An Indeterminate Universe of Sets

Chris Scambler

December 28, 2016

Abstract

In this paper, I develop a view on set-theoretic ontology I call Universe-Indeterminism, according to which there is a unique but indeterminate universe of sets. I argue that Solomon Feferman’s work on semi-constructive set theories can be adapted to this project, and develop a philosophical motivation for a semi-constructive set theory closely based on Feferman’s but tailored to the Universe-Indeterminist’s viewpoint. I also compare the emergent Universe-Indeterminist view to some more familiar views on set-theoretic ontology.

1 Introduction

The axioms of standard set theories are partly motivated by their association with the iterative conception of sets, according to which all sets exist in a well-founded cumulative hierarchy which is itself not a set. This picture is often itself motivated by describing a kind of set construction process, the output of which is the aforementioned cumulative hierarchy. The hypo- thetical process would go like this. In the beginning, there is nothing whatever, except you (or some other set-producing demiurge). At the zeroth stage of the process, form all the sets you can; since you have nothing around, this set will have nothing in it and so be the empty set. At any further successor stage, form all possible sets of things you had when done last time. At a limit stage, simply collect together all your work at previous stages. Every sequence of successor stages has a limit, and every limit has a further successor. There is, therefore, no end to the set construction process; according to the iterative conception, to be a set just is to be formed by you (or the demiurge) at some stage of said process.

This informal picture is suggestive, but imprecise; however, it is approximated formally in all standard set theories, in the the sense that in all such theories one can deﬁne

V0 = ∅; Vα = P(Vα); +1 S Vλ = α<λ Vα and then prove that every set is contained in Vα for some α. This formal deﬁnition captures the intuitive content of the informal picture painted above, and the fact that standard set theories

1 prove every set to be produced at some stage Vα represents a kind of formal endorsement of the informal picture.

There is something of a reﬂective harmony between the informal conception described in the previous paragraph and standard axiomatic set theories; while the informal conception com- mands serious consideration in virtue of its formalizability in standard set theories, at the same time standard set theories can seem arbitrary and unmotivated without the conceptual foundation and intuitive model provided for them by the iterative conception. As Kant might have put it, without formally axiomatized set theories the iterative conception is empty; but without the iterative conception, formally axiomatized set theories are blind.

In the later part of the twentieth century, it was discovered that there are a vast number of questions we can ask about cumulative hierarchy structures in the formal languages of standard set theories that those theories are demonstrably incapable of answering, if they are consistent. The Continuum Hypothesis (CH) and the Generalized Continuum Hypothesis (GCH) are particularly simple examples: they concern the jump in cardinality that occurs from one stage of the cumulative hierarchy to the next. According to CH, |Vω+1| is ℵ1 and thus only one inﬁnite cardinality greater than Vω; according to GCH, this is true of every stage. As is well known, Gödel showed that there was a model of the axioms of ZFC – the constructible universe, L – in which GCH holds; later, Cohen showed that we could ﬁnd models for the axioms of ZFC wherein |Vω+1| is ℵα for pretty much any α you care to name, subject to some restrictions. The method of Easton forcing allows GCH to be violated in equally radical ways.

The pervasive nature of the independence phenomenon raises hard philosophical problems concerning the relationship between standard set theories and their informal motivation in the iterative conception. The iterative conception tells us that at each successor stage of the set construction process, we form all possible sets of things that were available to us last time; on the face of it, there is only one way to do this, so one would think that problems like CH and GCH should have deﬁnite truth-values. But the results just discussed suggest that standard set theories at the very least cannot tell us how many sets are formed at each stage, and so cannot determine what those truth-values are; and if standard set theories, which make the informal conception precise, cannot tell us this, then what can?

There are (broadly speaking)1 two kinds of response to this situation in the literature.

Universe-Determinism: Standard set theories describe a unique universe of sets, the cumulative hierarchy. CH, GCH and all other set-theoretic propositions have unique truth-values, which are determined by the nature of that universe. Our present incapacity to solve such problems is a reﬂection of a substantive incompleteness in standard set theories.

Multiverse-Pluralism: The independence phenomenon reveals that standard set theories in fact describe many non-isomorphic universes of sets. The initial thought that the cumulative hierarchy was unique up to isomorphism was erroneous; in fact there are many such structures, each a legitimate and independent object of set-theoretic study. Since the truth-values of GCH, CH and many other set-theoretic propositions vary from universe to universe, these propositions do not have unique truth-values.

1Some views such as the “hyperuniverse” view of (Arrigoni and Friedman, 2013) are diﬃcult to place; however I think it fair to say that most currently held views ﬁt roughly into one or other of these categories.

2 The main goal of this paper is to develop and defend a third position, which cuts across these two. I call it

Universe-Indeterminism: Standard set theories describe a unique universe of sets, the cumulative hierarchy. However, the iterative conception and its formal approximations employ concepts that are inherently indeterminate. The indeterminacy arises in the idea that, at successor stages, we collect together all possible sets of things available to us at previous stages; formally, it arises through acceptance of the powerset axiom and its use in the recursive deﬁnition of the cumulative hierarchy. As a result of this indeterminacy, the universe of all sets does not determine truth-values for set-theoretic propositions including CH and GCH.

Universe indeterminism might seem like an unlikely view. Many authors who give arguments to the eﬀect that the universe of sets is unique up to isomorphism or even for the quasi-categoricity of set theory seem to take it that these results entail bivalence for corresponding classes of statements in the language of set theory; in the former case, all of them, in the latter case, for claims that can be bounded to particular ranks. These entailments are prima facie plausible, but not, I would suggest, so obvious as to not require argument – which they all too rarely receive.2 One of the motivating thoughts behind this paper is that the entailments might not hold. One might take the arguments for uniqueness or quasi-categoricity as evidence that our concept of set singles out a unique structure or some unique collection of structures, but nevertheless hold that the structure(s) thus engendered are inherently indeterminate and don’t settle all the questions one can ask about them in formal languages.3 This line might sound as daft as its negation sounded plausible, but I hope to show in this paper that in fact there is a viable line here that is worth considering.

To do so, I will take as a starting point commitment to the existence of a unique universe of sets. Given this, I will develop an axiomatization for set-theoretic truth in such a universe in a partially non-classical logic that a Universe-Indeterminist could take to formally flesh out their claims that certain set-theoretic propositions are indeterminate, and that is philosophically motivated by their ontological position. I shall also show how the position differs in detail from all brands of Multiverse-Pluralism, and briefly discuss the relative merits of the view.

2 Feferman’s Semi-Constructive Set Theory

The view that standard set theories deploy concepts that are inherently indeterminate has been championed by Solomon Feferman; in what follows, I will be heavily guided by his recent formal and philosophical work on semi-constructive set theories. Such theories, according to Feferman, provide a formal framework in which to assess the determinacy or otherwise of

2I have in mind (Martin, 2001) in the former, and (Kreisel, 1967) in the latter. There is an interesting recon- struction of an argument Kreisel might have given here in (Rumﬁtt, forthcoming) 3I will henceforth largely ignore issues of height indeterminacy; for simplicity’s sake, I will often talk as though I have reason to believe ‘the’ universe is of determinate ordinal height, but none of my arguments will turn on this. Everything I will say applies equally well to a Zermelo-style height potentialist, though the statements become slightly more complicated.

3 problems like CH; recent work by Michael Rathjen has raised the stakes by showing that CH ∨ ¬CH is not derivable in a certain semi-constructive set theory. These claims and results are striking, and worthy of close consideration; I also think, and will argue, that they are useful in the development of the Universe-Indeterminist position. Our starting point, therefore, will be an examination of Feferman’s Semi-Constructive set theories.

2.1 Axioms for the Theory SCS

Feferman’s Semi-Constructive set theories are predicative subsystems of standard set theory that restrict classical logic to formulas featuring only bounded quantifiers, employing only the weaker rules of intuitionistic logic to formulas featuring unbounded quantifiers – that is, to those featuring quantifiers ranging over the universe at large. The motivation for this restriction will be discussed in the next section; in this one, we simply give the axioms of the basic semi- constructive set theory – a system known as SCS – and derive some consequences of the axioms that highlight the relationship between SCS and the iterative conception of sets.

The axioms fall into two categories: (Semi-)logical and set-theoretic.

1. (Semi-)logical Axioms

(a) Axioms of intuitionistic predicate logic

(b) ∆0-LEM (Law of Excluded Middle): φ ∨ ¬φ, for ∆0 formulas φ

(d) Extensionality (e) Pair (f) Union (g) Inﬁnity: The set ω = T{x : ∅ ∈ x ∧ ∀y ∈ x[y ∪ {y} ∈ x]} exists. (h) ∈-induction: ∀x[∀y ∈ x[φ(y)] → φ(x)] → ∀x[φ(x)], for any φ

(i) ∆0-Sep: ∀a∃b∀x[x ∈ b ↔ x ∈ a ∧ φ(x)], for ∆0 formulas φ

(j) ACSet: ∀x ∈ a∃y[φ(x, y)] → ∃ f ∀x ∈ a[ f is a function with domain a and φ(x, f (x))], for any φ

Taken together, the axioms effect a restriction of classical logic to ∆0 formulas – formulas in which all quantifiers are of the form ∀x ∈ a or ∃x ∈ a for sets a.4 Such formulas are said to be “bounded” or “relativized” to a; their truth-value is computed relative to the domain induced by the set a. For formulas featuring unbounded quantifiers, only the weaker rules of intuitionistic logic are applied, on which φ ∨ ¬φ is not a theorem for arbitrary φ.

4 The following definitions are relevant to what follows. A formula is Σ0, Π0 or ∆0 iff all its quantifiers are bounded. A formula is Σn+1 (resp. Πn+1) iff it is of the form ∃x[φ(x)] (resp. ∀x[φ(x)]) where φ is Πn (resp. Σn). A T formula χ is ∆n iff there are Σn and Πn formulas φ, ψ such that T ` χ ↔ φ and T ` χ ↔ ψ. The superscripted T indicates all ∆n formulas are relative to a theory T for n > 0.

4 We now turn to the establishment of some basic facts about SCS, and especially its com- parison with relevantly similar classical and intuitionistic sub-systems of ZFC; the reason for doing this is to see how much standard set theory SCS endorses (at least in letter), which is actually a surprising amount.

KP(ω) (Kripke-Platek set theory with the axiom of inﬁnity) is SCS with ACSet replaced by 5 Σ1 collection and fully classical logic. The following theorem relates SCS and this familiar classical sub-system of ZFC.

Theorem 2.1 (Feferman). SCS and KP(ω) are equiconsistent.

Proof. (Sketch.) As in (Feferman, 2010). KP(ω) can be interpreted in SCS by a variant of Gödel’s N-translation, making use of the fact that SCS has ∆0-LEM and ∆0-MP to interpret 6 ∆0-Sep; SCS is interpreted in a functional set theory FSC in all ﬁnite types over V, which can in turn be interpreted in the system of Operational Set Theory OST that is, ﬁnally, known to be equiconsistent with KP(ω).

Theorem 2.1 shows SCS is equal in proof-theoretic strength to the relevantly similar classical system KP(ω). But at the same time, SCS has certain advantages over KP(ω). For example, the latter theory only proves the replacement schema for Σ1 formulas, while in SCS we have

Theorem 2.2 (Feferman). SCS ` Replacement for arbitrary formulas.

Proof. The replacement schema reads:

∀x ∈ a∃!y[φ(x, y)] → ∃b∀z[z ∈ b ↔ ∃x ∈ a[φ(x, z)]]

Suppose that ∀x ∈ a∃!y[φ(x, y)]. By ACSet, there is a function f with domain a and

∀x ∈ a[φ(x, f (x))].

Then the range of f is readily seen to exist (axioms (e), (f) and (i)) and meet the requisite condition (axiom (d)).

Thus, where KP(ω) only proves a very restricted version of the replacement schema, SCS proves the schema for arbitrary formulas. An exactly analogous result holds for the collection schema.

SCS also has important advantages over intuitionistic set theories. In exploring the differences, for concreteness we will compare SCS to KP(ω) with a purely intuitionistic logic (iKP(ω) hereafter), but similar comparisons can be made with stronger intuitionistic theories as 7 well. First, KP(ω) can be interpreted in SCS but not in iKP(ω), since ∆0-LEM and ∆0-MP are

5 Σ1 Collection is the following schema: ∀x∃y[φ(x, y)] → ∀a∃b∀x ∈ a∃y ∈ b[φ(x, y)], where φ must be a Σ1 formula. 6Feferman actually denotes the system FSC ↑ for the theory in all finite types, and FSC for an intermediary system having only 2 types and unrestricted quantifiers. 7I assume ∈-induction rather than foundation to hold “officially”, so to speak, in iKP(ω); foundation is demonstrably its contrapositive, given classical logic, so this makes little difference.

5 required for the interpretation (see (Rumﬁtt, 2015, 286 - 299) for discussion); thus SCS proves N-translations of all theorems of KP(ω), where iKP(ω) does not. Secondly and more importantly, the restriction to purely intuitionistic logic prevents the expression of truths fundamental to the iterative conception of set. Consider for instance the following

Theorem 2.3 (Diaconescu). ACSet implies ∆0-LEM over iKP(ω).

Proof. Suppose ACSet is true in iKP(ω), and let φ be a ∆0 formula. Using ∆0-Sep, deﬁne the sets A := {n ∈ {0, 1} : n = 0 ∨ (n = 1 ∧ φ)}, B := {n ∈ {0, 1} : n = 1 ∨ (n = 0 ∧ φ)}. Since ∀x ∈ {A, B}∃y ∈ {0, 1}[y ∈ x],

ACSet yields a function f : {A, B} → {0, 1} with f (A) ∈ A and f (B) ∈ B. Assuming φ, A = B by extensionality; on the other hand, if f (A) = f (B), then obviously φ must hold. Thus we get f (A) = f (B) ↔ φ. Since equality of natural numbers is decidable, we get φ ∨ ¬φ.8

Theorem 2.3 shows that admitting ACSet to the axioms of iKP(ω) forces us to transcend the intuitionistic logic to which iKP(ω) is restricted; on the other hand, since SCS includes ∆0- LEM, there is no such logical obstruction to ACSet also being included among its axioms. And indeed, this result is just the tip of the iceberg; other arguably more fundamental principles of iterative set theory imply some amount of classical logic. We have, for example:

Theorem 2.4. The linear ordering of the ordinals (LO) is equivalent to ∆0-LEM over iKP(ω).

Proof. ∆0-LEM → LO proceeds by the usual inductive proof, which makes use of LEM on the ∆0 formula α ∈ β. For the converse, let φ be a ∆0 formula and consider the set: α = {x ∈ {0} : x = 0 ∧ φ} α is readily seen to be an ordinal as α ⊆ 1.9 Assuming the ordinals are linearly ordered entails that either α < 1 or else α = 1; in case α < 1 we have ¬φ, and we have φ if α = 1. Whence φ ∨ ¬φ.

In standard set theory, the ordinals are the linear spine along which the powerset operator is iterated; but the previous theorem shows there is nothing particularly special about the ordinals in intuitionistic set theory. Here they are just hereditarily transitive sets, and do not have their ordinarily distinguishing order. Since in SCS we have ∆0-LEM, however, the ordinals are ordered as usual.

Another basic principle in iterative set theory that presents diﬃculties in intuitionistic systems is the axiom of foundation, though the situation is slightly more complicated here and serves to emphasise the importance of ∆0-MP (Markov’s Principle). 8This theorem was, as indicated, originally proved by Diaconescu in (Diaconescu, 1975); but the version of the proof given here is due to Goodman & Myhill in (Goodman and Myhill, 1975). 9In intuitionistic set theories, ordinals are hereditarily transitive sets; this condition is readily seen to hold of α.

6 10 Theorem 2.5 (Myhill). The well-foundedness of the ∈ relation (WF) implies ∆0-LEM over iKP(ω).

Proof. Let φ be a ∆0 formula and consider the set F = {x ∈ {0, 1} : x = 1 ∨ (x = 0 ∧ φ)}. F is non-empty, as it has 1 as an element. Given WF, it has an ∈-minimal element which by construction must either be 0 or 1. If it is 0, then φ. If it is 1, then since 0 ∈ 1, 0 < F by WF, so ¬φ holds. In either case, we have φ ∨ ¬φ.

Theorem 2.6. ∆0- LEM and -MP imply WF over iKP(ω).

Proof. Take φ(x, a) as x < a in ∈-induction; contraposing yields

¬∀x[x < a] → ¬∀z[∀y ∈ z(y < a) → z < a] which is intuitionistically equivalent to

¬¬∃x¬¬[x ∈ a] → ¬¬∃z[∀y ∈ z[y < a ∧ ¬¬(z ∈ a)]

∆0-LEM and ∆0-MP then yield an ∈-minimal element z of a if a is non-empty.

The well-foundedness of the membership relation, the linear ordering of the ordinals, and (we assume) the axiom of choice are all essential components of the iterative conception of set; under the iterative conception, set theory is the study of a well-founded, cumulative hierarchy formed by iterating the powerset operator along a linear spine provided by the ordinals. That these fundamental principles are all either equivalent to or imply principles of classical logic for at least ∆0 formulas seems to suggest that at least this much classical logic is simply presupposed by the iterative conception. Feferman’s SCS, in endorsing at least this much classical logic, is able to prove these fundamental truths of iterative set theory where intuitionistic set theories cannot, and this is surely a point in its favor. Even so, it is natural to ask for more; why should we investigate systems of set theory that restrict classical logic at all?

2.2 Feferman’s Motivation

Feferman offers a philosophical motivation for the axioms of SCS which I think is of consider- able independent interest, and worthy of close examination. However, I think (and will argue) that the best motivation for the restrictions to classical logic Feferman suggests is more directly associated with what I call the Universe-Indeterminist view; eventually, I will show how to modify Feferman’s motivation so as to motivate a semi-constructive set theory closely related to SCS that is tailored to axiomatizing such an indeterminate universe. As my motivation is ultimately only a small modification of Feferman’s, however, it is proper to examine Feferman’s closely beforehand here. In this section, I develop Feferman’s motivation for SCS, and in the next I offer a critical analysis of it. From that analysis, other principles will emerge that from the basis of the Universe-Indeterminist view.

Feferman’s motivation for SCS is based on the following principles:

10Speciﬁcally, I have in mind a formal claim to the eﬀect that every non-empty set has an ∈-minimal element.

7 FP1 A conception of the set-theoretic universe on which a collection is a deﬁnite totality if and only if it forms a set

FP2 A conception of logic according to which “what’s deﬁnite is the domain of classical logic, what’s not that of intuitionistic logic” (Feferman, 2011, 23)

If one accepts FP1 and FP2, the axioms of SCS may be justified as follows. By the reasoning of Russell’s paradox, the universe V of all sets is not a set; so, by FP1, it is not a definite totality. By FP2 intuitionistic logic is therefore appropriate when reasoning with formulas featuring unbounded quantifiers, which range over the indefinite totality V. On the other hand FP1 and FP2 together quickly justify the use of classical logic when reasoning with formulas whose quantifiers are all bounded to particular sets. These arguments conceptually motivate the (semi-)logical axioms; axioms of the inuitionistic predicate calculus govern reasoning featuring unbounded quantifiers, whilst ∆0-LEM and ∆0-MP underwrite the use of classical logic when reasoning over particular sets. The restriction of separation to ∆0-Sep is motivated by the desire to restrict the schema to definite properties, by which we mean (in the present context) those whose defining formula ranges only over definite totalities (Feferman, 2011, 25).11 Set-theoretic axioms (d) through (f) are understood as motivated simply by the desire to do set theory.

The same justification transfers in part to ∈-induction and ACSet, in that we will simply take it as an assumption that ∈-induction and the axiom of choice are fundamental principles of iterative set theory. However, as was emphasised in statement of the axioms, these schemata are notably unrestricted. ACSet, for instance, underwrites the existence of choice functions for suitable formulas of any complexity as opposed only ∆0; as such, one might worry that the axiom is too strong to be justified by FP1 and FP2, and that only a weaker form of choice, say for ∆0 formulas, should be employed. In fact, the axiom can be justified. Intuitively, its apparent strength is tempered by the fact that it is a conditional axiom; this prevents our inferring the existence of the choice functions without our first having met the stringent requirement of giving a constructive proof of the antecedent. Moreover, the conditional is itself justifiable on many of the standard approaches to the semantics for intuitionistic logic. On the Heyting semantics for intuitionistic logic, for example, a claim of the form φ → ψ is construed as a method for converting any proof of φ into a proof of ψ. Each instance of choice, on this reading, is supposed to provide a method for turning a constructive proof of ∀x ∈ a∃yφ(x, y) for the relevant a into one for the existence of a choice function. Finding such a method is close to trivial. For, on the Heyting semantics, a proof of ∀x ∈ a∃yφ(x, y) just is a procedure which, for a given x in a, produces a y standing in the φ relation to x; to find the choice function, then, we can simply take the function whose values at x in a are exactly the y produced by the given proof. This informal idea is formally substantiated by Feferman’s interpretation of SCS in the Functional Set Theory

11It might seem strange that full replacement is justified, but only partial separation. There are two things to note here: first, it is important to note the conditional nature of replacement as opposed the straightforwardly existential form of separation. Whereas separation categorically asserts the existence of the separation set, in replacement we are only told that a replacement set exists under certain conditions, which are intuitively strengthened by the intuitionistic background logic for higher complexity formulas. I discuss these issues further in the next paragraph in the context of choice; many of the same considerations carry over. Also, it is worth noting that adding separation for higher complexity classes yields more classical logic, given choice, via a modification of the argument for Theorem 2.3. Thus justifying full separation would amount to justifying classical logic for all formulas. This is not the case for replacement.

8 FSC (encountered in the proof of theorem 2.1), an intuitionistic theory in all finite types over V 12 featuring only bounded quantifiers and that is constructively justifiable. ACSet is thus justified by a combination of factors; on the one hand, we have the (assumed) conceptual support for the axiom of choice on the iterative conception; on the other, we have a formal result showing

ACSet interpretable in the constructively justifiable set theory FSC, so that we can be certain it does not entail an unjustified extension of classical logic. Similar considerations justify the full ∈-induction schema; this too can be interpreted away in FSC, and is justified insofar as we conceive of sets as constructed in a stepwise manner as the iterative conception dictates. We shall apply this two-fold method of justification again to a different axiom in section 3.3.

The only outstanding axiom is axiom (g), the axiom of infinity.The axiom underwrites the existence of ω, the totality of all finite Von Neumann ordinals, as a set; given FP1, the question of whether or not axiom (g) can be justified translates into the question of whether or not ω is a definite totality. Addressing this question will require discerning exactly what Feferman means by the terms ‘definite’ and ‘indefinite’, a question which we shall address fully in the next section.13 For now, let us assume that ω is indeed a definite totality in whatever sense Feferman has in mind, so that the presence of axiom (g) among the axioms of SCS is also taken to be justified under Feferman’s motivation.

If this promissory note can be redeemed, Feferman can philosophically motivate the axioms of SCS from his principles FP1 and FP2. But what significance does he attach to the resulting system? According to Feferman, theories related to SCS are philosophically signif- icant because they provide a formal framework for assessing the determinacy of set-theoretic statements, where a sentence φ is formally determinate in a semi-constructive system SCS + θ – which may be a proper extension of SCS by some set existence axiom(s) – if and only if SCS + θ ` φ ∨ ¬φ (Feferman, 2011, 23).14 The underlying thought is that in restricting classical methods of reasoning to quantificational statements whose domain forms a definite totality (or set, under FP1), we prevent our assuming LEM holds for statements by an unjustified over- extension of classical logic; LEM holding for a statement φ is thus a substantive fact, following from the definiteness of the totalities quantified over in the statement of φ. If one accepts the above motivation, then, a proof that LEM fails for some φ in a justified semi-constructive set theory may contribute to philosophical debate concerning the determinacy of φ. In section 4, we consider a recent formal result germane to this project, namely Michael Rathjen’s proof that SCS + P(ω) exists 0 CH ∨ ¬CH (Rathjen, 2016a). One might naturally hope for such results to offer substantive contributions to debate concerning set-theoretic indeterminacy, perhaps pro- viding some kind of evidence for Feferman’s position on the indeterminacy of CH; we will examine these possibilities closely later in the paper.

My own view, however, is that there is another equally interesting avenue of thought suggested by the axioms of this theory and Feferman’s motivation for it. The theory SCS endorses

12The interpretation is one of the main results of (Feferman, 2010). FSC (which Feferman actually denotes FSC ↑) stands in much the same relation to SCS as Gödel’s theory T (employed in the “Dialectica” interpretation of (Gödel, 1958)) does to Primitive Recursive Arithmetic. For more on the latter see (Avigad and Feferman, 1998). 13Feferman himself and other predicativists take the axiom of infinity as self-evident; since ω is inductively defined, it is crystal clear, indeed the canonical instance of a clear or definite totality. I am sceptical that appeals to clarity of this kind are very helpful, and this is why I take it that the axiom is as much in need of justification as other set existence axioms, like the powerset axiom. 14Feferman actually uses the term ‘definite’ rather than ‘determinate’; I have opted for the latter to avoid confu- sion with ‘definite’ in the sense of indefinite extensibility, to be discussed later.

9 classical logic for claims featuring bounded quantifiers, which yields sufficient strength to develop a fair amount of set theory and to retain certain fundamental principles of the iterative conception of sets. At the same time, SCS refrains from endorsing arbitrary instances of LEM for claims about the universe at large. On a deflationary view on the nature of truth, where φ is straightforwardly equivalent to True(φ), this means also refraining from endorsing arbitrary instances of True(φ) ∨ ¬True(φ); this line of thought suggests any envisioned universe whose range is the universal quantifiers of SCS is an indeterminate one, of which some propositions are neither true nor false. Rathjen’s argument shows that CH could be an example of one such proposition. The theory, then, seems to offer a formal tool for developing the Universe- Indeterminist position, namely that there is a unique but indeterminate universe of sets. In the next section I will argue that a slight extension of SCS is well-suited to exactly this task.

3 Indeﬁniteness and Indeterminacy

In the last section, we saw that Feferman offers a motivation for the axioms of SCS in terms of his two principles FP1 and FP2. We are still owed, however, a justification for the axiom of infinity, and I think that consideration of the justification Feferman might offer for this axiom illuminates some difficulties for Feferman’s motivation. In the first subsection of this section, I discuss those difficulties; in the second, I suggest an alternative motivation which evades them, and which leads to an amendment of SCS that turns out to be central to the Universe- Indeterminist position.

3.1 Indeﬁniteness

Feferman’s motivation places an emphasis in the distinction between ‘definite’ and ‘indefinite’ totalities, and before we move on we must understand what this distinction amounts to. As we are concerned with Feferman’s motivation, we will elaborate the distinction from within the framework of his favoured foundation for mathematics, conceptual structuralism.15 According to the conceptual structuralists, the basic objects of mathematical study are conceptual structures like hω, S c, <, 0, +, ×i. These are held to be abstracted and refined from basic conceptions – in this case that of a discrete, linearly ordered, simply infinite sequence with a least member – which in turn are derived from everyday experiences like counting, matching, combining and so on. Importantly, conceptual structuralism is an anti-realist foundation for mathematics in the sense that conceptual structures are held to be social constructions, and do not enjoy mind- independent existence.16

Since conceptual structures do not enjoy mind-independent existence, we cannot assume that every mathematical proposition will have a truth-value. However, unlike the intuitionists (who share such a metaphysics), the conceptual structuralist does not feel compelled to straight-

15A parallel discussion along Dummettian lines, interpreting ‘indeﬁnite’ as ‘indeﬁnitely extendible’, might also be given; very similar conclusions would apply to Feferman’s motivation so construed to those we draw at the end of this section. 16This account of Feferman’s view is rather terse, but hopefully faithful: the interested reader is directed to the papers (Feferman, 2009),(Feferman, 2011), and (Feferman, 2014).

10 forwardly equate provability and truth. Instead, truth in conceptual structures beyond present provability is held to be intimately related to the “clarity” of a structure’s basic conception (Feferman, 2011, 11). Thus, according to Feferman:

The conception of the structure hω, S c, <, 0, +, ×i is so intuitively clear that ... there is no question in the minds of mathematicians as to the deﬁnite meaning of such statements and the assertion that they are true or false, independently of whether we can establish them in one way or the other. (Feferman, 2014, 5)

For this reason, Feferman holds that arithmetical claims are bivalent and classical logic is justi- ﬁed for arithmetical reasoning. But on the other hand:

the intuitive distinction ... between sets and sequences is [important since] sequences hav[e] a greater clarity than sets. For example, we have a much clearer conception of arbitrary sequences of points on the Hilbert (or Dedekind, or Cauchy- Cantor) line, or at least of bounded strictly monotone sequence, than we do of arbitrary subsets of the line.... We have a clearer conception of what it means to be an arbitrary infinite path through the full binary tree than of what it means to be an arbitrary subset of ω, but in neither case do we have a clear conception of the totality of such paths, resp. sets. In both cases, our conception allows us to reason that there is no enumeration of all infinite paths through the binary tree, resp. of all subsets of ω, simply by the appropriate diagonal argument. But it is an idealization of our conceptions to speak of 2ω, resp. P(ω), as being definite totalities. And when we step to P(P(ω)) there is a still further loss of clarity, but it is just the definiteness of that that is needed to make [determinate] sense of CH.(Feferman, 2011, 21)

In constrast to ω, then, Feferman does not think that P(ω) or (a fortiori) P(P(ω)) are associated with clear conceptual structures.17

How does the distinction between definite and indefinite play out in conceptual structuralist terms? The latter passage intimates a connection between the clarity of a structure’s basic conception on the one hand, and the definiteness of the totality engendered by it on the other, as for instance the lack of clarity in the conception of all paths through the binary tree is seemlessly interchanged with the indefiniteness of the totality of 2ω. Some have taken this to be all there is to Feferman’s position: they have read Feferman as arguing that a totality is definite insofar as its basic conception is “intuitively clear”. Those who have read Feferman in this way have rightly protested that any distinction between definite and indefinite thus engendered will be unlikely to prove helpful in debates on set-theoretic indeterminacy. As Peter Koellner has put it, intuitions of clarity are “too subjectivistic” and not “robust” enough to be useful in the present dialectical context (Koellner, 2011, 14); those who find CH determinate will likely profess to finding the totality P(P(ω)) definite on grounds of an intuition of clarity that Feferman simply doesn’t share, and it is hard to see how either party in this debate could get the upper hand.

I do not think that this is the best way to read Feferman. Elsewhere, Feferman writes that a totality is definite if and only if quantification over that totality is ‘a definite logical operation’,

17The structure behind P(ω), for instance, would be hω, P(ω), ∈i.

11 by which he means that quantifying over a determinate property always yields a determinate (bivalent) statement.18 There is also textual evidence that suggests Feferman holds that a conceptual stucture is clear if and only if its associated totality is definite.19 These remarks suggest an alternative reading, wherein a totality is definite when and only when the principle of bivalence holds for quantificational statements over that structure, with this latter feature of a structure being equivalent to the “clarity” of its basic conception. So understood, we seem to get a much stronger case for Feferman’s distinction between the definite and the indefinite, one that does not straightforwardly bottom out in subjective intuitions of clarity. After all, all the problems known to be independent of arithmetic are of the Gödelian metamathematical vari- ety;20 there seems to be little reason to doubt, therefore, the “definite meaning” and bivalence of statements of arithmetic, which in turn entails that ω is definite totality.21 Axiom (g), therefore, can justifiably be included among the axioms of SCS. On the other hand there are substantive problems of second- and third-order arithmetic that are known to be independent of the relevant axiom systems; an example for second-order arithmetic is the question of whether the projective sets of reals are Lebesgue measurable, and in third-order arithmetic we have, among others, CH. Thus, we have found reasons for thinking that ω is definite, but that P(ω) and P(P(ω)) are not, that are not straightforwardly based on “intuitions of clarity”.

Although this does seem preferable as a line to the one attributed Feferman by Koellner, even so one might worry that this line of argument remains unconvincing in the present dialectical context. After all, the independence of such problems from our present theories of the relevant domains does not entail they fail to be bivalent; our axiomatizations may simply be incomplete. Indeed, some have suggested that the question of the Lebesgue mesurability of the projective sets has since been settled with recourse to large cardinal axioms, and hold out similar hopes for CH (See for example (Woodin, 2001), (Koellner, 2006)). So in order to work out which of these totalities are ‘definite’ from Feferman’s point of view, we shall have to work out whether such independent problems really are instances of bivalence failure. This difficulty has ramifications for Feferman’s project, which itself is directed at the resolution of precisely these issues. Recall that Feferman thinks semi-constructive set theories can provide a formal framework for assessing the determinacy of set-theoretic statements, and defines a set- theoretic statement φ to be formally determinate relative to a semi-constructive system SCS + θ where SCS + θ ` φ ∨ ¬φ. We have already alluded to Rathjen’s proof that SCS + P(ω) exists 0 CH ∨ ¬CH, and indicated that this might inspire optimism with regards Feferman’s project. But at the same time, it is easy to prove that SCS + P(P(ω)) exists ` CH ∨ ¬CH; since CH can be stated with quantifiers bounded to P(P(ω)), the proof is a simple application of ∆0-LEM. Now, under FP2, the question of which of these set existence axioms to endorse reduces to the question of which, if either, of these totalities is definite. If that question itself comes to turn on

18See the appendix to (Feferman, 2014). 19On this connection, for instance, Feferman writes that “truth in full [bivalence] is only applicable to perfectly clear structures” (Feferman, 2011, 11); given the definition of definiteness just considered, this entails that a totality is definite only if its associated conceptual structure is clear. In the first passage of this section, Feferman suggests that clarity of a structure entails the definiteness of its associated totality, where he says that the clarity of hω, S c, <, 0, +, ×i entails the definiteness of ω. Putting these together, we arrive at the hypothesis that Feferman holds a structure to be clear if and only if its associated totality is definite. 20Not all of them share the obvious metamathematical nature of Gödel’s theorems, as examples like the Paris- Harrington thereom show. Still, there is little doubt about the truth-values of any of these results in the standard model, in stark contrast to those examples condsidered below. 21Here we think of ω as the domain of N, the standard model of arithmetic.

12 the bivalence of problems like CH, then one might worry that the semi-constructive framework is unlikely to advance debate on such questions; and as we have just seen, one’s stance on the deﬁniteness of totalities like P(P(ω)) does indeed seem to turn on one’s antecedent views on the determinacy of problems like CH.

These issues are very tricky; indeed, pursuing them leads to some fundamental questions in the philosophy of mathematics. The natural rejoinder in defense of Feferman is that the set-theoretical realist, who thinks that e.g. P(ω) is a definite totality over which classical quantification is legitimate, is obliged to give some argument in favour of the definiteness of such totalities if their position is to rest on anything more secure than blind faith. In (Feferman, 2011, 20) Feferman makes precisely this move. He argues that the only way for the set-theoretical realist (at present) to make their case is with appeal to categoricity results for structures like hω, P(ω), ∈i in full second order logic,22 and then turns to the familiar claim that appeal to full second-order resources is “evidently question begging” in the present context. There is a rich history of debate on this question which we cannot rehearse here: the interested reader is directed toward the Kreisel / Weston debate ((Weston, 1976), (Kreisel, 1967)), chapter 9 of (Shapiro, 1990), and Toby Meadows’ (Meadows, 2013) recent paper on the philosophical significance of categoricity theorems.

It seems to me that Feferman’s motivation for SCS in terms of definiteness, then, leads us into thorny issues with a very rich history in the philosophy of mathematics. While Feferman’s motivation for SCS in terms of definiteness certainly presents a useful and provocative contribution to that debate, I am not convinced that the merits of semi-constructive systems are exhausted by that contribution. Indeed, I will argue that by replacing the notion of definiteness with the semantic notion of indeterminacy, modified versions of Feferman’s two principles can form the cornerstone for a Universe-Indeterminist position in the ontology of set theory and the basis for a novel argument for the indeterminacy of CH. In the next section, I will begin to develop this approach in detail.

3.2 Indeterminacy, Part I

In the last section, I suggested we explore alternatives to the notion of indeﬁniteness as employed in Feferman’s motivation for SCS; in this subsection and the next, I will consider the results of adopting principles analogous to Feferman’s, but in which the concept of deﬁniteness is replaced by that of determinacy of sense. In the end, I’ll argue that this motivation yields a slightly stronger theory than SCS that is suited to explicating the Universe-Indeterminist position discussed in the introduction.

The concept ‘determinacy of sense’ has been fruitfully employed in recent work by Ian Rumfitt. In (Rumfitt, forthcoming), Rumfitt explicates determinacy of sense making use of the intuitive idea of a legitimate interpretation of a statement, where interpreting a statement S (in a given context) as saying that P is legitimate just when a speaker of the language in question would not reveal themselves incompetent were they so to interpret S (in the relevant context). If, for example, a beginning English speaker were to interpret “John is very sympathetic” as saying that John is fun, they would thereby reveal themselves to be an incompetent speaker

22cf. (Kreisel, 1967).

13 of English; such an interpretation is recognizably illegitimate to competent speakers of the language. With the rough-and-ready, intuitive idea of a legitimate interpretation in hand, we can deﬁne determinacy of sense as follows:

DET: A statement is of determinate sense only if all its legitimate interpretations are materially equivalent.

DET is perhaps easiest to grasp through its contrapositive. Given DET, a statement will be of indeterminate sense when it has materially inequivalent though equally legitimate interpretations; in such a circumstance, a competent speaker of a given language cannot be reprimanded for interpreting it in one of at least two materially inequivalent ways. For instance, if one over- hears the statement “Sally went to the bank” in the absence of further context, then assuming the ﬁnancial institution is appropriately far away from the water’s edge there are two materially inequivalent though equally legitimate ways that speaker may interpret the statement, which latter is concluded to be of indeterminate sense. Thus, lexical ambiguity is a species of indeterminacy of sense. Similarly if one says of borderline-Barry that he is bald, then assuming he genuinely is a borderline case one might legitimately interpret that assertion as saying something true or equally as saying something false; thus vagueness too is a species of indeterminacy of sense.

It is important to note that DET only provides a necessary condition for determinacy of sense. It does not follow from DET that any expression whose legitimate interpretations are materially equivalent is in fact of determinate sense. For instance, if it turns out that the money bank is on the river bank, then “Sally went to the bank” might have each of its legitimate interpretations materially equivalent. But it does not follow that “Sally went to the bank” is of determinate sense. This is no counterexample to DET, which only endorses the converse; any expression with such materially diverging expressions cannot be of determinate sense. This point will prove of central importance in section 3.3.

In order for analyses of the above kind to work with regards a particular expression or class of expressions, one must ensure that all and only the legitimate interpretations of a statement (or class of statements) are employed; we may not interpret “went to the bank” as “had ten thousand cats”, nor may we ignore the legitimate river-bank interpretation. In many cases, especially ones of ordinary vagueness, it will itself be indeterminate which the legitimate interpretations are, making precise analyses in particular cases diﬃcult if not impossible to carry out. However, to apply this analysis to the class of set-theoretic statements, similar indeterminacy in the class of legitimate interpretations is not obviously going to arise; it seems in principle possible that we should be able to precisely delimit a class of such legitimate interpretations by isolating a class of privileged models of some set theory to serve the role of the legitimate interpretations. If we can successfully do this, then we can say exactly when a set-theoretic statement is of determinate sense - it may be determinate when its truth value is the same in all such interpretations, but will be indeterminate in sense otherwise.

Ultimately, we aim to motivate restrictions to the logic of set theory in terms of a restriction of classical logic to set-theoretic statements of determinate sense; given DET, we must therefore attempt to delimit the class of legitimate interpretations of set-theoretic statements. As we have already suggested, we shall want to take models of axioms for set theory as interpretations of set-theoretic statements, so we shall be delimiting a class of such models. But which? There are many plausible answers to this question, and many correspond to views which have traditionally

14 been motivated by quite diﬀerent arguments. The Universe-Determinist, for example, can in the present context be understood as advocating the (possible) existence of a unique, maximal interpretation of set-theoretic discourse, which is the only legitimate one; anything short of it is counted as deviant. On the other hand, one can view a Multiverse-Pluralist like Joel Hamkins as suggesting that any model of the axioms of ZFC (or perhaps even some weaker base theory, like second-order arithmetic) is as legitimate an interpretation of a given set-theoretic statement as any other - and indeed, is a universe of sets in its own right - so that statements whose truth- value varies between such interpretations are, when understood in full generality, indeterminate in sense. In motivating the axioms of SCS, however, we adopt the following:

STMH (Standard Transitive Model Hypothesis): Any standard transitive model of the axioms of (ﬁrst-order) ZFC is a legitimate interpretation of a set-theoretic statement.

The STMH is obviously not at all trivial, and we will return to its status in section 4. For now, I contend merely that even if the STMH is not obviously true, it is not obviously false either; it is at least a plausible thought. A standard transitive model M (of the axioms of ZFC) is a structure hM, Ei |= ZFC, where M is a transitive set and E is ∈, the real membership relation. In such a model, sets “are who they think they are” in terms of membership in the sense that for all y in M, we have (xEy)M ↔ x ∈ y. Given that sets are completely constituted by their members, one might thus argue that a standard transitive model gets the subject matter right in this important respect. Put another way, supposing set theory is the study of the ∈ relation and its behaviour, the STMH seems reasonable; in standard transitive models, ∈ is always captured correctly, but other relations and predicates are left to fend for themsleves. Such intrinsic plausibility arguments are, I am willing to grant, fairly weak; one might give similar arguments in favour of the other two candidates just mentioned above, as well, plausibly, of myriads more. Nevertheless they are a good starting point, since if the idea could not be defended on its own grounds, it would be difficult to defend at all. A deeper defence, though, must clearly go beyond such arguments. One might question, for instance, why first-order ZFC has been chosen as a base theory, why we admit countable models, or why such models need to be well-founded, and so on. I will return to these difficulties in 4, after I have developed my positive proposal in more detail.

Given the STMH and DET, one can go some way toward arguing for the restrictions to classical logic imposed by the axioms of SCS in terms of an attempt to restrict classical logic to set-theoretic statements of determinate sense. The following deﬁnition is standard:

Deﬁnition 3.1 (Absoluteness). Let M be a class of models for some language L. A formula φ(x~n) is said to be absolute for M just in case whenever M, N ∈ M, a~n ∈ M ∩ N, we have M |= φ[a~n] if and only if N |= φ[a~n].

Note that, if the models in question are all models of some theory T, and φ ∈ T is a theorem, φ is trivially absolute for those models. Indeed, absoluteness is only non-trivial for open formulas, or sentences independent of T.

Reﬂecting on the deﬁnition of absoluteness, one sees that it really amounts to this: a formula φ is absolute if the truth-value of φ for particular objects a~n is the same for every model of the class. Given this, DET and STMH, taken together, entail that a set-theoretic expression is of determinate sense only if it is absolute for standard transitive models of ZFC (we shall

15 return to this principle, calling it P1 in 3.3). To prevent applying classical principles of logic to set-theoretic statements of indeterminate sense, then, we shall want to effect restrictions of classical logic to formulas provably absolute for standard transitive models of ZFC, and many axioms of SCS can be viewed as motivated by just such a restriction. After all, ∆0 formulas are absolute for standard transitive models, whilst formulas featuring unbounded quantifiers are in 23 general not absolute (but see Theorem 3.2). The restriction of classical logic to ∆0 formulas, effected by the (semi-)logical axioms, can thus be justified in so far as such a restriction prevents the application of classical principles for non-absolute formulas, which are held to be of indeterminate in sense by the lights of DET and the STMH. As before, the set-theoretic axioms (d) through (f) are justified by the desire to do set theory, and the justification of ∈-induction and ACSet remains unchanged; the restriction to ∆0-Sep can now be understood as motivated by the desire to not separate collections satisfying indeterminate (non-absolute) conditions. Thus, all the restrictions imposed by the axioms of SCS can be supported on terms of a restriction of classical logic to set-theoretic expressions of determinate sense. However, since DET only gives a necessary condition for determinacy of sense, any motivation for SCS given on its terms will necessarily be only a partial one. If we are to view the axioms of SCS as motivated by the desire to restrict classical logic to set-theoretic expressions of determinate sense, we not only require criteria for ruling out the applicability of classical logic, like the STMH and DET, but we will also need criteria justifying the employment of classical logic in other cases. We need some reason to think that ∆0 formulas, for instance, are determinate in sense, and that classical logic should apply to them: for at present, we have only argued that LEM should not apply to non-absolute formulas. In the next section, I will address this deficit.

Before moving on, we should pause to remark that even though we have at present only a necessary condition for determinacy of sense, nevertheless our motivation for SCS offers a sharper method for excluding axioms from our favoured semi-constructive set theory than was available on Feferman’s motivation. In the previous section, we saw that the powerset axiom presented difficulties in this regard: the question of whether or not to include an axiom asserting the existence of P(P(ω)) or P(ω) among those of SCS was seen to reduce to the question of whether the collections in question were definite totalities, but questions of the latter kind in turn seemed to reduce to the very questions of bivalence in set theory we were hoping to advance with the semi-constructive framework. Given a motivation for SCS based on DET and the STMH, though, it is plain that we should not include such axioms among those of our favoured semi-constructive set theory. Since e.g. P(ω) is a Π1 set – its definition features an unbounded universal quantifier - neither the definition of the set nor all quantificational statements bounded to it are absolute for standard transitive models of ZFC.24 Were we to introduce an axiom asserting that set’s existence, along with defined symbols for the set in question, P(ω) would become capable of employment in ∆0 formulas in the expanded language, and the principles of classical logic we are attempting to restrict to absolute formulas would become applicable to those non-absolute quantificational statements bounded to P(ω). In effect, that is, we would trick the system into an unjustified extension of classical principles to a class of non-absolute, and hence indeterminate formulas. We should thus not endorse axioms underwriting the existence of infinite powersets on the present motivation for SCS, when the theory is understood as

23see e.g. (Barwise, 1975, 10) 24 1 However, Π1 formulas (in the analytical hierarchy) are absolute for standard transitive models of ZFC; this shows some quantiﬁcational statements bounded to P(ω) do not violate DET, and this will prove very important in the sequel.

16 motivated from DET and the STMH.

So far, so good; however, our lack of any suﬃcient conditions for determinacy of sense means that we have at present no capacity to justify the inclusion of axiom (g), the axiom of inﬁnity, in SCS, so at present our motivation is still incomplete. Accordingly, we must now turn to the question of under what conditions a set-theoretic statement is, in fact, of determinate in sense.

3.3 Indeterminacy, Part II

In the last section, we began to show that at least the axioms of SCS could be motivated by the desire to restrict classical logic to set-theoretic statements of determinate sense. One can isolate the following two principles, closely analogous to Feferman’s, from that motivation:

P1: A set-theoretic statement is of determinate sense only if it is absolute between standard transitive models of ZFC

P2: What’s determinate is the domain of classical logic, what’s not that of intuitionistic logic

In the remainder of this section, we establish that we can fully motivate at least the axioms of SCS in terms a restriction of classical logic to set-theoretic expressions of determinate sense, but also show that in fact these principles coupled with a Universe-Indeterminist view on ontology motivate a slight strengthening of SCS which we call SCS+.

While P2 is directly analogous to FP2, the analogy between P1 and FP1 is imperfect since FP1 provided necessary and sufficient conditions for definiteness, but P1 only offers a necessary condition for determinacy of sense. This explains the deficit we were left with in the last section; while our motivation for SCS could justify the exclusion of certain axioms and the restriction of classical logic for non-absolute formulas, it could not justify the inclusion of the axiom of infinity or the extension of classical logic to ∆0 formulas. In order to remedy this situation, we need to know when a set-theoretic expression is of determinate in sense; we need a sufficiency condition to supplement the principle P1.

The obvious sufficiency condition is simply to render P1 a biconditional, and to say that a set-theoretic expression φ is determinate in sense if (and only if) it is absolute for standard transitive models of ZFC. The condition would justify the use of classical logic for ∆0 formulas, which are absolute, and would allow for the justification of the axiom of infinity’s inclusion in SCS as follows. Since ω is a ∆0-definable set and hence absolute for standard transitive models of ZFC, quantificational formulas bounded to ω (with parameters from ω) – commonly called arithmetical formulas – are likewise absolute.25 Given the sufficiency condition under consideration, we should have that arithmetical formulas were of determinate sense; since we are motivating SCS as a theory which offers principles of classical logic (only) to such formulas, we will thus want to underwrite principles of classical logic for arithmetical formulas. By endorsing the axiom of infinity we do just that: we will render arithmetical formulas ∆0, since

25See e.g. (Kunen, 2013, 123 - 124) for proof.

17 they would be bounded to the set ω, and hence extend principles of classical logic to them using the (semi-)logical axioms, as desired.

So far, so good. However, it turns out that adopting this sufficiency condition would completely trivialize the present project. The following argument, due to Jonathan Payne, makes the point especially forcefully. Let φ be any formula. Then φ ∨ ¬φ is true in any model of ZFC, and hence is absolute. Given the sufficiency condition under consideration, it follows that LEM for φ is of determinate sense, and hence (by P2) that we should want SCS to have (φ ∨ ¬φ) ∨ ¬(φ ∨ ¬φ) provable. If this is to hold, we shall have to add the latter as a new axiom schema for all φ. But the second disjunct is intuitionistically false; hence the first disjunct must be true, and we have inferred LEM for arbitrary φ.

A similar point stands to be made about theorems of ZFC. In ZFC these are all equivalent to ∆0 formulas; since theorems are true in all models, they will be equivalent to any true ∆0 formula one likes to take. Thus any theorem of ZFC would be determinate in sense, given this suﬃciency condition, and it would be hard indeed to deny such statements ought to be determinate truths, even if the analogous argument to that of the previous paragraph does not go through (since in general the negations of theorems of ZFC will not be outright inconsistent with the axioms of SCS). Thus, a compelling case could be made for the addition of all theorems of ZFC as axioms to SCS, given the suﬃciency condition under consideration. However, a great many theorems of ZFC are of course not theorems of SCS, including some fairly basic ones. It is easy, for example, to see that Lω1 provides a model for the axioms of SCS in which ω1 does not exist, so that the existence of the latter cannot be provable in SCS.

Each of these results contravenes the motivation under consideration. For P1 and P2, taken together, entail that classical logic is only applicable to absolute formulas; but Payne’s argument shows that by making P1 into a biconditional, we end up admitting LEM for arbitrary formulas φ that are not absolute. But the diﬃculties here cut deeper, in fact running right to the heart of what is special about the Universe-Indeterminist view and its diﬀerences from the Multiverse-Pluralist view: For it turns out that, in strengthening P1 or DET to biconditionals, one essentially reduces the view under consideration here to a variant of multiversism.

To see this, it helps to compare the Universe-Indeterminist and Multiverse-Pluralist views on truth and ontology. Natural approaches to determinacy and truth for Multiverse- Pluralists are broadly supervaluational in flavour: Universes are models V |= ZFC (or some similar theory), and φ is defined to be a multiverse truth when and only when V |= φ for all V ∈ M, where M is a collection of models (the multiverse). The basic syntactic unit of interest for multiverse truth is the sentence, and a sentence φ is determinate when it or its negation is a multiverse truth, but indeterminate otherwise. Now, suppose we take the collection M to be the collection of standard transitive models of ZFC. Then a sentence φ will be determinate if and only if it has the same truth-value in all standard transitive models. This conception of determinacy is exactly the one we would get were we to strengthen P1 to a biconditional in our motivation for SCS. In this sense, then, adopting the converse of P1 as a sufficiency condition reduces the view under consideration to a species of multiversism.

The Universe-Indeterminist has an altogether diﬀerent approach to the metaphysics of set theory, and that diﬀerence manifests itself in her approach to truth and determinacy. As there is only one universe of sets for the Universe-Indeterminist, that is, unicity of truthmakers is

18 important. On the multiverse picture, a theorem of ZFC – the existence of ω1, say – was a determinate truth because it held in all universes of the multiverse. The fact that the entities satisfying the definite description “least uncountable ordinal” differ in extension from universe to universe is, on a multiverse metaphysics, not particularly troubling; after all, the entities in question are envisaged as belonging to wholly distinct universes. But for the Universe-Indeterminist, if ω1 exists it must be unique; for there can only be one least uncountable ordinal in the universe V. In contrast to the Multiverse-Pluralist, then, the plenitude of differing witnesses for this existential claim in legitimate interpretations inhibits her taking it as a determinate truth.

The theoremhood of a proposition φ in ZFC should not be sufficient, then, to secure the determinacy of that proposition for the Universe-Indeterminist. From one point of view, this makes perfect sense; after all, in the previous section we pointed out that material equivalence of all legitimate interpretations of a sentence is often woefully inadequate for securing the determinacy of sense of the statement in question, and accepting this sufficiency condition in the case of set theory seems tantamount to accepting that a set-theoretic statement is determinate in sense whenever its legitimate interpretations are materially equivalent. Since determinacy of sense is more fine grained than material equivalence, it is natural to look for a more fine grained sufficiency condition for determinacy of sense.

Our proposed rejection of theorems of ZFC as indeterminate in sense may nevertheless seem odd in so far as we have accepted that standard transitive models of ZFC are legitimate interpretations of set-theoretic propositions, and thus seem to have implicitly committed ourselves to the correctness of ZFC as a theory of sets. But in fact this is entirely in keeping with the Universe-Indeterminist perspective. On that view, one accepts that standard set theories are motivated by the iterative conception and attempt to axiomatize the universe of sets; at the same time, however, the Universe-Indeterminist is of the view that the underlying conception is unclear, and that as a result not all concepts employed in articulation of the theory are determinate. Even theorems of ZFC which employ such concepts, then, may fail to be determinate truths of the universe if their formulation involves such indeterminate concepts.

We still need a sufficiency condition for determinacy of sense; to arrive at one, it helps to consider the pathological examples more closely. In both Payne’s case and the case of ω1’s existence, it is evident that although these expressions are absolute for standard transitive models of ZFC, the absoluteness of such expressions is only provable by assuming means that transcend those justified by the lights of P1 and P2. For instance, proving that (φ∨¬φ)∨¬(φ∨¬φ) is absolute where φ itself is not requires the unjustifiable assumption that φ ∨ ¬φ is provable, which of course we cannot help ourselves to in the present context. Similarly, to prove that “ω1 exists” is absolute, one simply has to prove the theorem – an exercise which requires the use of an axiom asserting the existence of P(ω), which we have already seen cannot be justified on the present account. This suggests that the natural sufficiency condition for a Universe-Indeterminist to take is one that captures exactly those formulas which are absolute for transitive models of ZFC, and which are provably such without going beyond means she can justify. In the present context, this means that they should be provably absolute for models of her favoured semi-constructive set theory; in the last subsection, we saw that SCS placed an outer bound on those means. Thus we arrive at the following sufficiency condition for determinacy of sense:

P3: A set-theoretic proposition is determinate in sense if it is absolute for models of ZFC, and its absoluteness is provable without transcending means available in SCS.

19 Since all models of ZFC are of course models of SCS, this deﬁnition will capture exactly those formulas which are both absolute for models of ZFC, and provably so without using means going beyond those justiﬁed by P1 and P2.

The following basic fact allows P3 to be sharpened somewhat.

T Theorem 3.2 (Feferman-Kreisel). φ is ∆1 iﬀ φ is absolute between end-extensions of models of T.

SCS The theorem entails that, given P3, we want to know if SCS proves LEM for all ∆1 formulas φ; at present, I have not seen deﬁnite results on this matter.26 However, we do have the following result, which suﬃces for present purposes:

+ 27 Theorem 3.3. SCS and SCS = SCS+∆1-LEM + ∆1-MP are of equal proof-theoretic strength.

Proof. (Sketch) This theorem is really an easy corollary of work done in (Feferman, 2010); there is an obvious interpretation of SCS in SCS + ∆1-LEM + ∆1-MP, and SCS + ∆1-LEM +∆1-MP can be interpreted in the ﬁnite type set theory FSC as in Theorem 2.1, since Feferman has shown how to prove a natural translation of ∆1-LEM in FSC and the method given for this translation is readily extended to ∆1-MP. Roughly, the translation of ∆1-LEM in FSC works by showing that translations of ∆1 formulas φ(x) are provably equivalent functional equations of the form f (x, s(x)) = 0 in FSC, where s(x) is a choice function, and since such equations are decidable in FSC (and MP holds for them)28 we get translations of each of these principles for ∆1 formulas in FSC. The theorem then follows easily using the interpretation of FSC in OST employed in theorem 2.129

SCS Given these facts, we can add ∆1 -LEM to SCS with equal justiﬁcation as that supplied for ACSet earlier; for we have both a demonstration that the resulting theory is interpretable in a constructively justiﬁable system, as well as the conceptual motivation given by the connection just outlined between determinacy of sense and absoluteness. Moreover, Michael Rathjen (Rathjen, 2016b) has recently proved the following theorem:

Theorem 3.4 (∆1 Principle:). If LEM is provable for φ in SCS, then φ is equivalent to a ∆1 formula and hence absolute for standard transitive models.

This shows that the theory of SCS+, which inherits this property and indeed strengthens it to a biconditional, is formally adequate to the Universe-Indeterminist’s position as outlined here; it is exactly the theory justiﬁed by P1, P2 and P3 in the sense that any more or less classical logic would violate one or other of these principles.

26In (Rathjen, 2016b), Rathjen presently has this result of SCS together with global choice. The status of the full principle seems at present unclear. 27 By ∆1-LEM and -MP, I mean exactly the schemas given as semi-logical axioms (b) and (c) but with φ extended to range over SCS-provable ∆1 formulas. 28In the context of FSC, this means that ¬¬∃x f (x) = 0 → ∃x f (x) = 0. 29 That FSC proves a functional translation of ∆1-LEM is actually found in some slides for a talk Feferman gave in 2012; see his website for details. Also, and perhaps more importantly, there is an ambiguity in my use of FSC. Here, I am using it to refer to the system in 2 types over V for the translations; this system is interpreted in full FSC, which Feferman denotes FSC ↑, by using a variant of the Gödel Dialectica interpretation. The latter theory is the one Feferman shows can be interpreted in OST in (Feferman, 2010).

20 It is time to flesh out the connection between SCS+ and Universe-Indeterminism. In SCS+, we have that where ∀x[φ(x) ∨ ¬φ(x)] is used in the proof of some sentence ψ in SCS, then φ(x) is absolute for standard transitive models and hence of determinate sense. The following, then, seems like a coherent line of thought by which a Universe-Indeterminist might set out their position in detail. They hold that there is a unique universe of sets V, and that statements about V fall into one of three categories: those which can be demonstrated only using classical logic for expressions that have a determinate sense, and so single out a determinate set-theoretic property; those that can be refuted in this way; and those which fall into neither of the previous two categories. The first two are what she takes for the determinate truths and falsehoods of V, respectively; sentences that fall into neither category are indeterminate. SCS+ offers a formal theory, justified by her principles, of where the line between determinate and indeterminate should lie: theorems of SCS+ being the determinate truths, anti-theorems determinate falsehoods, and neither theorems nor anti-theorems those that are indeterminate. The formal means available to the Universe-Indeterminist allow them to be quite precise about their position here: we’ll soon see that on this view of set-theoretic truth and ontology, for example, neither CH, nor ¬CH, nor CH ∨ ¬CH will be determinate set-theoretic truths.

In the next and ﬁnal section, we will explore in more detail boundary between determinate and indeterminate that arises on the Universe-Indeterminist’s conception, and discuss the philosophical merits of the emerging position.

4 The Universe-Indeterminist View

With P1 and P3 in tow, we now have necessary and suﬃcient conditions for set-theoretic determinacy of sense; and now that we have such conditions, we are in a position to explore the emerging boundary between determinate and indeterminate in set theory.

The view we are considering has it that a set-theoretic statement φ is determinately true when φ is a theorem of SCS+; determinately false when it is an anti-theorem; and indeterminate otherwise. Thus, it explicates determinacy in terms of provability or refutability in SCS+. Interestingly, the substance of Feferman’s original view – that ω was a definite totality to which classical logic was applicable, but P(ω) was not – is vindicated on the present account: for we saw in the last section that on the present motivation for the Universe-Indeterminist position, we can derive an explicit justification both for the inclusion of the axiom of infinity, and the exclusion of the powerset axiom. At the same time, a host of other cases are decided with regards their determinacy. Being a bijection, an injection or a one-one function are all predicates given by ∆0 formulas and hence determinate; classical logic is therefore applicable when reasoning about such predicates, e.g. we may assume all functions are either bijections or not. Other predicates we may demonstrate to be absolute include being the transitive closure of a set, or being the product or sum of two ordinals; these latter require some work, since they cannot be given + explicitly by ∆0 formulas, but the work can be done within SCS making use of Theorem 2.2. Parts of generalized recursion theory and second-order arithmetic also rank among the theorems of SCS+; how much of each is presently unknown, though some harder work could in principle settle the question.30 On the other hand, predicates such as having a given (infinite) cardinality

30 One can show, e.g., that all theorems of ACA0 are theorems of SCS without too much work; since arithmetical

21 or coﬁnality and being the powerset of another (inﬁnite) set are non-absolute predicates, for which classical principles cannot be presumed to apply on the Universe-Indeterminist position; we shall see a result of these failures in a short while when we come to Theorem 4.1.

Thus a sharper boundary emerges between those statements that are and those that are not determinate in sense; on the Universe-Indeterminist view, we are able to justify the particular axioms included, but also those excluded from our set theory of choice, SCS+. The sharpness of this division ought to inspire hope that the Universe-Indeterminist position can contribute seriously to debate on set-theoretic indeterminacy. If one accepts P1, P2, and P3 we shall have clear principles for the demarcation of the determinate from the indeterminate in set theory; it will turn out that, for instance, that CH is indeterminate on this view.

We are now in a position to take a closer look at the Universe-Indeterminist position on set-theoretic ontology. According to the iterative conception of set, the universe V of all sets th comes in stages Vα for each ordinal α; the α stage of the hierarchy is recursively defined by S Vα = β<α P(Vβ). The powerset operator is thus essential to the development of the iterative universe V. Many commentators have argued, though, that the stipulation that the powerset of an infinite set contain all subsets of that set might be underdetermined (as in (Devlin, 1993, 120-21)) or vague (as in (Väänänen, 2014), section 3.2); indeed, on the present conception of determinacy, this recourse to absolutely general universal quantification renders the powerset 31 predicate inherently indeterminate, since the definition of a set’s powerset is Π1. If one endorses such arguments, and takes the powerset operator to be indeterminate, then which sets get formed at each level of the hierarchy will again be indeterminate; it will be indeterminate what holds and what does not in the universe V. Since, however, absolute formulas keep the same truth-values in all standard transitive models, their truth or falsity remains quite unaffected by such indeterminacy; however the powerset operation is determined, the truth-values of absolute propositions will remain constant. As in SCS+ one endorses classical methods of reasoning for such formulas and attempts to prohibit it otherwise, one can argue that theorems of that system are the determinate truths of the envisaged indeterminate universe. The theory of SCS+, that is, can be viewed as characterizing the universe as it must be, or as a collection of sentences to be read as the determinate core of truths about the indeterminate universe V. 32

We now consider two objections to the emerging view in order to clarify it. The ﬁrst objection is that a theory like SCS+ is ill-suited to its purported role as a theory of the determinate core of truth in an indeterminate universe, since the resulting determinate core would be extremely meagre. SCS+ cannot prove the existence of uncountable sets, so that the existence of e.g. ω1 will not rank among the determinate set-theoretic truths on the present account; one might naturally argue that this was a rejection of classical set theory tout court, amounting to a wholesale abandonment of classical set-theoretic mathematics both in its properly mathemati- formulas are ∆0, the arithmetical comprehension axiom is essentially just an instance of ∆0-Sep. The exact amount of second-order arithmetic that holds in various semi-constructive set theories is an interesting open question; further work here is needed. 31Feferman himself seems to express a very similar sentiment here (Feferman, 2014, 8), but the indeterminacy in the unrestricted all in the deﬁnition of powersets is not cashed out in terms of a failure of absoluteness. 32It is worth remarking that I am here viewing SCS+ as a theory in the logical sense - a collection of sentences - rather than as a theory in the (non-algebraic) mathematical sense with an intended semantic interpretation. The reasons for this should emerge as we continue, but roughly I think that the theory in the mathematical sense with which we are concerned is ZFC; only some concepts employed there are, in Feferman’s sense, vague, and the theory of SCS+ results from obviating vague concepts in deductions.

22 cal and foundational guises. This charge can be resisted; on the one hand, the minimality of the determinate core oﬀers foundational advantages, while on the other fully classical methods of set theory are necessary for acquiring determinacy results in SCS+, and hence are essential to its philosophical signiﬁcance. Consider by way of example the following

Theorem 4.1 (Rathjen, 2016). The theory T = SCS + P(ω) exists 0 CH ∨ ¬CH

Proof. (Sketch!) It is known that SCS has a realizability model in L[U], the constructible L[U] universe relative to an arbitrary predicate U; the realizers are in the (codes for) Σ1 partial L[U] recursive functions, which are uniformized by a Σ1 predicate in L[U]. Let M |= ZFC + ¬CH, and code P(ω)M into a set of ordinals A so that P(ω)L[A] = P(ω)M. Suppose

(*) T ` CH ∨ ¬CH

Then by the realizability theorem, one of the disjuncts is realized; by choice of L[A] it must be ¬CH. Working in ZFC, force from M to M[G] |= ZFC + CH where P(ω)M = P(ω)M[G]. Then, M in M[G], there is a bijection from P(ω) to ℵ1; code this bijection into a set of (suﬃciently large)33 ordinals B, so that L[A∪B] |= CH. On the assumption (*) plus the realizability theorem, we infer that L[A ∪ B] realizes CH. However, using various properties of the constructible universe/ the realizability theorem, we can also show L[A ∪ B] retains a realizer for ¬CH. This yields the desired contradiction; we can derive that the ﬁrst value of the realizer for CH ∨ ¬CH is both 0 and 1 on the basis of the foregoing. Whence (*) has been reduced to absurdity.

The above sketch is extremely sketchy – in full, the proof runs to several pages. Neverthe- less, it suffices to demonstrate the utility of classical set-theoretic methods in attaining determinacy results in extensions of SCS (including SCS+). Classical forcing techniques carried out in full ZFC are essential to the result; it is the existence of the model M of ZFC, as well as its forcing extension M[G] guaranteed to exist on the basis of those axioms (given M) that allow us to infer a contradiction from the assumption (*). In the proof, we also assume that P(ω) is a set, again transcending methods strictly justifiable on the present account. However, these temporary assumptions can be justified as a method of gleaning positive facts about the relative determinacy of given statements. If, for instance, the following

Conjecture 4.2. T ` PD ∨ ¬PD

could be ratiﬁed, there would be a clear sense in which PD were “more” determinate than CH, since its determinacy could be proved from a weaker theory. Thus, exploring the full range of set-theoretic possibilities in classical set theories ought not to be taken as a project at odds with accepting arguments of the kind outlined above; rather, pushing the boundaries of indeterminate set-theoretic concepts like powerset and cardinality can nevertheless yield positive facts about them, which we of necessity study in classical ZFC.

We need not, then, repudiate the study of classical set theory. And indeed, we can on the present account positively encourage the exploration of the higher inﬁnite; relative consistency

33 M Basic properties of the constructible universe tell us there is an ordinal % such that L[A]% contains P(ω) , A and the realizer for ¬CH. All ordinals in B must be above %.

23 results gleaned in higher set theory, after all, will form the basis for relative determinacy results in the spirit of Theorem 4.1. The same sort of liberalism offered by a Multiverse Pluralist such as Hamkins, who (rightly!) encourages the study of the full diversity of set-theoretic possibilities, can thus also be offered on the present account. Indeed, in the same spirit, the fact that there is such a minimal core of determinate set-theoretic truths on the present account can even be understood as an advantage. On the Universe-Determinist account, according to which there is a maximal universe of sets in which all questions are answered, when one wishes to employ consequences of GCH in one context, and ¬CH in another, one must view oneself as working in a “conceited” model; one in which one is working under presumptions that are strictly and literally false, if useful to a certain degree. On the Universe-Indeterminist view, this is unnecessary; each of GCH and ¬CH are consistent with the theory of SCS+, and might be viewed, much as they are on the multiverse account, as equally legitimate ways of filling in the gaps, so to speak, around that determinate core; they are equally legitimate though incompatible ways the universe might be. Thus, the study of classical set theory, far from being repudiated, can be encouraged on the present account.

The second objection we will consider is that any apparent improvement offered by our alternative motivation, which motivates SCS+ in terms of a restriction of classical logic to expressions of determinate sense, is really illusory since it rests on the STMH, the hypothesis that the legitimate interpretations of a set-theoretic statement were all and only the standard transitive models. We saw in section 3.2 that other positions could readily be motivated by taking different stances on the relevant class of legitimate interpretations. For instance, one could reasonably maintain that the standard transitive model constraint was too weak; after all, if we wish to describe the set-theoretic universe V, surely the least we can require is that an interpretation be of the form Vκ, an initial segment of the universe? The powerset operator turns out to be absolute for such structures, so that the picture we get of set-theoretic indeterminacy is radically different; we should, on such an account, wind up with CH being a determinate problem.34 Moreover, there are other plausible positions too, as we saw in section 3.2 The natural objection is that I have not really made the debate any more precise or tractable that it was before, but have merely pushed the bump in the rug back to the question of which the legitimate interpretations are.

I do not wish to give the impression I think the question of which the legitimate interpretations of a set-theoretic statement are is an easy one; but I do think the debate is more tractable than that concerning the definiteness or otherwise of a given totality. For instance, against the above objector, one can point to the acceptance of Cohen’s countable transitive models of ¬CH and ¬AC; these have been accepted by practicing set-theorists as non-deviant, despite their being countable structures. Like arguments can be offered against those who suggest transi- tivity or non-well-foundedness are unimportant; indeed, non-well-founded models of ¬AC had been known since the late nineteenth century and disregarded as unimportant. Likewise, the preference for models of the first-order axioms of ZFC is supposed to reflect the practicing set-theorists preference for the same. The suggestion here is that the truth or otherwise of principles like the STMH will be resolved by examining set-theoretic practice itself; by examining set-theoretic practice, that is, we can hope to isolate which features of models render them legitimate objects of study for set-theorists. The STMH offers a broad selection of such interpretations, which is plausible in that it can account for the extreme diversity of models studied by

34See (Rumﬁtt, forthcoming) for an argument here, based on Kreisel’s (Kreisel, 1967)

24 set-theorists - from deﬁnable inner models and forcing extensions to universes containing large cardinals - while at the same time respecting the set-theorist’s preference for well-founded and transitive structures.

Of course the meagre evidence provided here does not entail the STMH is correct; but it does show the sort of evidence that might be offered in support of it, evidence of a kind that does not seem available with respect indefiniteness debates. And, at the very least, it seems that the debate that might arise over principles like the P1, P2 and P3 will not descend into the debate concerning the determinacy of CH and other independent problems which originally motivated our enquiry. Working out whether P3 is true, for instance, will require a closer examination of the plausibility of the Universe-Indeterminist’s view on ontology, together with an assessment of the STMH and DET; assessing the STMH will, in turn, involve assessment of set-theoretic practice. This latter point is also important, since it shows that sort of evidence to which against which the proposed principles must be judged will be properly set-theoretic. In approaching the problems thrown up by the independence phenomenon in set theory, ultimately the data to which a philosopher must turn will have to come from set-theoretic practice itself; in motivating the axioms of SCS+ and our view on set-theoretic indeterminacy from an argument based on the STMH, the position presented here satisfies this requirement.

5 Conclusion

Let me review what I take to be the main points of the paper. First, I have developed a view I call Universe-Indeterminism, according to which there is a unique universe of sets that does not determine truth-values for all set-theoretic propositions. Truth in this universe is axiomatized by the system SCS+, which does not have unrestricted LEM. For the Universe-Indeterminist, the range of its unbounded universal quantiﬁers is the indeterminate universe; since the universe is indeterminate, unbounded quantiﬁcational statements are not treated classically; we do not hold φ ∨ ¬φ for arbitrary claims of this kind. We have seen that in SCS+, CH is a concrete example of such an indeterminate statement.

We have also seen that the view differs from both the Universe-Determinist and Multiverse- Pluralist positions. In particular, it differs from the former in rejecting the view that all statements of set theory are bivalent; nevertheless, it agrees with it that there is a unique universe of sets described by standard set theories. It differs from the latter, on the other hand, by rejecting the view that there are many universes described by standard set theories, though it agrees that some set-theoretic statements do not have truth-values. One might initially worry that the dispute between the Universe-Indeterminist and the Multiverse-Pluralist was termi- nological; don’t they both reject determinacy in truth-value because of the plethora of interpretations of set-theoretic statements? This worry is revealed to be misguided by considerations from section 3.3, where we saw that the ontological differences between the Multiverse- Pluralist and the Universe-Indeterminist were manifested in a first-order difference of opin- ion concerning the existence of uncountable sets. The Multiverse-Pluralist’s supervaluational tendencies led them to accept the existence of ω1 as a determinate truth of the multiverse, whilst the Universe-Indeterminist was led to reject the claim that ω1 exists as indeterminate. Thus, the Universe-Indeterminist’s position leads to a scepticism about uncountability that the

25 Multiverse-Pluralist will not in general subscribe to.

The Universe-Indeterminist view oﬀers, I think, an intriguing new possibility on the ontology of set theory, cutting across the traditional Universe-Determinist / Multiverse-Pluralist divide. It is important to recognize that there are such views; that commitment to the existence of a unique universe of sets described by axiomatic set theories and studied by practicing set theorists does not thereby commit one to holding that all set-theoretic propositions have determinate truth-values. Perhaps instead the logic of truth for the universe of sets is non-classical, in something like the way described here. One upshot of this paper, then, is that one need not accept a multiverse ontology if one wants to resist the claim that all set-theoretic propositions are bivalent.

Acknowledgements

I would like to thank Walter Dean, Hartry Field, Nick Jones, Alex Paseau and Jonathan Payne for their criticisms and suggestions. Special mention is due to Neil Barton, Ian Rumﬁtt, and Scott Sturgeon; without their encouragement this paper wouldn’t exist, and without their com- ments it would not be a fraction as good.

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