬁnes a set—it could be a new parameter added to the language, for instance.

149 presumably meant to be both mathematical in character and intrinsic to the objects in question, we require that properties be positively stable to deﬁne a set:

∀α(A(α) → A(α))

This prevents accidental or empirical definitions of a set such as: let A hold of a real number generator α if some specified lawless sequence λ is defined up to an odd argument and let A not hold of α if λ is defined up to an even argument. If A is a formula that satisfies these two constraints, namely respecting identity and positive stability, then we can say that A defines a set of real numbers. Something more still needs to be said about when two sets decompose the continuum. The most stringent requirement would be as follows:

∀α ∈ R((A(α) ∨ B(α)) ∧ ¬(A(α) ∧ B(α)))

This version requires either A or B to already and always hold of every real number generator. On the other hand, we might allow that it is not yet determinate whether a given real number generator α is in A or B, provided that α is inevitably in A or is inevitably in B:

∀α ∈ R((IA(α) ∨ IB(α)) ∧ ¬(A(α) ∧ B(α)))

Under either of these ﬁrst two notions of a decomposition, it is easy to see that there is no decomposition of the continuum.

150 Proposition 6.18. If A and B deﬁne sets of real number generators and ∀α ∈ R((IA(α) ∨ IB(α)) ∧ ¬(A(α) ∧ B(α))), then either ∀α ∈ RIAα ∨ ∀α ∈ RIBα.

Proof. Suppose that we have Aα and Bβ. Given some λ ∈ L, which is not currently

deﬁned on 100, deﬁne γ so that if λ(100) = 0, then ∀xγ(x) = α(x), and if λ(100) > 0, then ∀xγ(x) = β(x). Then ♦γ = α ∧ ♦γ = β, from which it follows that ♦Aγ ∧ ♦Bγ. But this contradicts the fact that IAγ ∨ IBγ.

These ﬁrst two ways of understanding what it means to decompose the continuum are fairly strong, and as a result the fact that the continuum cannot be so decomposed is not very interesting. But there is a weaker way of understanding a decomposition of the continuum. This alternative deﬁnition of a decomposition allows more indeterminacy: not only is it presently indeterminate whether α is in A or B, it is furthermore not yet determinate whether α eventually ends up in A or eventually ends up in B, provided only that α eventually ends up in one or the other of A and B:

∀α ∈ R(I(A(α) ∨ B(α)) ∧ ¬(A(α) ∧ B(α)))

Philosophically, it is perhaps not entirely obvious which of these conditions has the best claim to capturing (in our modal framework, of course) the intuitionistic conception of a decomposition of the continuum. However, the following example suggests that the first two conditions are too strong: For a fixed lawless sequence λ, let A hold of all real number generators if λ(33) = 0 and let B hold of all real number generators if λ(33) > 0. This seems to be a perfectly reasonable way to decompose the continuum, but A and B so defined do not satisfy either of the first two conditions above if λ(33) = f. Nevertheless, we can dodge the deeper philosophical question of

151 what the ‘right’ definition of a decomposition of the continuum is by working with the third, weakest condition and showing that it suffices for technical purposes. In what follows, a decomposition of the continuum should always be understood in the third and weakest sense. In what follows, we will show that if A and B decompose the continuum, then both A and B are topologically open. Since “A and B decompose the continuum” is the antecedent of this theorem, working with the weakest definition of “decompose the continuum” gives the strongest result. That result, however, will only be obtained under an additional assumption, which can be called ‘local dependence’.

Deﬁnition 6.19 (Local Dependence). Local Dependence is the claim that for any

choice sequences α1, ..., αk there is a lawless sequence λ such that:

0 0 0 0 ∀x1∀x1...∀xk∀xk[♦(α1(x1) = x1 ∧ ... ∧ αk(xk) = xk) →

0 0 0 0 0 ∀y∀y (♦λ(y) = y → ♦(α1(x1) = x1 ∧ ... ∧ αk(xk) = xk ∧ λ(y) = y )]

When any choice sequences satisfy this condition we can say that they are independent of each other.

The motivating idea is that, while choice sequences may in general depend on the values of some lawless sequences, they can only depend on some speciﬁed lawless functions. The universe of lawless sequences is too heterogenous, and has too little structure, for any choice sequence to depend on what happens in the entire universe of lawless sequences. This hypothesis is called local dependence because it formalizes the intuition that choice sequences cannot depend globally on the universe of lawless sequences. We can note also that the k-variable version of Local Dependence deﬁned

152 above actually reduces to the 1-variable version by considering the choice sequence

formed as the k-tuple of the αi’s , namely α(x) = hα1(x), ..., αk(x)i.

Lemma 6.20. Assume Local Dependence, and suppose A and B deﬁne sets that decompose the canonical continuum, i.e.:

0 ∀α(A(α) → α ∈ R ) ∧ ∀α∀γ(A(α) ∧ α = γ → A(γ)) ∧ ∀α(A(α) → A(α))

(and likewise for B) and:

0 ∀α ∈ R (I(Aα ∨ Bα) ∧ ¬(Aα ∧ Bα))

Then if Aα and Bβ, it follows that Iα#β.

Nx Proof. Suppose for reductio that Aα, Bβ, and ¬Iα#β. Abbreviate α(x) = 2x and

Mx β(x) = 2x (assuming they are deﬁned). Then by Lemma 6.9 (1), there is a future

path along which Nx − Mx ≤ 2. Moreover, (¬α = β), so along that same path it is 1 true that ∀x∃y > x♦ |α(y) − β(y)| > 2y−1 . Deﬁne γ by:

 Nx+Mx  d 2 e  x if α(x + 1) > α(x) or β(x + 1) > β(x)  2  ∀x : γ(x) 6= f → γ(x) :=    b Nx+Mx c  2  2x otherwise

To see that R0γ, it suﬃces to check that γ(x + 1) − γ(x) ≤ 2−x−1. This is easy to verify by considering the eight possible combinations of values for α(x + 1) and

0 0 1 γ so that γ (x) = γ(x) as long as |α(y) − β(y)| ≤ 2y−1 for all y ≤ x + 2; and if x0 is 0 the ﬁrst value where this fails, then ∀y ≥ x0, put γ (y) = α(y) if λ(y) = 1 and put γ0(y) = β(y) if λ(y) 6= 1. Clearly by construction we have that R0γ0. Recall that along the path on which α and β approach each other arbitrarily closely, it is always possible for α and β to diverge. And since λ is independent of α and β, it follows that whenever α and β possibly diverge, it is possible for γ0 to

0 0 0 follow α and possible for γ to follow β. So in other words, ¬I¬(♦γ = α ∧ ♦γ = β). 0 0 0 0 0 0 0 But [(Aγ ∨ Bγ ) → (¬♦γ = α ∨ ¬♦γ = β)]. Hence I(Aγ ∨ Bγ ) → I(¬♦γ = 0 0 0 α ∨ ¬♦γ = β)], from which we get ¬I(Aγ ∨ Bγ ), a contradiction.

Lemma 6.21. Assume Local Dependence, and suppose A and B decompose the canonical continuum (as in Lemma 6.20), R0α, and Aα. Then ∃x∀y > x[IA(α(y) − 2−y)∧ IA(α(y)) ∧ IA(α(y) + 2−y)].

Proof. Suppose not, so that ∀x∃y > x[♦B(α(y) − 2−y)∨♦B(α(y))∨♦B(α(y) + 2−y)]. Let λ and η be lawless with λ independent of α, and let f be a function that agrees with λ on its initial deﬁned segment. Deﬁne a sequence γ such that if λ and f agree up to x, then γ(x) = α(x), and if k is the least argument on which λ and f disagree, then γ(k) = α(k) − 2−k if η(k) = 0, and γ(k) = α(k) if η(k) = 1, and γ(k) = α(k) + 2−k if η(k) > 1. For all y > k, γ(y) = γ(k). Clearly R0γ. Now as long as λ and f agree, it is possible that γ could go on to agree with some

−k α(k) ± 2 which is possibly in B, and thus ♦Bγ. So as long as λ and f agree, ¬Aγ.

154 On the other hand, γ and α will only be apart if λ and f disagree. So, as long as λ and f agree, it is not inevitable that γ#α, and hence, by Lemma 6.20, ¬Bγ. Thus, as long as λ and f agree, ¬(Aγ ∨ Bγ). Since ¬I∃xλ(x) 6= f(x), it follows that ¬I(Aγ ∨ Bγ), contradicting the claim that A and B decompose the canonical continuum.

With this lemma in hand we can now oﬀer a reﬁned and stronger version of Lemma 6.20:

Lemma 6.22. Assume Local Dependence, and suppose A and B decompose the canonical continuum (as in Lemma 6.20), and that R0α and R0β. If Aα and Bβ, then

−y ∃y|α − β| > 2 , i.e., α#β.

Proof. Suppose Aα and Bβ. Then by Lemma 6.21 there is an x1 such that ∀y >

−y −y x1[IA(α(y) − 2 ) ∧ IA(α(y)) ∧ IA(α(y) + 2 )], and likewise there is an x2 such

−y −y that ∀y > x2[IB(β(y) − 2 ) ∧ IB(β(y)) ∧ IB(β(y) + 2 )]. Let x = max(x1, x2).

−y Assume for reductio that ∀y♦|α − β| ≯ 2 . So it is possible that |α(x) − β(x)| ≤ 1 N−1 2x−1 . For concreteness (and without loss of generality), say α(x) = 2x and β(x) = N+1 N−1 1 N+1 1 2x . But then we would have that IA( 2x + 2x ) and IB( 2x − 2x ), which is impossible.

Note that Lemma 6.21 implies that every point in A is a limit point of other members of A. Thus a point in A must either be internal, or else a limit point of B.

Lemma 6.23. Assume Local Dependence and that A and B decompose the canonical continuum. Then there is no neighborhood in which A and B are both dense in the

rationals. More exactly, the following is impossible: There are rationals p1, p2 such

155 that for every q1, q2, if p1 < q1 < q2 < p2, then there exist a, b ∈ Q such that

q1 < a < q2 ∧ IAa and q1 < b < q2 ∧ IBb.

r Proof. Suppose for reductio that there are such p1, p2. For λ ∈ L and p1 < 2x < p2, r define a sequence δ so that δ(x) = 2x , and for y > x, δ is a projection of λ defined so 1 that if λ(y) = 0, then δ(y) = δ(y − 1) − 2y ; and if λ(y) = 1, then δ(y) = δ(y − 1); 1 and otherwise δ(y) = δ(y − 1) + 2y . Now for every y > x, there exist a, b such that IAa ∧ IBb and δ(y) − 2−y < a, b < δ(y) + 2y. Since δ is defined by projection from a lawless sequence, it can approach a arbitrarily closely, and likewise for b. Thus necessarily there are a, b such that Ia ∈ A and Ib ∈ B, but ¬δ#a and ¬δ#b. Since

this would be impossible if Aδ or Bδ, it follows that ¬(Aδ ∨ Bδ), which is obviously absurd.

Putting these results together, we can show that if A and B decompose the continuum, then A and B must be open sets in the sense that for every element α of A there is a rational interval around α such that every canonical real number generator in that interval is also in A (and likewise for B).

Theorem 6.24. Assume Local Dependence and that A and B decompose the canon-

0 ical continuum. Then if Aα, there are p1, p2 such that p1 < α < p2 and for all γ ∈ R ,

if p1 < γ < p2, then IAγ.

Proof. The proof will proceed in two parts. Before proceeding to the details, I will provide a rough sketch. We know from Lemma 6.21 that every point in A must either be interior or a limit point of both A and B. So the ﬁrst step will be to show that if α is a limit point of both A and B, then there must be a neighborhood around α such that A and B are both dense in Q. But since this is ruled out by Lemma 6.23, we

156 infer that α is an interior point of A ∩ Q. The second step will then be to argue that if A contains all the rationals in a neighborhood, then it contains all the canonical real number generators in that neighborhood. Let us proceed to the details. First, assume for reductio that for all p there is a β such that |α − β| < p and IBβ, but that A and B are not codense in the rationals around α. Then:

¬∃p1, p2[p1 < α < p2 ∧ ∀q1, q2[p1 ≤ q1 < q2 ≤ p2 → ∃a, b(q1 < a, b < q2 ∧ IAa ∧ IBb)]]

Thus for all p1, p2 around α, there are q1, q2 such that the open rational interval between q1 and q2 is entirely contained in the eventual A’s or the eventual B’s.

Moreover we can assume that for every p1, p2 around α there must be some interval

(q1, q2) contained in B, since otherwise the interval (p1, p2) itself would be a rational neighborhood around α that is entirely contained in A, which is our desired conclusion. Now let λ ∈ L be independent of α, and let f be a function that agrees with λ on the initial defined segment. Define η to agree with α as long as λ and f agree; if k is the least argument on which λ and f disagree, then define η to be a projection of λ as follows: if λ(y) > 2, then η(z) = η(y) for all z ≥ y; otherwise, if λ(y) = 0, then

1 η(y) = η(y−1)− 2y ; and if λ(y) = 1, then η(y) = η(y−1); and ﬁnally if λ(y) = 2, then 1 η(y) = η(y − 1) + 2y . Now, as long as λ and f agree, ¬η#α, so ¬Bη. Furthermore, as long as λ and f agree, it is possible for them to disagree, and it is then also further

possible for η to end up arbitrarily close to a rational in some interval (q1, q2) ⊆ B, from which it follows ¬Aη. Thus as long as λ and f agree, ¬(Aη ∨ Bη). And since λ and f do not inevitably disagree, it follows that ¬I(Aη ∨ Bη), which is absurd. This

completes the reductio, so we have shown that there are p1, p2 such that p1 < α < p2

and for all q in the rational interval (p1, p2), IAq.

157 Figure 6.1: A simple decomposition

α A B β ··· o ···

Now for the second part of the proof we want to show that for all γ ∈ R0 such

0 that p1 < γ < p2, IAγ. Take an arbitrary such γ. Let k be such that ∀η ∈ R (η ∈

γ(k) → p1 < η < p2). If ♦Bγ, then by Lemma 6.21, we would have ♦∃xIBγ(x). But this is ruled out the by ﬁrst half the proof, since γ(x) will be rational.

While this theorem falls somewhat short of the desired result that there is no non-trivial decomposition of the continuum, it is nevertheless still an interesting anti- classical result. Moreover, it has a few interesting corollaries, and it suggests a plausibility argument for the full claim that there is no decomposition of the continuum. I will close this section by ﬁrst outlining that plausibility argument and then presenting a few corollaries of Theorem 6.24. Suppose that A and B non-trivially decompose the continuum, with Aα and Bβ. Since A and B must both be open, the simplest case to picture is if A is to the left of B, as depicted in Figure 6.1. Now if this is the decomposition, then for every

N N+1 x there must inevitably be some 2x ∈ A and 2x ∈ B. If there were a sequence N+1 γ that always took the value 2x , it would always be within B; but for any x, a sequence that agreed with γ up to x but subsequently always went leftward would be in A. A projection π of a lawless sequence could follow this path of γ indeﬁnitely, but also could at any point go leftward. So as long as π is following the path of γ, π would be neither in A nor in B. There are some diﬃculties in making this argument

158 Figure 6.2: A general decomposition

A ··· B A B ··· A ··· B ··· ) ( o o ) ( ) ( ···

rigorous, but it provides a reasonable plausibility argument against the possibility of this simple decomposition, where A is entirely to the left of B. Consider now the more general and slightly more complicated case where A and B decompose the continuum and are interspersed between each other, as pictured in Figure 6.2. We can consider two cases. First, suppose there are α ∈ A and β ∈ B that are in neighborhoods of A and B adjacent to each other. That is, suppose:

I∃α∃β[Aα ∧ Bβ ∧ ∀γ[α < γ < β →

0 0 0 0 0 0 (Aγ → ∀γ (α < γ < γ → IAγ )) ∧ (Bγ → ∀γ (γ < γ < β → IBγ ))]]

If this is the case, then we can argue similarly to the simple case by considering a γ that is always within striking distance of the border between these neighborhoods of A and B. On the other hand, if there are no such adjacent neighborhoods, then for any

γ to the right of a neighborhood A0 ⊂ A there is an η < γ with ♦Bη. Then we

would expect that a projection of a lawless sequence near the border of A0 could

always move to the right of A0 and end up within a neighborhood B0 ⊂ B around some η ∈ B. Again, there are diﬃculties in making this argument rigorous, but it does provide a reasonable plausibility argument against the more general possibility of a decomposition. To close this section I will now enumerate a few corollaries of Theorem 6.24. The ﬁrst one comes from extracting the available rigorous proof from

159 the plausibility argument against the simple decomposition.

Corollary 6.25. Assume Local Dependence and ΞISC and that A and B are in Ξ and decompose the canonical continuum. Then there are no α ∈ A and β ∈ B which are adjacent in the following sense:

0 0 0 ∀γ[α < γ < β → (Aγ → ∀γ (α < γ < γ → IAγ ))∧

0 0 0 (Bγ → ∀γ (γ < γ < β → IBγ ))]

Proof. If there were, then we know that for x suﬃciently large (which can be found

n n+1 from α and β), for all y > x there is inevitably an n such that 2y ∈ A and 2y ∈ B. n+1 Let γ(y) = 2y . Now for λ ∈ L independent of γ, let f be a function that agrees with the deﬁned portion of λ. Deﬁne η so that as long as λ and f agree η follows

1 γ, and if λ and f ﬁrst disagree on k then for all z > k put η(z) = η(z − 1) − 2z . Thus, as long as λ and f agree, η agrees with a choice sequence that is in B, but if λ and f ever disagree then η will go into A. Thus as long as λ and f do not disagree, ¬(Aη ∨ Bη). Since λ and f do not inevitably disagree, we have ¬I(Aη ∨ Bη), which is impossible.

The second corollary similarly establishes a limitation on what a decomposition of the continuum can look like.

Corollary 6.26. Assume Local Dependence, that A and B decompose the canonical continuum, and that for some α ∈ A and β ∈ B, the regions of A and B around α and β are adjacent in the sense of the previous corollary. Then necessarily there is no function f such that if ∀xγ(x) = f(x), then γ ∈ R0 and ∀η[(α < η < γ → Aη) ∧ (γ < η < β → Bη)].

160 Proof. If there were such an f, then we could find a λ ∈ L that agrees with f on its defined segment, and then define γ to follow λ as long as λ and f agree, and to go left if λ and f first disagree on an odd argument and to go right if λ and f first

disagree on an even argument. Then as long as λ and f agree, ♦Aγ ∧ ♦Bγ, hence ¬(Aγ ∨Bγ). And since it is not inevitable that λ and f disagree, we would then have ¬I(Aγ ∨ Bγ), which is absurd.

The next two corollaries are slightly more interesting; they each show that Theo- rem 6.24 has further anti-classical consequences. The ﬁrst one will be that if there is a non-trivial decomposition of the continuum, then there exist sets which are bounded above but which do not have any least upper bound. In other words, the temporal- potential continuum does not have the least upper bound property. By an upper bound of a set A I mean a γ ∈ R (or in R0 as determined by context) such that

∀α ∈ Aα ≤ γ; a least upper bound an upper bound γ such that, necessarily for all η ∈ R, if η < γ, then η is not an upper bound on A.

Corollary 6.27. Assume Local Dependence, and that there are sets A and B which decompose the canonical continuum, with Aα and Bβ. Then there is a set A0 which necessarily has no least upper bound.

Proof. If A is bounded above, then we can take A0 = A. If not, then consider A0γ :↔ γ < β ∧ Aγ. Obviously A0 has an upper bound. Moreover, it is easy to see that A0 and B will also decompose the continuum. So if γ is an upper bound on A0, then Bγ, and hence by Theorem 6.24 there is a p < γ which is also an upper bound on A0. So no upper bound can be least.

This corollary gives us a dilemma of two anti-classical results: either it is impossible to decompose the continuum, or the least upper bound principle fails. The

161 possibility of the least upper bound principle failing is not necessarily surprising in our setting, since the principle is not constructively provable. In fact, Weak Church’s Thesis entails that there exist bounded monotone sequences of rationals which have no least upper bound.15 This fact does not make Corollary 6.27 prolix, however, since Weak Church’s Thesis is not a theorem of MC. Whether Weak Church’s Thesis is consistent with MC will depend on the precise formulation of the thesis. Any sensible formulation of Weak Church’s Thesis will be refuted in MC1, however. The ﬁnal application of Theorem 6.24 takes us back to the question of whether there are discontinuous functions. While Theorem 6.24 does not quite allow us to prove that there are no discontinuous functions, it does allow us to prove a slightly weaker result. Recall from above that a formula F deﬁnes a function (via canonical real number generators) when:16

0 0 ∀α ∈ R ∃γ ∈ R F (α, γ) ∧ ∀α∀γ(F (α, γ) → F (α, γ))∧

0 ∀α, β ∈ R∀γ1, γ2 ∈ R (α = β ∧ F (α, γ1) ∧ F (β, γ2) → γ1 = γ2)]

Adapting the standard epsilon-delta criterion, we will say that a function F is continuous when:

0 0 ∀α ∈ R ∀p > 0∃q > 0∀β ∈ R [|α − β| < q →

∀γ1∀γ2(F (α, γ1) ∧ F (β, γ2) → |γ1 − γ2| < p)]

15Weak Church’s Thesis is an axiom asserting that every inﬁnite sequence is not not recursive. See (Beeson, 1982, 51). 16Recall that the term ‘deﬁnes a function’ is a little misleading, as explained above.

162 Accordingly, F is discontinuous if:17

∃α ∈ R0∃p > 0∀q > 0∃β ∈ R0[|α − β| < q∧

∀γ1∀γ2(F (α, γ1) ∧ F (β, γ2) → |γ1 − γ2| ≮ p)]

Let us say that F is sharply discontinuous at α if:

0 ∃p > 0∃p1∃p2[p1 < α < p2 ∧ ∀β ∈ R (p1 < β < p2 →

∀γ1∀γ2(F (α, γ1) ∧ F (β, γ2) → |γ1 − γ2| < p ∨ |γ1 − γ2| > p)]∧

0 ∀q > 0∃β ∈ R [|α − β| < q ∧ ∀γ1∀γ2(F (α, γ1) ∧ F (β, γ2) → |γ1 − γ2| > p)]

Informally, the idea is that, in some neighborhood around α, the function F either takes a value that is quite close to F (α) or takes a value that is distinctly further away. There is a sharp dividing line, as it were, and the values that witness the discontinuity all fall outside that line.

Corollary 6.28. Assume Local Dependence. If F deﬁnes a function on canonical real number generators, then F has no sharp discontinuities.

Proof. If there were such a sharp discontinuity in the interval (p1, p2) around α, then that interval could be decomposed into the set A of arguments that F mapped to values within p of F (α), and the set B of arguments that F mapped to values at least p greater or less than F (α). Then since F is sharply discontinuous at α, α would not be interior in A. 17Note that discontinuity is not the negation of continuity. The negation of continuity is possible discontinuity.

163 Now it is also easy to see that if A and B decompose an interval, then there are A0 and B0 that decompose the continuum. Given α, β in the interval such that Aα and Bβ, (assuming without loss of generality that α < β), deﬁne A0γ :↔ γ < α ∨ (α < γ < β ∧ Aγ) and B0γ ↔ γ > β ∨ (α < γ < β ∧ Bγ).

6.7 Conclusion

In this chapter I introduced the modal theory of choice sequences MC. It is obtained by starting with the modal theory of lawless sequences MCLS and forming new choice sequences by projection. A Kripke model assured us that MC is consistent, and it proves simple intuitionistic principles such as the Bar Theorem and Kripke’s Schema. The really interesting work, however, comes when we deﬁne the notion of a real number generator and the usual arithmetic operations on them. This allowed MC to capture modal analogues of several results characteristic of intuitionistic analysis. The behavior of lawless sequences allowed us to mimic a weak Brouwerian counterexample, showing that it is unprovable that every real number generator is determinately rational or determinately irrational. The indeterminacy inherent in the universe of choice sequences also allowed us to prove that the natural order < on real number generators is not linear and that there exists a monotone bounded sequence of rationals which is not necessarily Cauchy (and hence not a real number generator). Two of the hallmarks of intuitionistic analysis are the continuity of all functions on the real numbers and the indecomposability of the continuum. On these counts, our results are only partial. In §5, we saw that MC1 proves that no function has a deﬁnable discontinuity. (MC1, recall, results from MC by allowing second-order parameters to occur in the sequence comprehension scheme). In §6, we considered what a decomposition

164 of the continuum would have to look like and concluded that, under the assumption called Local Dependence, if A and B decomposed the canonical temporal-potential continuum, then every point in A is interior in A and every point in B is interior in B. While weaker than an indecomposability result, this is still of significant interest. In particular, it entails (still under the assumption of Local Dependence) that either there is no decomposition of the canonical temporal-potential continuum or the canonical temporal-potential continuum lacks the least upper bound property. It also entails a slightly stronger continuity result, namely that (under the assumption of Local Dependence) no function is sharply discontinuous in the sense defined above. Clearly, much work remains in studying MC. Obvious questions include whether full continuity and indecomposability results can be proved. I conjecture they can, but suspect that they will require new techniques beyond those used here. Further mathematical work can involve investigating further areas of intuitionistic mathematics such as differential and integral calculus, topology, and measure theory.

As with MCLS there remain also many interesting metamathematical questions about the proof-theoretic and consistency strength of MC. While Kripke models nicely capture the intuitive picture of branching future times, other tools have proved more fruitful in the interpretation of intuitionistic theories, and it would be natural to try to apply them here. In particular, is it possible to develop a realizability semantics for MC, in the vein of Rin and Walsh(2016)? The model in §1 demonstrated the consistency of MC, and can be easily extended to MC1. It would be non-trivial, however, to extend the method used there to construct a model for MC +ΞISC for classes Ξ that allow choice sequence quantiﬁers to occur in the comprehension formula A. Thus the consistency question for those theories remains open.

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172 Appendix A

Frame Conditions For Intuitionistic S4.2

In this appendix I show that the frame conditions appealed to in Theorem 2.8 are the right ones for intuitionistic S4.2. That reﬂexivity and transitivity are the condition for S4 is well known, see, for instance, (Simpson, 1994, 56). The following result shows that adding I-convergence gives the right condition for S4.2. (This theorem is a special instance of the more general Theorem 2.1 in Plotkin and Stirling(1986)).

Theorem A.1. Let F = hW, ≤,R,Di be an intuitionistic modal frame satisfying

(F1) and (F2) and such that R is reﬂexive. Then ♦φ → ♦φ is valid in F iﬀ F is I-convergent.

Proof. (⇒) Assume that F is not I-convergent, so that there are some w, v, u such that wRv and wRu, but for no s, t do u ≤ sRt and vRt. We want to show that for some choice of valuation function V , the resulting model M based on frame F

does not validate ♦φ → ♦φ. It will suffice to define V so that, for some atom P , M, u P and M, v 1 ♦P , since this will make M, w ♦P but M, w 1 ♦P . For this we can let V make P true in only those worlds t such that for some s, u ≤ sRt. (As is easy to check, given the reflexivity of R, this also guarantees that M meets the

the condition on intuitionistic models that if M, u P and u ≤ s then M, s P ). This will ensure that M, u P ; it remains to show that M, v 1 ♦P . Given the

173 definition of V , it will suffice to show that if u ≤ sRt, then not vRt. But this is simply the assumption that F is not I-convergent. (⇐) Assume F is I-convergent; since ≤ is reflexive, it follows that R is convergent.

Let M, w 1 ♦φ. So there exist t, u such that w ≤ tRu and M, u 1 ♦φ. We want to show M, w 1 ♦φ, for which it suﬃces to show that for all v where wRv, M, v 1 φ. Fix v. By (F1), there is a s ≥ v such that tRs. By convergence of R, there is an r such that uRr and sRr. But since M, u 1 ♦φ, it follows that M, r 1 φ. So v ≤ sRr and M, r 1 φ, and hence M, v 1 φ.

174 Appendix B

The Need for Axiom S8

Other than S8, the axioms of MCLS all have to do with the behavior of a single sequence in isolation. The concept of lawlessness, however, can be characterized as having to do not just with how a single lawless sequence behaves in isolation, but also as how a group of lawless sequences behave collectively. Not only should there be no laws that govern how a single sequence grows in time, there should also be no laws that govern the interrelations of lawless sequences. And without S8, this collective lawlessness may be absent.

To see this, we can observe that if we drop S8, there is a countermodel in MCLS to the following two-variable version of the open data principle:

[A(α, β) ∧ ¬∀xα(x) = β(x) → ∃x∀γ(α(x) = γ(x) → A(γ, β)]

Our countermodel will be constructed in a manner similar to the Kripke model from

§5.2.1. Recall that the underlying frame of this model is N

model for MCLS without S8 can be deﬁned as in §5.1. We will proceed in two steps. For the ﬁrst step, assign to the root world the singleton of the empty sequence.

And if we have assigned a set Sn to the world n (n will be some sequence number),

175 then we assign to the world n_hmi the set:

_ {s hmi : s ∈ Sn} ∪ {hi}

Intuitively, whenever we move from one world to the next, we define the new set of sequences by extending each sequence in our original set by one value and then adding the empty sequence to the new set as well. Let β be the sequence whose first element is (first) defined at h1i, and let α be the sequence whose first element is defined at h1, 1i. Recall that at each world n_hmi, we extend each choice sequence that was defined at n by adding concatenating m to its initial defined segment. Now for the second step. To the model defined in this way, add a new sequence γ as follows: let γ be introduced at the same node as α. We extend γ at new worlds in a

diﬀerent way than we extend α and β. Write γn for the segment of γ deﬁned at node

_ n. For m > 1, γ will be extended in the same way as α and β, i.e. γn_hmi = γn hmi.

_ _ But put γn_h0i = γn h1i and γn_h1i = γn h0i. It will help to have an illustration. A partial diagram of the sequences α, β, and γ in this model follows. To reduce clutter, I will omit h and i, =, and commas. Thus,

β112 abbreviates that the initial (non-f) segment of β is h1, 1, 2i. (The top node in the portion of the model illustrated here is the element h1i of N 1 → α(x) = β(x)). Then at the node h1, 1i in this model, A(α, β) is true, as is ¬∀xα(x) = β(x). Moreover, at this node we have ∀x(α(x) = γ(x)). But obviously A(γ, β) is false at this node. Hence the

multivariable version of the open data principle fails in MCLS. The reason for this failure is that the sort of lawlessness incorporated in the axiom

176 Figure B.1: Sketch of a Kripke model for MCLS without S8.

β = 1

β11, α1, γ1

β110, α10, γ11 β111, α11, γ10 β112, α12, γ12

. . β1100, α100, γ111 β1101, α101, γ110 β1110, α110, γ101 β1111, α111, γ100 .

......

S5 ensures that there is no law that governs how the next value of a given sequence is to be chosen. But it is left open that there is a global law governing how the values chosen for one series relate to the values chosen other series—and it is exactly this sort of interrelation that we exploited in our countermodel to the multiple-variable open data principle. To avoid this remaining sort of lawlikeness, we need to add a few axioms. Intu- itively, we want to say that no choice sequences share any necessary relations, so that the values of one choice sequence are not in any way bound to the values of any other choice sequences. Thus we are led to S8.

177 Appendix C

More Stability Properties

Recall that a formula A is positively stable if it satisﬁes:

~ ~ ~ ∀~α∀X∀~x[A(~α, X, ~x) → A(~α, X, ~x)]

And a formula is negatively stable if it satisﬁes:

~ ~ ~ ∀~α∀X∀~x[¬A(~α, X, ~x) → ¬A(~α, X, ~x)]

Throughout this appendix we can work in either MCLS or MC.

Proposition C.1. The following formulas are positively stable, where t may be any term:

1. t1 = t2, when at most one of t1 nor t2 contains a choice parameter, and either

both t1 and t2 contain f or neither does

2. t1 < t2

3. t ∈ X

4. A

5. A ∧ B and A ∨ B, when A and B are positively stable

178 6. ∃xA, ∃XA, and ∃αA when A is positively stable

7. ∀xA, ∀XA, and ∀αA when A is positively stable

8. A† and A?.

Proof. 1. If neither t1 nor t2 contains a choice parameter or f, then t1 = t2 is a

theorem of MCLS (because it contains Robinson Arithmetic), and hence so is t1 =

t2. It is easy to show by an induction on the number of function symbols and choice

variables that if t is any term containing f then t = f is a theorem of MCLS. So if

both t1 and t2 contain f, then we have t1 = t2.

If t1 contains a choice parameter but t2 does not, and neither contain f, then S3 entails the stability of t1 = t2.

2. Since f ≮ f (by f4), t1 < t2 can only be true if t1 and t2 both denote numbers.

Since ti = xi is positively stable by (1) and x1 < x2 will be necessary whenever true, t1 < t2 must be positively stable.

3. t ∈ X can be true only if t denotes a number rather than f. By (1), t = x is stable, so by A9, t ∈ X is stable.

4. Since our modal logic is S4, we have A → A. So A is positively stable. 5. From A → A and B → B we can easily derive A∧B → (A∧B), so A∧B is positively stable if A and B are.

Similarly, from A → A and B → B we obviously have A → (A ∨ B) and B → (A ∨ B); so if we assume A ∨ B, arguing by cases we can infer (A ∨ B). So we have A ∨ B → (A ∨ B), so A ∨ B is positively stable if A and B are.

179 6. If A is positively stable, we have ∀x(A → A). By standard quantiﬁcational logic we have ∃xA → ∃xA, from which we can derive ∃xA → ∃xA. Likewise for ∃X and ∃α.

7. Again, if A is positively stable, we have ∀x(A → A). By standard quan- tiﬁcational logic we have ∀xA → ∀xA, from which we can derive ∀xA → ∀xA. Likewise for ∀X and ∀α. 8. This follows from claim 4 and Lemma 5.4.

Proposition C.2. The following formulas are negatively stable:

1. t1 = t2, when neither t1 nor t2 contains any choice parameters

2. t1 < t2 , when neither t1 nor t2 contains any choice parameters

3. t ∈ X, when t does not contain any choice parameters

4. ♦A

5. A ∧ B and A ∨ B, when A and B are negatively stable

6. ∃xA, ∃XA, and ∃αA when A is negatively stable

7. ∀xA, ∀XA, and ∀αA when A is negatively stable.

Proof. 1. If ti contains f, then ti = f is a theorem of MCLS, and hence so is ti = f.

If ti does not contain f, then for some n, ti = n and thus ti = n will be theorems

of MCLS. So when ti does not contain any choice parameters, its referent will never

change. So t1 6= t2 → t1 6= t2.

2. As noted in (1), the referent of t1 and t2 will never change, so t1 < t2 → t1 <

t2.

180 3. Again, the referent of t will never change. If t denotes a number then A9

secures the claim. If t denotes f, then f8 secures the claim. 4. If we have ¬♦A, then we get ¬A, from which we can infer ¬A, whence ¬♦A. So we have ¬♦A → ¬♦A. 5. If we assume ¬(A ∧ B), then either ¬A or ¬B. Since A and B are negatively

stable we have ¬A → ¬(A ∧ B) and ¬B → ¬(A ∧ B). Thus arguing by cases, we get ¬(A ∧ B) → ¬(A ∧ B). Similarly, if we assume ¬(A ∨ B), then we have ¬A and ¬B, from which we can

get ¬A and ¬B by the negative stability of A and B. From this we can infer ¬(A ∨ B). Thus we have ¬(A ∨ B) → ¬(A ∨ B). 6. If A is negatively stable, we have ∀x(¬A → ¬A). By standard quantiﬁcational logic we have ∀x¬A → ∀x¬A, from which we can derive ¬∃xA → ¬∃xA. Likewise for ∃X and ∃α.

7. Again, if A is negatively stable, we have ∀x(¬A → ¬A). By standard quantiﬁcational logic we have ∃x¬A → ∃x¬A, from which we can derive ¬∀xA → ¬∀xA. Likewise for ∀X and ∀α.

Also of interest are the sentences that are eventually stable; they change their minds only ﬁnitely many times, eventually settling on either truth or falsity. It is easy to see that not every sentence is eventually stable. For instance, consider the formula that is true just in case the maximal initial segment of α is of odd length. As α becomes deﬁned on new values, this formula will alternate between truth and falsity forever. We can, however, characterize the class of eventually stable formulas.

1 1 We can recursively deﬁne ± (A) := ♦(A ∧ ♦¬A) and ∓ (A) := ♦(¬A ∧ ♦A) (n+1) n (n+1) n n and ± (A) := ♦(A ∧ ∓ (A)) and ∓ (A) := ♦(¬A ∧ ± (A)). Put ﬂip (A) :=

181 ±n(A) ∨ ∓n(A). Now we can say that a formula A is eventually stable if for some n, flipn(A) is not satisfiable. Note that eventual stability is not the same thing as being pre-determined, that is, even if φ is eventually stable, whether A will ‘eventually’ settle on being true or false may remain open indefinitely. More precisely, we can sometimes prove that A

is eventually stable, even if we can refute A ∨ ¬A. For instance, ∃xα(x) = 0 is eventually stable, because if it is ever made true it can never ﬂip back to being false. On the other hand, as long as ∃xα(x) = 0 is false, we cannot say whether it will ever become true or not.

Proposition C.3. If A is positively stable or negatively stable, then A is eventually stable.

Proposition C.4. The following formulas are eventually stable:

1. t ∈ X

2. t1 < t2

3. t1 = t2

4. A

5. ♦A

6. ¬A, whenever A is eventually stable

7. A ∨ B, A ∧ B, and A → B, whenever A and B are eventually stable.

Proof. 1. This follows from Proposition C.1 (3). 2. This follows from Proposition C.1 (2).

182 3. If neither t1 nor t2 contains a choice parameter, this follows from Proposition

C.2 (1). If at most one of t1 or t2 contains a choice parameter, and either both or

neither of t1 and t2 contain f, this follows from Proposition C.1 (1).

So ﬁrst consider the case where t1 includes α but not f, and t2 contains f and may or may not contain a choice parameter. Then t1 = y is positively stable. But, as noted in the proof of Proposition C.2 (1), t2 = f; so it is not possible that t1 6= t2 ∧ ♦(t1 = t2).

Now consider the possibility that t1 and t2 both contain choice parameters, and neither contain f. Both t1 = y and t2 = x will be positively stable, so it is not possible that t1 = t2 ∧ ♦(t1 6= t2) 4. This follows from Proposition C.1 (4). 5. This follows from Proposition C.2 (4). 6. If A is eventually stable, then for some n, flipn(A) is not satisfiable. Note that flipn(A) is:

♦(A ∧ ♦(¬A ∧ ♦(A ∧ ...)) ∨ ♦(¬A ∧ ♦(A ∧ ♦(¬A ∧ ...))

By rearranging the main disjuncts replacing A with ¬¬A, we can see that this is equivalent to:

♦(¬A ∧ ♦(¬¬A ∧ ♦(¬A ∧ ...)) ∨ ♦(¬¬A ∧ ♦(¬A ∧ ♦(¬¬A ∧ ...))

And this latter formula is ﬂipn(¬A). So ¬A is eventually stable. 7. For A ∧ B to change truth value from one world to another, at least one of A and B must change truth values as well, so that A ∧ B cannot change truth values 2n

183 times without at least one of A and B changing truth values n times. So if flipn(A) and flipm(B) are unsatisfiable, then so is flip2(n+m)(A∧B). Now the eventual stability of A ∨ B and A → B follow from the definibitility of ∨ and → from ∧ and ¬.

Corollary C.5. Every quantiﬁer free formula is eventually stable.

It is also interesting to note that under the ♦ translation from Chapter 2, any formula φ♦ is eventually stable—though I do not know that there is any particular significance to this fact. Note that in none of the proofs leading up to Corollary C.5 did we appeal to the axiom S5. Thus these lemmas are all true of the theory MC as well as MCLS. Notably absent from Proposition C.4 is that claim that ∀σA or ∃σA are eventually stable whenever A is eventually stable. (Given that ¬A is on the list, adding either of ∀σA or ∃σA would immediately yield the other as well). The reason for this lies in our definition of eventual stability. The form of this definition is: for some n, for all variable assignments ν, flipn(A) is not true under ν. But this leaves open the possibility that for every n there is a ν that satisfies flipn(A). If so, then we can piece together a single assignment ν0 that, for every n, satisfies flipn(∀σA). This also means that the difference between open formulas and their universal closures is important. One example where this difference manifests is:

A(α, x) := (α(x) 6= f ∧ α(x + 1) = f) → α(x) = 0

This formula is eventually stable, since (for any model of MCLS) there is no variable assignment that makes true ¬A ∧ ♦(A ∧ ♦¬A). To see why, ﬁx some variable assignment. If ¬A ∧ ♦(A ∧ ♦¬A) is true at a world w, then it must be true at w that (α(x) 6= f ∧ α(x + 1) = f) ∧ α(x) 6= 0. But if α(x) 6= f, then there is some y such

184 that α(x) = y and y 6= 0. Then α(x) = y, by S3, and hence α(x) 6= 0. So the only way for ♦A to be true at w requires that there be a world v (where wRv) that makes false α(x + 1) = f. Hence there is a z such that v makes true α(x + 1) = z and hence α(x + 1) = z. So v makes true φ, and makes false A ∧ ♦¬A. So w cannot make true ¬A ∧ ♦(A ∧ ♦¬A). But w was arbitrary, so this formula is not satisﬁable. On the other hand, it is easy to see that ∀xA(α, x) is not eventually stable. If α has arbitrarily many zeros with at least one non-zero entry between them, then ∀xA(α, x) will change truth value arbitrarily many times.

185