The Continuum Hypothesis in Algebraic Set Theory
T. P. Kusalik
Department of Mathematics and Statistics McGill University Montreal,Quebec
December 31, 2008
A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Master of Science
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•*• Canada Acknowledgements
I first must thank my supervisor. Michael Makkai, for helpful suggestions, an swers to my many questions, and perseverance in reading my sometimes lengthy drafts. I thank Robert Seely and the Montreal Category Theory Research Cen ter seminar for allowing me to give a talk on a preliminary version of this thesis, and giving me much helpful feedback. I thank the organizing committee of the Category Theory 2008 conference for providing me with a chance to present a more complete version of my research, the Department of Math and Stats for providing me with the funding to attend this conference, and the many researchers (including Jeff Egger, Peter Johnstone, and Peter Lumsdaine) who gave me helpful suggestions while there. I thank the women of the Math and Stats office for much-needed administrative support, and spare keys to my of fice when needed. I thank NSERC for funding my degree, and AGSEM for fighting for my right to get paid for my work. I thank my family, friends, and lovers for helping me maintain my sanity. And, last, but certainly not least, I thank the poor and repressed people who make up the majority of this world for the hard, underpaid work they have put in to allow us Westerners to spend our lives engaged in academic research.
1 Abstract
In [23], Lawvere and Tierney proved the consistency of the negation of the continuum hypothesis with the theory of Set-like toposes. In this thesis, I generalize the Lawvere-Tierney result in two directions. Lawvere and Tierney's result relies upon the law of excluded middle and the axiom of choice, and I provide a formulation and proof of the consistency of the negation of the continuum hypothesis which abandons this assumption. Moreover, I generalize the work that's been done on the continuum hypothesis and its consistency from the context of topos theory presented in the Lawvere-Tierney proof to the context of algebraic set theory.
Abrege
Dans [23]. Lawvere et Tierney ont demontre la compatibilite de la negation de l'hypothese du continu a.vec la theorie des topos qui ressemblent au Set. Dans cette these, j'universalise le resultat de Lawvere-Tierney dans deux directions. Le resultat de Lawvere-Tierney compte sur le principe du tiers exclu et Faxiome du choix. et je fournis une formulation et une demonstration de la consistance de la negation de l'hypothese du continu qui abandonne cette assomption. Aussi, j'universalise tous ces resultats sur l'hypothese du continu et sa consistance de la contexte de la theorie du topos a la contexte de la theorie algebrique des ensembles.
2 Contents
1 Introduction 6 1.1 History 6 1.2 Generalizing the Lawvere-Tierney Result 7 1.3 Outline 9
2 Heyting Pretoposes 11 2.1 Axioms of a Heyting Pretopos 12 2.2 Internal Language of a Heyting Pretopos 15 2.2.1 Kripke-Joyal Semantics 16 2.2.2 Informal Conventions 18 2.3 Elementary Toposes 19 2.3.1 The Topos of Sets 20 2.4 Consistency Proofs 22 2.5 Construction of New Heyting Pretoposes 24 2.5.1 Pretoposes within Set Theory 24 2.5.2 Internal Functor Categories 24 2.5.3 Booleanization 32 2.5.4 Preservation of Nondegeneracy 36
3 Algebraic Set Theory 38 3.1 Motivation for the Notion of Category of Classes 39 3.2 Axiomatic Description of a Category of Classes 40
3 3.3 Cosmological and Insular Categories of Classes 44 3.4 Set Theory in a Cosmological Category of Classes 46 3.5 The Topos of Small Objects 49 3.6 Constructing new Categories of Classes 52 3.6.1 Preservation of (Univ) 55
4 Cardinality 59 4.1 The Cantor-Bernstein Theorem 60 4.2 Mono-cardinality and Epi-cardinality 62 4.3 Internal and External language 63 4.4 Combining Epi- and Mono- Cardinality 65 4.5 Relationship with Excluded Middle and the Axiom of Choice . . 67 4.6 Cantor's Theorem 68 4.6.1 A counterexample 70
5 The Continuum Hypothesis 72 5.1 Subquotients and PERs 73 5.2 Formulating (CH) and (NCH) 76 5.3 Different choices for R 77 5.3.1 Real Numbers as Dedekind Cuts 80 5.3.2 Real Numbers as Cauchy Sequences 81 5.4 The Consistency of the Continuum Hypothesis 82
6 The Consistency of (NCHP(N)) 84 6.1 Cohen's Proof 85 6.2 Boolean Toposes with Choice 88 6.3 Connection with Classifying Toposes 91 6.4 Generalization to an Arbitrary Topos 92 6.4.1 Preservation of Cardinal Inequalities 94 6.5 Generalization to Algebraic Set Theory 96 6.5.1 Remarks on the Generality of this Construction 97
4 7 The Consistency of (NCHM) 99 7.1 The Poset P and the map g 100 7.1.1 Definitions of P and g 100
7.1.2 g Factors Through Rm 102 7.1.3 g is Mono 104 7.2 Construction of a Natural-Distinguishable object A 106
7.3 The Case of Rd 107 7.4 Generalization to Algebraic Set Theory 108
8 Consequences and Corollaries 109 8.1 The Diagonal Functor 109
8.2 The Construction for (NCHP(N)) 116 8.2.1 Church's Thesis 117 8.2.2 The Cauchy Reals 119
8.3 The Construction for (CHKJ 119 8.4 Conclusion 122
5 Chapter 1
Introduction
1.1 History
The continuum hypothesis, or (CH), is one of the oldest undecidable questions of modern set theory. Georg Cantor formulated it originally as a "hypothesis" to either be proved or disproved, although couldn't arrive at either a proof or disproof. In 1939, Godel [8] proved, using his method of "constructible sets", that the continuum hypothesis was consistent with the axioms of standard ZFC set theory. However, it wasn't until 1963 that it was finally proven, by Paul Cohen [6, 7], using a novel method of "forcing", that the negation of the continuum hypothesis, or (NCH), is also consistent with the axioms of ZFC, and thus that the continuum hypothesis is undecidable. The notion of "topos" was originally defined by Grothendieck [l] as part of a project of research into the foundations of algebraic geometry. The notion of "Grothendieck topos" is a sort of "category of sheaves" appropriate for alge braic geometry which generalizes the notion of the category of sheaves over a topological space. Lawvere [19, 20] and Tierney soon discovered that a topos in fact captures many of the same properties that are found in the category of sets, and formulated the notion of "elementary topos" to capture all such categories which can form a suitable context for higher-order mathematical
6 constructions. Lawvere and Tierney [23] noticed that, when the universe of sets is replaced with a suitable topos (a Boolean topos with choice), Cohen's "forcing" technique simply becomes an internal version of Grothendieck's sheaf construction. "Forcing"-style consistency proofs become much more elegant in this topos-theoretic context. 30 years later, the question of how to develop set theory in a category- theoretic context was revisited by Joyal and Moerdijk in [15]. They discovered that, in a category with certain structure (known as a "category of classes"), an object V of this category which behaves like the universe of sets can be defined algebraically. This led to the founding of the field of "algebraic set theory". While much of the research in algebraic set theory is focused upon studying the universe-of-sets object V in cases in which the standard set theoretic assump tions are substantially weakened, for me, the prime importance of algebraic set theory is that it provides a categorical framework to carry our constructions requiring a stronger set-theoretic structure than topos theory. The main dif ference is that in algebraic set theory, unlike in topos theory, there exists an object of the category of classes corresponding to the universe of sets, and thus properties of the universe of sets can then be formulated in an internal way in the category in question.
1.2 Generalizing the Lawvere-Tierney Result
Lawvere and Tierney's main result in [23] is the construction, from a given (nondegenerate) Boolean topos with choice (actually, an additional condition of "two-valuedness" is assumed, but is not essential to the proof, so I will disregard it in this thesis), a new (nondegenerate) Boolean topos with choice satisfying (NCH). This then shows that (NCH) is relatively consistent with the theory of Boolean toposes with choice. My goal in this thesis is to generalize the Lawvere-Tierney proof in two directions. I will first prove the relative consistency of (NCH) with the theory of elementary toposes (where neither
7 the law of excluded middle nor the axiom of choice is assumed), and then with a strong version of algebraic set theory. Both these proofs will use modified versions of the internal sheaf constructions used by Lawvere and Tierney. One of the reasons that topos theory is an appropriate setting for a rela tive consistency pr6of is that the theory of elementary toposes with natural numbers object is in some sense "the same" as higher-order intuitionistic logic with natural numbers type (to be more precise, toposes are interpretations of higher-order intuitionistic logic - for more on this connection, see [17]). In par ticular, this means that the proof of the relative consistency of (NCH) with topos theory proves the relative consistency of (NCH) with higher-order in tuitionistic logic. Algebraic set theory is not known to be equivalent to some logical theory in the same way, however, it also forms an appropriate setting for consistency proofs as a category of classes is somehow the minimal extension of a topos which allows for quantification over the class of all sets. For more on what is meant by a "consistency proof" in these contexts, see section 2.4. One may wonder to what extent this result is "new", and not just a corollary of Cohen's result. Both topos theory and algebraic set theory are strictly weaker than classical ZFC set theory, and thus if (NCH) is consistent with ZFC, then it is also consistent with topos theory and algebraic set theory. However, Cohen's result is only a relative consistency result in that it only shows that if ZFC is consistent, then ZFC + (NCH) is consistent. Similarly, the Lawvere- Tierney result assumes the consistency of the theory of Boolean toposes with choice. While the consistency of topos theory implies the consistency of the law of excluded middle, we have no proof that it implies the consistency of the axiom of choice, thus assuming either the set-theoretic or topos-theoretic axiom of choice is an additional assumption. The proof given here, unlike Cohen's (or Lawvere and Tierney's). makes only the assumption of the consistency of topos theory itself. In the case of algebraic set theory, it turns out that the consistency of the strong version of algebraic set theory that I give here in fact implies the
8 consistency of ZFC, and thus a relative consistency proof for algebraic set theory is in fact a corollary of Cohen's result. However, even if my main results could have been proven without my work, my approach is still useful in two other ways. Firstly, my proof is a direct proof of relative consistency in algebraic set theory, that uses only categorical notions, rather than a proof that uses the full machinery of ZFC implicit in Cohen's proof. This will be an advantage if one wishes to generalize my methods to prove the relative consistency of statements which, unlike (NCH), may not be consistent with ZFC. Secondly, my proof uses only a "presheaf" construction, rather than the "sheaf" construction used in the Lawvere-Tierney proof, which means that it will preserve certain properties of the base category which may not necessarily be preserved in the Lawvere-Tierney proof. This means that if we put certain conditions on our topos or category of classes, we can produce a new topos or category of classes which satisfies (NCH) and still satisfies the same conditions. This is an important fact that cannot be reproduced in the Lawvere-Tierney method.
1.3 Outline
I begin this thesis with a chapter on Heyting pretoposes, a class of of category which includes both toposes and categories of classes. This chapter will mostly be concerned with defining terms which will be used throughout the thesis, and proving some facts about the general constructions that I will use. Topos theory is also treated briefly in this chapter, but is not described in depth, as it should be familiar to many readers. The next chapter deals with the definition of the framework of algebraic set theory that will be used throughout the thesis. I define a notion of "eos- mological category of classes" to denote a category which contains an object behaving like the universe of sets, and a notion of "strong category of classes" such that both elementary toposes and cosmological categories of classes turn
9 out to be strong categories of classes. I also prove a number of basic theorems about strong and cosmological categories of classes that I will use later. The third and fourth chapters deal with the definitions and basic theorems necessary for formulating the Continuum Hypothesis and its negation in a strong category of classes. Since the logic of a strong category of classes is in general only intuitionistic, notions which are easy to define in classical set theory become much more complicated. The third chapter deals with the definitions and theorems basic to the notion of cardinality, and the fourth chapter uses this notion of cardinality to formulate the Continuum Hypothesis and its negation. The fifth and sixth chapters contain the presentation of my main proofs. The fifth chapter deals with Cohen's proof and Lawvere and Tierney's refor mulation, and the direct generalization of such a proof to the more general contexts of an elementary topos or a cosmological category of classes. How ever, due to reasons discussed in the fourth chapter, the version of (NCH) proven to hold in this chapter does not cause the continuum hypothesis to fail in as strong a way as we'd like. Thus, the sixth chapter repeats the same proof in a somewhat different context to yield a proof of the consistency of a stronger version of (NCH). The seventh chapter will deal with preservation of certain properties under presheaf constructions; It turns out that a large class of internal language for mulas will hold in a strong category of classes £C"P exactly when it holds in £. Since the greater part of the proofs in chapters five and six use only inter nal functor category ("presheaf") constructions, it follows that the continuum hypothesis can be negated while preserving certain structures. This leads to relative consistency results for axiom systems incompatible with ZFC, which thus cannot be corollaries of Cohen's result. Also, the preservation of certain structures under presheaf constructions will imply that there are certain rel ative consistency results which cannot be proven using the methods of this thesis.
10 Chapter 2
Heyting Pretoposes
The categories occurring in both topos theory and algebraic set theory are what are known as Heyting pretoposes. A Heyting pretopos is a "category which has enough structure that intuitionistic typed first-order logic can be interpreted within it". This means that, in some sense, the objects of a Heyting pretopos will correspond to "types" and the arrows will correspond to "terms" in such a way that first-order statements "about" terms can be translated into properties of the category in question. Each variable will take a type, and we can think of arrows into an object X as terms of type X. Formulas
Xi, X2:... ,xn of types X\, X2...., Xn respectively will correspond to subobjects of the object Xi x X% x ... x Xn. Before explaining how this identification of terms with arrows and formulas with subobjects will work. I will have to define the categorical structure of a Heyting pretopos. The first section of this chapter will define the notion of "Heyting pretopos" which will be used throughout this thesis. The second will show how the cat egorical structure can be used to interpret logic through a mechanism known as the "internal language", and will present the (Kripke-Joyal) semantics for this language. The third section will deal with a distinguished class of Heyt ing pretoposes, the elementary toposes, and will make some definitions to be used later. The next section will deal with a definition of what is meant by
11 "consistency" in a pretopos framework, and will explain the general strategy used in my consistency proofs. The final section is concerned with the internal functor category and Booleanization constructions that I will use to create new Heyting pretoposes from old ones. These constructions will be defined, and some preliminary results about them will be proved.
2.1 Axioms of a Heyting Pretopos
In order to allow logic to be interpreted within a category, the category in question will need to satisfy certain axioms. The first set of conditions on a category £ will define what is known as a "coherent category" (referred to in [21] as "logical category"), which will allow us to interpret the logical symbols T, J., =, A, V, 3.
(Cl) £ has all finite limits
This axiom allows us to construct the conjunction of two subobjects as their pullback and the equality relation as the diagonal map A—> A x A. Since we now have all pullbacks by axiom (Cl), we will denote the pullback f of a subobject S of A along a map B >- A by f*(S).
(C2) Every object A has a minimal subobject J-A, and for any two subobjects S, R of a given object A, there exists an object S V R which is the smallest subobject of A containing both S and R. Moreover, these constructions are stable under pullback: for any arrow f : B—> A, /*(J_^) = _LB and r(SVR) = f*(S)Vf*(R).
This axiom, of course, is the one that defines the logical operation V, and the logical constant _L.
(C3) For any map A *-B, there exists a factorization of f A—:—^C >• B where im(/) is mono and f is surjective in that the only
12 subobject of C through which f factors is C itself. Moreover, these images are stable under pullback: for any g : D—>• B,
This axiom allows us to define existential quantification. Whenever we have a subobject S of A x B, we can define 3a
Definition 2.1.1. We call any category satisfying axioms (Cl); (C2); (C3) a "coherent category".
Of course a coherent category doesn't allow us to interpret all of first-order logic, but only the fragment consisting of T, _l_, A, V, =, 3, known as "coherent logic". If we want to interpret all of first-order logic, we need an additional axiom: f (V) For any map A >- B and any subobject S of A, there exists a subobject R = Vf(S) of B which is the largest subobject of B whose pullback f*(R) factors through S.
This axiom of course defines the universal quantifier as the operation which sends a subobject S of A x B to the subobject ya(£A(S) = V7r2(S') of B. ==>• , -i and <^=^ can all be defined using this universal quantifier. For details, see [21]. Now, the axioms (CI), (C2), (C3), (V), define only what is known as a "Heyting category", not a "Heyting pretopos". In order to make the category in question into a pretopos, we need to add additional structure.
(PI) 8 has finite disjoint coproducts. This means that there exists an initial object 0 and, for any pair A, B of objects, a coproduct A + B such that the coproduct injections pull back to yield the square below:
0 ^A
h
W w B ^A + B 12
13 To define the next axiom, we need to define, categorically, what it means for R>—^A x A to be an "equivalence relation". I will not do so here, but refer the reader to ([21], p. 117) or ([13], p. 16) for a definition.
(P2) For any equivalence relation R I A in £, R has a coequalizer whose kernel pair is R, itself.
Definition 2.1.2. A "Heyting pretopos" is a category satisfying (Cl), (C2); (C3), (V), (PI), and (P2).
All of the categories considered in this thesis will be Heyting pretoposes. For brevity's sake, we will often say "pretopos", when we mean "Heyting pretopos". Grothendieck's original definition of "pretopos" in [1] was rather different than the one above (taken from [21]).
Definition 2.1.3 (Grothendieck). A "pretopos" is a category £ such that:
1. £ has all finite limits.
2. £ has finite disjoint coproducts which are stable under pullback.
3. Every epimorphism is the coequalizer of its kernel parr.
4- Every equivalence relation has a coequalizer and is the kernel pair of its coequalizer.
It can be easily see that a category £ is a "Heyting pretopos" in our sense exactly when it is a "pretopos" in Grothendieck's sense and, in addition, sat isfies (V). For a proof see [21]. One thing to note about all of the axioms above is that they are all preserved by taking slice categories. Given an object C of a category £, if £ is a coherent category, a Heyting category or a Heyting pretopos, then so is the slice category £/C. This will turn out to be useful when discussing internal functor categories later.
14 The reader may wonder why we use a Heyting pretopos as our categorical framework rather than just a Heyting category. While most of the results of this chapter will in fact not require axioms (PI) and (P2), an important result of the next chapter will. It will turn out that the existence of quotients of equivalence relations will be accessary (but of course not sufficient) for the preservation of the category-theoretic structures we are considering under sheaf constructions in theorem 3.6.1.
2.2 Internal Language of a Heyting Pretopos
The structure of a Heyting pretopos is such that each one comes equipped with an "internal language" - a theory in intuitionistic typed first-order logic - which allows us to talk about the objects and arrows of the category in question as if they were types and terms. The "types" of the internal language consist of all objects of the Heyting pretopos, and any variable must be assigned a type. From now on, I will use y, x\, x^, • • • :xn,... to denote variables and y, Xi, X2,..., Xn their respective types. I first define a notion of "term" and then a notion of formula. The terms of type A correspond to arrows into A For any arrow / : X\ x .. . x Xn —>• A r we have a term /~'(x-1.... , xn) (the "name" of /) of type A with free vari ables {xi,.... xn}. Conversely, given a term T{X\. .... xn) of type A with free variables {xi,..., xn}, we have a corresponding arrow ||r(xi,...,x„)|| :
Xi x ... x Xn—^ A
Formulas with free variables {xi,.... xn} correspond to subobjects of X\ x
... x Xn. Given a subobject, S ^->- Xi x ... x Xn, we have a corresponding r n formula S (x\...., xn), and corresponding to every formula 4>(x\..... xn). we have an "interpretation" ||<#(xi,..., xn)\\ >—*- X\ x ... x Xn. •
The next step is to define the "logical symbols" T. _L, A, V, =>, B^gy, Vyey. = as operations which yield formulas from terms and/or other formulas. I will not do so here, as the definitions, while tedious, are intuitively obvious to anyone
15 who has studied a little categorical logic. I refer the reader to [21], [13], [18], or [17] for rigorous definitions of the internal language and its interpretation. While the interpretation of formulas as subobjects is an important part of the interpretation of logic, there is still a need to evaluate formulas at given terms in order to determine their truth or falsity. Above, I mentioned that terms of type X are given by arrows into X. This means that if we want to evaluate a formula 0 with free variables of types Xi, X2,..., Xn, we need an : arrow a = (a\, a>2, • • -, cy-n) A—^Xi x X2 x ... x Xn to act as a tuple of terms at which we can evaluate
Definition 2.2.1. If (f) and a are as above, the symbol lh is defined such that
A lh (f>[ai, Q2, • • •, Qtn] if and only if the arrow \\a\\ : A —>- X\ x I2 ... x' Xn factors through the subobject \\
For a closed formula d> (a formula with no free variables) in a Heyting pretopos £, I write £ lh
2.2.1 Kripke-Joyal Semantics
In a standard classical presentation of the semantics of first-order logic, the semantics of the logical operators is usually given by tying their meaning to the corresponding logical operator in the metalanguage. For example, for a first-order structure M we say that M \= cp A ip iff M f= 0 and M \= '. In this way, we define the meaning of A in terms of the meaning of "and". A similar thing can be done in categorical logic, via a theorem known as the "Kripke- Joyal semantics". In this case, semantics is a theorem, rather than a definition, because the interpretation of formulas has already been defined via subobjects.
Theorem 2.2.1. // 4>{x) and tp{x) are formulas with free variables among
16 x = (xi, X2,..., xn) of types X\)X2-1. . ., Xn, and if a = (a\, a>2, -. •, an) ^s an
arrow A —*- Xx x X2 x ... x Xn then:
1. A Ih 0[«] A rp[a] iff A Ih 0[a] and 4 Ih V[a]
2. A Ih 3. A Ih (p[a\ =>• ijj[a] iff for any arrow p : B—^A such that B Ih 4- A II—<4>[a] iff whenever p : B—>• A is such that B Ih 4>{ap] then B ~ 0. 5. A Ih 3yeY(j)[a, y] iff there exists an epip : B—^^4 and an arrow (3 : B—>Y such that B Ih cj)[ap, f3]. 6. A Ih \/y&Y I will not prove that such a semantics works here. See ([18], pp. 305-309) for a proof in the case where £ is an elementary topos. While the proof that is given assumes that £ is a topos, this assumption is not needed as the proof uses only properties of £ (such as epis being preserved under pullback) that hold in any pretopos. In fact, the full pretopos structure of £ is needed here - for example we need that all epis are the coequalizer of their kernel pair to conclude that they are preserved under pullback. The clause in the semantics above for 3 reveals an intimate connection between the existential quantifier and epimorphisms. For example, an object A satisfies A Ih 3aej\(a = a) if and only if the unique map A—>• 1 is epi. Definition 2.2.2. We call an object A of a Heyting pretopos £ "inhabited" if and only if the unique map A —>• 1 is epi. 17 2.2.2 Informal Conventions The internal language as rigorously defined is often times too cumbersome to act as a useful tool for proving statements about Heyting pretoposes. Thus I will make the following notational conventions. I will often leave out some arrows from formulas Also, if I have a subobject S >-^~ X\ x ... x Xn, I will often write S lh <$> for S lh 4>[m\. I will also often times leave out the rn and when I have a map / : X—*~A, we will often times just write f(x) for the term of type A with free variables {.x, y} given by rf o -K^, and will just write x for the term ridxn(x). Sometimes, the types of the free variables occurring in a formula may not be clear. Thus, if variables xi,.... xn have types X\,..., Xn, I will often write ||xi G X2,...:xn e Xn : (}){x1)...,xn)\\ or ||xi,...,xn : 0(xi,..., x„)|| for ||0(xi,...,Xn)||. Not only will I abbreviate the formal internal language to make it easier to work with, but I will often times treat the internal language as an informal language in which to do mathematics. For example, suppose I have closed formulas 4>i, 4>2, • • •, 4>m Jp, and am trying to prove that for any Heyting pretopos £ in which £ lh ^, for each i, I also have that £ lh 'ijj. I will then assume 4>i, fa, • • • as premises and prove I/J via an informal argument in intuitionistic first-order logic. If such an informal argument were to be made rigorous, it would become a proof that (pi A d)2 A ... 4>n => ib is a theorem of intuitionistic first-order logic, and thus that we must have £ lh fa A 4>2 A ... 18 2.3 Elementary Toposes One of the first examples of a Heyting pretopos that comes to mind is the category Set of sets (which, of course, can only be defined once we fix a universe of sets). In fact the category of sets is a very special kind of pretopos called an elementary topos. Elementary toposes have been very much studied, and much is written about them in [13], [18], [14], among other places. Most of the structure of an elementary topos is already present in a Heyting pretopos. The only additional structure necessary to make a Heyting pretopos £ into an elementary topos is to add a power-object V(A) for each object A. Definition 2.3.1. We call a Heyting pretopos £ an elementary topos if, for every object A of £, there is an objectV(A)7 and a relation EA =—^AxV(A) such that for every subobject S of Ax B there is a unique map such that the square below is a pullback. S >EA Ax B —^ A x V(A) {id,{aeA:S(a)}) K ' In general, the logic of the internal language of an elementary topos is a little bit stronger than that of a Heyting pretopos in that the power-object structure allows us to define some extra logical operators via canonical arrows in the topos. For every object A we have the relation G^ >->A x V(A), and have for any formula 19 same way as set theory does. For more on the relationship between topos theory and higher-order logic see [17]. 2.3.1 The Topos of Sets There are also a number of conditions satisfied by the topos Set of sets (as suming the axioms of ZFC for the universe of sets), that are not satisfied by toposes (or Heyting pretoposes) in general. Let us start by considering the subobject lattice of each object in a Heyting pretopos £: the lattice of subob- jects of 1 can be regarded as a sort of "lattice of truth-values". In a Heyting pretopos, the lattice of subobjects of any given object X is in general only a Heyting algebra. However, in the topos of sets, all the subobject lattices are in fact Boolean algebras, and the lattice of subobjects of 1 is the Boolean algebra {0,1}. Thus, I define the law of excluded middle as below. (EM) The lattice of subobjects of any object X is a Boolean algebra. Definition 2.3.2. We call £ a Boolean pretopos if it is a Heyting pretopos and satisfies (EM), and call it a Boolean topos if it is an elementary topos and satisfies (EM). We call a Boolean pretopos "two-valued" if there are in fact only two subobjects ofl. The next condition satisfied by Set is the axiom of infinity, which can be formulated using the universal property of a natural numbers object. (Inf) There exists an object N with maps 0 : 1 —^N and S : N—^N which is a "natural numbers object" in that for any object X and maps a : 1 —*-X, f : X —>• X there exists a unique map g making the following diagram commute: 20 1 -—-N ^^N a 9 | \' Definition 2.3.3. We call an elementary topos a "topos with natural numbers object" if it satisfies (Inf). In this thesis, I will often times, for brevity's sake, say "topos" when I re ally mean "topos with natural numbers object". This is because the continuum hypothesis, along with many other questions I aminterested in, can't be formu lated without the axiom of infinity. Unless we say otherwise, all of our toposes satisfy (Inf) We can also define the axiom of choice in an arbitrary topos. The existence of power-objects implies the existence of exponentials CB which act like the set of functions from B to C (see [13] or [18] for a rigourous definition of exponen tials). We use these exponentials to define the second, "internal" formulation of the axiom of choice. (AC) For any epimorphism g : A —^ B. there is a section f : B —^ A such that g o f = id. (IAC) For all objects A and B, £ lh VeeAB((Vaeyl36eB a = e(b) ==> 3seBA(\faeA e(s(a)) = a))). (IAC) is, in some sense, just a translation of (AC) into the internal lan guage, although, as an internal language condition, it is actually an axiom schema not a single axiom, as there is no single internal language statement which will allow us to talk about all objects of a topos 8. We would need algebraic set theory, presented in the next chapter, to formulate (IAC) as a single axiom. These axioms above allow us to define a notion of "Set-like topos". 21 Definition 2.3.4. We call an elementary topos satisfying (EM), (Inf). and (IAC) a "Boolean topos with choice". 2.4 Consistency Proofs My goal in this thesis is to prove some relative consistency results about certain types of Heyting pretoposes. In order to do so, we must first understand what is meant by "consistency". In terms of logic, a theory is consistent exactly when it does not entail a contradiction. In this sense, we could call a Heyting pretopos £ "inconsistent" if £ lh _l_ and "consistent" otherwise. This distinction is in fact made by the following definition. Definition 2.4.1. A Heyting pretopos is "degenerate" if® ~ 1, and "nonde- generate" otherwise. It turns our that the degenerate pretoposes are exactly those whose internal language is inconsistent. Every degenerate pretopos is equivalent to the one with one object and one arrow, hence I often speak of the degenerate pretopos. However, what I want to do in this thesis is not to prove that a given Heyt ing pretopos has a consistent internal language, but to prove that a "theory" about Heyting pretoposes is in fact consistent. By a "theory" about Heyting pretoposes, I mean a certain class of Heyting pretoposes given by adding addi tional axioms to those defining "Heyting pretopos". For example, the "theory of elementary toposes" (topos theory), or the "theory of Boolean toposes with choice" is such a theory. Definition 2.4.2. If A is a collection of additional conditions that a Heyting pretopos is asked to satisfy, we say that A is "consistent" if and only if there is some nondegenerate Heyting pretopos £ satisfying A. My goal in this thesis is to prove the relative consistency of a certain axiom, the negation of the continuum hypothesis (which I will denote by (NCH)). 22 (NCH) will be seen to be a condition on a Heyting pretopos, so given a set A of conditions, the set A + (NCH) will be a stronger set of conditions. Then, what I want to do is prove that if A is consistent, then A + (NCH) is also consistent. In other words, I want to be able to prove that if we have some nondegenerate Heyting pretopos satisfying A, then there is also a nondegenerate Heyting pretopos satisfying A + (NCH). In order to preform such a relative consistency proof, I will assume that a Heyting pretopos 8 satisfying A is given, and then construct from it a new Heyting pretopos £' satisfying A + (NCH). The way in which I construct new Heyting pretoposes from old ones will be discussed in the next section. The notion of "consistency" of a set of axioms A defined above can be seen analogous to the logical notion of "semantic consistency" (the existence of a model) as opposed to the notion of "syntactic consistency" (the non-derivability of a contradiction). If we wanted to provide a proof of "syntactic relative consistency", we would need to have an argument that shows that if a certain theory doesn't prove a contradiction, then a stronger theory would also not prove a contradiction. If we were going to use the model-theoretic method of constructing a new model from an old one to perform such a proof, we would need a way of deducing, from the assumption of syntactic consistency, the existence of a model. In other words, we would need a completeness theorem. For certain sets of axioms A on a Heyting pretopos, completeness results are known. For example, it is known that if T is a theory of higher-order intuitionistic logic (as defined in [17]), then T is consistent exactly when there is a nondegenerate topos with T as its internal language. This would allow me to translate my proof of semantic relative consistency for topos theory into a proof of syntactic relative consistency for higher-order intuitionistic logic if I so desired. However, here I am not concerned with syntactic consistency, and will be content with any method which allows us to construct new models from old ones. 23 2.5 Construction of New Hey ting Pretoposes As Heyting pretoposes will form the "models" that I will use in my consistency proofs, I need to have methods of constructing models. In this section I will both discuss the Heyting pretoposes that can be constructed within set theory, as well as ways of constructing hew pretoposes from old ones without assuming the consistency of ZFC. 2.5.1 Pretoposes within Set Theory The primary examples of Heyting pretoposes are categories of sets. For any cardinal K we can construct a category Set Proposition 2.5.1. For any infinite K, the category Set I will not prove this theorem here. I will just note that all of the construc tions required in the definition of a Heyting pretopos can be performed (in an obvious) way, and that none of these constructions will allow one to construct sets of cardinality > K from sets of cardinality < n. While this method does provide us with an infinite set of examples of Heyt ing pretoposes, these examples are far from enough, as every one of them is both two-valued and Boolean. In order to consider Heyting pretoposes in their full generality, we need ways to construct non-Boolean Heyting pretoposes from the Boolean ones we already have. 2.5.2 Internal Functor Categories Anyone familiar with topos theory knows that, for any category C, the functor category Set is a topos. Since a topos is a special case of a Heyting pretopos, Setc°P will in fact always be a Heyting pretopos. What is not immediately clear is that it is possible to replace Set with an arbitrary Heyting pretopos £ and 24 get a new Heyting pretopos £c°v via a construction analogous to that of Setc P. In order to make things as general as possible, we do not want to require the existence of a universe of sets, so the category C (which is a set), must be replaced by some internal entity in £. I will in fact make use of the notion of "internal category". An internal category consists of an object Co of objects, an object C\ of arrows, domain and codomain arrows d.c \ O^—^OQ, an arrow i : Co—^C\ which sends every object of C to the identity map on that object, and an object C2 — C\ xCo C\ consisting of all the composable pairs of arrows with a composition arrow o : C2 —*- C\. These objects and arrows must satisfy a number of internal language conditions given in ([13], p. 48). Given a category C, a functor F : Cop —>- Set is a choice of a set F(C) for each object C of C and an arrow F(f) : F(D) —> F(C) for every arrow / : C —>• D in C satisfying the appropriate "functoriality" condition. We can also encode this functor as a "discrete fibration": a function f0 : F0—>• Co, where F0 = Ucec0 F(C), and fo(x) = C exactly when x € F(C). We can then _1 recover each F{C) as the preimage /0 (C). If we consider the pullback square: we note that the set F\ is the set of all pairs (/, x) such that x is in the domain of F(f). So a map dF : F\ —>- F0 which sends (f.x) to F(f)(x) will encode how F acts on arrows of C. Thus I define an internal ^-valued functor F over the internal category C = (Co, C\, C2, c, d. i, o) to be a pair of objects F\. FQ of : £ together with arrows: /0 ^0 —^ C0, j\ : F\ —>• C\, and dp, cp : F\ —^ F0 satisfying certain internal language conditions (again given in [13], p. 49). I will call F0 the "object of objects" of F, and Fi the "object of arrows". An internal natural transformation (an arrow between internal functors) 25 F a > G is a pair of arrows <7o : -Po —*~Go, ai : F\ —>• G\ such that appropriate set of diagrams (given in [13]) commute. I call a0 the "object function" of a, and Definition 2.5.1. Given an internal category C in a Heyting pretopos £, we let £C°P denote the category with objects internal £-valued functors and arrows internal natural transformations defined above. The notation Setc°P then agrees with its ordinary meaning as a functor category. A category of the form Setc P is also often times referred to as a "presheaf category", with objects being "presheaves". Thus, I will use "inter nal presheaf" so as to be synonymous with "internal £-valued functor", and "internal presheaf category" will be synonymous with "internal functor cate gory" . It should be noted that an internal functor F = (F0, F1; /0, /i, cF, dF) is determined by the data of FQ, fo and dF, since iq, f\ and cF are determined uniquely so that the square above is a pullback. Also, given internal functors F = (F0, F1. /o, fi,cF, dF) and G = (G0, G1,g0, #i, cG, dG), an arrow a : F—^G is uniquely determined by its object function o$. This implies that a is epi whenever cro is epi, and is mono whenever GQ is mono. We will see later that the converse also holds. The following theorem is proven in [18] and [13]. Theorem 2.5.2. If £ is an elementary topos, than so is £C°P. If, in addition. £ satisfies (Inf) then so does £C°P. This theorem should generalize to the case of Heyting pretoposes. If £ is a Heyting pretopos, then £C"P should also be a Heyting pretopos. Such a result is quoted as "well-known" in ([3], p. 11), although I have yet to find a proof in the literature. The proof is long, but straightforward, thus I will provide it below. 26 Theorem 2.5.3. If 8 is a Heyting pretopos and C is an internal category, then 8C°P is also a Heyting pretopos. Proof. Consider the functor U : 8C°P - -8/Co which sends an internal functor F to its object-over-Co of objects FQ - Co, and an arrow a : F —*- G to its object function a0. We know that 8/Co is a Heyting pretopos, and we will use this fact in proving that 8C°P is a Heyting pretopos. I will first prove that U creates finite limits, coproducts, and coequalizers. Let us first consider the example of pullbacks. Suppose that a : F —>- H and r : G —>- H are given, and consider the pullback square in 8/C0 below. x I wish to show that there is a unique dP : Po c0Ci which makes P = (P0, p0, dP) into an internal functor with a and (3 internal natural transformations, and that c v this unique dP makes P the pullback of the corresponding square in 8 ° . The arrow functions Q\ and 0i can be induced from a0 and f30 by pullback along Fi —^ F0 and G\ —^-*- Go- Then, the unique dp is given by the universal property of the pullback PQ since we must have that a0dp = dpa\ and Podp = dcPi in order for a and p to be internal natural transformations. To show that this dp so defined does in fact make (P0,dp) into an internal functor, we must verify that a number of diagrams involving dp commute. For example, we must show that the diagram below commutes. dp x C'o id X PQ C0 CI xCo Ci >- P0 xCo C\ ldXf7nc° 'c0 X PQ C0 CI dp Fn 27 But, by the fact that dP commutes with a and ft, both dp(dp x id) and dp(idx o) are maps / : PQ XCO C\ XCO C\ —^PQ making the two squares below commute. X PQ C0 C\ XCO CI Pn agXid QO Po XC CI X CI + F 0 CO dp(dpXid)=dp(idxo) 0 x x Po c0 Ci c0 Ci • Pn /?o x id /3o Go x Ci x C\ Go Co Co dc(dc xid)=dc(idxo) Because Po was defined as a pullback, we have that there is a unique such '/, thus the two maps we wish to be the same are in fact the same. The same method works to prove that all the other diagrams necessary to show that (Po, dp) is an internal functor also commute. The last thing we need to show (for the creation of pullbacks) is that P, a, (3 is actually a pullback in £C"P. Let us suppose we have an internal functor X with internal natural transformations 7 : X—^F, S : X—*-G. Then, because P0 is a pullback, there is a unique map / : X0 —>• P0 such that a0(f) = 70, Po(f) = <5o- All we need to show then is that / is in fact an internal natural transformation. To show this we need to show that a certain diagram com mutes, and the proof that this diagram commutes will be perfectly analogous to that for the diagrams necessary to show P to be an internal functor. We can now repeat this procedure for terminal and initial objects, binary coproducts and coequalizers. The object of objects P0 of the limit or colimit will always be created using the limit or colimit in £/CQ (which corresponds to the limit or colimit in £). Then, dP will be constructed using the universal 28 property of such a limit or colimit. In the case of colimits, it is important that the colimits in question are stable under pullback, since otherwise dp cannot necessarily be constructed. For example, given internal functors F and G, the coproduct F + G must have object of objects F0 + G0, and object of arrows (F0 + G0) xCo C\. The arrows dF : F0 xCo d—^ F0 and dG : G0 xCo Cx —^G0 induce an arrow dF + dc : F0 XQ, CI + G0 xCo C\ —*~ F0 + Go, and such an arrow in fact has domain (F0 + Go) xCo C\ as long as coproducts are stable under pullback. Thus, stability under pullback is important here. So, we now have that U creates finite limits and colimits, which means that we can deduce that, since (CI) and (PI) hold in £/CQ, they will also hold in £C°P. The creation of limits shows that the existence of limits in £ /Co implies the existence of limits in £C°P, and also that if certain colimits in £ /Co are stable under pullback, since pullback in £C°P is constructed using pullback in £/Co, those same colimits are also stable under pullback in £/CQ To see that (C3) holds in £C°P, we let a : F—^G be any internal natural transformation. The kernel pair of OQ will have a coequalizer r0 in £/CQ, SO by the creation of limits and colimits, there is a unique T\ making r into the coequalizer of the kernel pair of o in £C°P. Also, it follows from the creation of limits and colimits that r will be the coequalizer of its own kernel pair, and hence will be surjective. Thus, to prove (C3), we only need to find a mono p such that o = por. By, (C3) in £/C0, a0 factors as PO°T~O where H0> ^Go is the smallest subobject of Go through which cr0 factors. Consider the diagram below, where pi, T\ are defined via pullback along CH and cG. TO u PO The big square commutes because a is natural, and the left-hand square commutes since r is natural. Thus, the two routes around the right-hand square 29 agree when composed with TJ, thus since T\ is a pullback of an epi, hence epi, the right-hand square commutes. Thus, p is an internal natural transformation, which is mono because p$ is, and we have found a factorization for a. (C2) follows from (C3) and (PI) since the sup of a pair of subobjects can be constructed as the image of their coproduct. To prove (P2), we prove that the condition of "being an equivalence relation" is preserved by the functor U. The fact that all finite limits exist in £/C0 and that U creates finite limits shows that U preserves finite limits. Also, the functoriality of U says that if a subobject S of an internal presheaf F factors through another subobject R, then So —*- F0 factors through R0 —>- F0. This means that any property defined only in terms of pullbacks and "factoring through" is preserved by U. The notion of "being an equivalence relation" is in fact such a property. Thus c v if R g A is an equivalence relation in £ " , then Rp I A0 is an equivalence relation in £/C0, and thus has a coequalizer whose kernel pair is R0l which thus creates a coequalizer in £C°P whose kernel pair is R. To prove (V), a slightly longer argument is necessary. For any subobject S C P of an object A of £ ° , and any arrow a : A—^B. we want a subobject Va(S) of B which is the largest subobject R of B such that the pullback of R along a factors through S. We can construct VCT(S) as the subobject of B whose object of objects is the subobject Vcr(S)0 of B0 given in the internal language by: \\y eB0:Vbe Btfx G A0((cB{b) =yA dB(b) = a0(x)) =* SQ(x))\\ We need to prove three things: that this is in fact a subpresheaf of B, that its pullback along a factors through S, and that any subpresheaf of B whose pullback factors through S is a subobject of V First, to show that V<7(S) is in fact a subpresheaf (i.e. that dB restricts to dv„(S), we need that £ Ih V6eBl(cB(6) e VCT(S)0 => dB(b) G VCT(S)0). So, suppose such a 6 is given with cB(b) G V(7(S). Then, let b' G i?i and x G AQ be given such that cB(b') — dB(b) A dB(b') = o"o(x). Then, since A is an 30 internal functor, b and b' have a "composite" bob', such that CB(6O6') = CB(6) and dB(b o 6') = dsib') = o"o(x). So, since cg(6) £ V<7(S)0, a: G So, and thus 4WeVff(S)0. Secondly, I will show that £ lh VxeJ40(a"o(x) G Vo.(S)0 =>- x G So). So, sup pose x £ A0 is such that a0(x) G Vcr(S)0- Then, we can take b = (a0(x), id) G B BQ XC0CI — BI,SO that d {b) = cr0(x), and thus a; satisfies (CB(6) = yAdsib) = X S. Then, we wish to show that £ lh \/X£B0( E Ro => x £ Vo-(S)o). Let r G i?o- B B And let 6 G B1 and x G A0 be given such that c (b) = r and d (6) = o{x). Then, since R is a subpresheaf, dB(b) G RQ as well, and thus a(x) G RQ. Since a_1(R) factors through S, this means x G So, so m fact it follows that r G V We call constructions in an internal functor category £C°P "pointwise" ex actly when they are accomplished by lifting the corresponding construction in £/CQ along U. In the above proof, the only structure in £C°P which was not constructed pointwise was the universal quantifier. To see the motivation for this term "pointwise'', consider the case £ = Set. Set/Co is simply the cate gory whose objects are a choice of a set for each element of Co and arrows are a choice of a function between the corresponding sets for each element of CQ. Set/Co is just the product of Co-many copies of Set. Suppose we have two functors F.G : Cop —>• Set. Then, we can define their "pointwise product" by taking a functor that sends every c in Co to F(c) x G(c). The "pointwise product" just consists of C0-many products in Set, thus it is the product in Set/Co. Now, we have seen above that an arrow in £C"P is epi or mono provided that its object function is epi or mono respectively. A map / in a Heyting pretopos is epi if and only if it is the coequalizer of its kernel pair, and is mono if and 31 only if its kernel pair is the identity. Since coequalizers and kernel pairs in £C°P are constructed using coequalizers and kernel pairs in £/CQ, a map is mono or epi in £C"P exactly when it is mono or epi in £/CQ, which occurs exactly when it is mono or epi in £. We thus have the following corollary. Corollary 2.5.4. /// is an arrow of £C°P, and /o the corresponding "object function" in £, then f is epi iff /o is epi and f is mono iff /o is mono. This corollary will prove extremely useful later on. 2.5.3 Booleanization The above construction of internal functor categories allows us to construct a non-Boolean pretopos from the Boolean pretopos Set. In other cases, we want to construct a Boolean pretopos from a non-Boolean one. Within any Heyting pretopos, we will be able to define a full subcategory, called the category of -i-i-sheaves, which will be Boolean. This is a generalization of the category of -i-i-sheaves on a topos. defined originally by Lawvere and Tierney (also in [18] or [13]). For any subobject S of an object A in a Heyting pretopos £, we can consider r 1 r [ the formula S~ (x)J and form the formula -i-> S~ (a). This formula will have its interpretation as a subobject of A which we will denote by ->^S (which of course contains S, since rS~l(a) =>• -^^rS'1(a) as formulas) of C. Intuitively, an object B of £ is a -i->-sheaf if it "can't differentiate between" S and ->- Theorem 2.5.5. If £ is an elementary topos. then £^ is a Boolean topos, and if £ satisfies (Inf) then so does £-,-,. 32 One would hope that this this theorem generalizes to the case of Heyting pretoposes. We do in fact have the following. Theorem 2.5.6. For any Heyting pretopos E, and any ->-i-sheaf A, the lattice of subobjects of A in £_,-, forms a Boolean algebra. In order to understand what's going on here, I will first prove a necessary lemma. Lemma 2.5.7. A subobject S of a ^^-sheaf A is a ->-i-sheaf exactly when -i-i5 ~ S as subobjects. Proof. Suppose S is a ->-<-sheaf. Then, we must be able to extend the identity / map S—^S to a map -i-iS" *-S. Let m. denote the inclusion S^^-A and n the inclusion —>—'S* =—^ A. Since S factors through —i—i^S, n must be the unique (because A is a sheaf) map extending m. Since / extends the identity, m o f also extends m, so n = m o /, and thus n factors through m. This means -I-IS1 C S, and since for any S, S C -i->5, we have that S ~ ->->S. Now, suppose -i-i/S — S. Let R be a subobject of another object B off, and consider a map / : R—^S. Let g : ->->R—*-A be the unique extension of mo f : R—^ A where m is defined as above. Let 3gR denote the image of R under g, so that, as subobjects of A, 3gR < S. This implies that -i-i3ffi? < ->-iS. Also, by intuitionistic logic, we have that, for any g and R, 3g-i->R < ~^^3gR, so we > have 39(-i-iit ) < -i- Now, the theorem will follow from a second lemma. Lemma 2.5.8. For any object A of a Heyting pretopos £, the set of all subob jects S of A such that -i->S — S (the lattice of all -^^-closed subobjects) forms a Boolean algebra. 33 Proof. Suppose we have two subobjects S and R such that -v-iS1 ~ S and -1-1R ~ R. We construct their meet as the subobject S/\R and their join as the subobject ->(->SA^R). SARis -.-.-closed, since -i->(S/\R) => (-•-•S)A(-I-I.R) => SAi? by assumption on S and i?. —<(—iS"A—ii?) is -.-.-closed since it is a negation and triple negation is the same as single. S A R is clearly the meet here since it was the meet in the lattice of all closed subobjects. However, to prove that -<(-i5 A ~iR) is the join, we must prove that for any C — -,-|C, C > S and C > R iff C > ->{-^S A -.#)._ Clearly, if C > -<-iS A -./?), then since 5 V i? =>: -(-5 A -.fl), we have C > 5 and (7 > R. Now if C > S and C > /?, -.C < -5 and ^C < -./?, so ^C<-5A -.i?, soC = -.-.C > -^(-i5 A -7?). So the -.-.-closed subobjects at least form a lattice. To see that they in fact form a Boolean algebra, note that ~>S is a complement for S, since S A ->S = _L and the join of S and -.5 is ->(-IS/\->-IS), which is -I(-ISAS) (since 5 is -.-.-closed), which is -.(-L) which is thus T. So, we do in fact get a Boolean algebra. • While the Booleanization of a Heyting pretopos always yields a category which satisfies (EM), the resulting category is not always a Heyting pretopos itself. While in certain special cases (such as toposes and categories of classes), the category S^ will carry stronger structure, in general, it will not. Wrhile £-,-, always has all finite limits, £-,-, will not in general even have coproducts. Consider the Heyting pretopos Set^ where Ki is the first uncountable cardi nal. Let C be the category (actually a poset) whose objects consist of the set N together with a distinguished element *, and for which the only non-identity arrows are unique arrows /„ : n —>• * for each n G N. Since C has a countable set of objects and a countable set of arrows, it forms an internal category in f~*0'D f^OT) Set<^1 So, we can consider the internal functor category Set 34 coproducts. Proof. First I will show that any functor F in £ is a ->-i-sheaf if and only if F is a product diagram. First, suppose F is a ->-i-shea£. Then, for any set S, let A(S) be the diagonal functor with constant value S. Let Ts denote the subpresheaf of A(S) which takes value S on N, but takes value 0 on *. We can see that -1-^5 ~ A(S), so for any map a : Ts—^F, there is a unique extension to a map r : A(S) —>- F. Since T„ = F(fn)T* for any n, an extension of a to r is exactly a choice of a function T* : 5*—^F(a). Thus, the existence of a unique r is nothing more than the existence of a unique T* such that F(/n)r» = an for each n. Thus F must be a product diagram. Now, suppose we have a functor F which is a product diagram, and let S be a subpresheaf of some presheafnon-isomorphic A. We wish to extend a map a : S —>- F to a map r : -1-15" —*• F. For any n it can be seen that (-.-.5)(n) = 5(n), and (-.-.£)(*) = {x G 4(*) : Vn4(/n)(x) 6 5(n)} (see [18], pg. 273 for a characterization of -i->S in a functor category). So, we must have Tn = °~n for each n. To define r*, we note that, since r and a agree on S\ we , must have F(/„)r*(x) = r^4(/„)(x) = cr>l(/n)(x) for each x G (-•-i5 )(*). Thus, since i?(*) is a product, r is defined uniquely by this property, so F is in fact a sheaf. Now, I shall show that there are two functors of this form (functors which are ->->-sheaves), which cannot have a coproduct. We note that if functors F and G are -1-1-sheaves, any natural transformation a between F and G is given by a choice of an : F(n) —^ G(n) for every n. Let F(n) = 1 for every n, and let G(0) = 0 and G(n) = N for all n > 1. Then F(*) = 1 and G(*) = 0. F and G both have a countable number of countable objects, so they can both be represented by discrete fibrations in Set 35 that (F + G) is not in fact a sheaf which can be represented by a discrete fibration in Set So, we can see that, in some sense, the Booleanization construction is signif icantly harder to make work than the internal functor category construction. In any Heyting pretopos, we can construct an internal functor category which will again be a Heyting pretopos, but the Heyting pretopos structure will not be preserved by Booleanization. We will see that we need to introduce a stronger structure on £, either that of topos theory, or that of algebraic set theory, in order to make the Booleanization construction go through. 2.5.4 Preservation of Nondegeneracy The reason I am concentrating on the above two methods of building new pretoposes out of old ones is that they will both come into play in the proof of the consistency of the negation of the continuum hypothesis. In order for a relative consistency proof to work, we must create a new Heyting pretopos which will be nondegenerate as well as satisfying the given collection of axioms A. Thus, the following proposition is important. Proposition 2.5.10. If £ is nondegenerate and C is inhabited, then £C°P and £_,_, (if it is in fact a Heyting pretopos) are again nondegenerate. Proof. Let us first prove it for £C°P. Since Co is inhabited, the internal functors given by the discrete fibrations CQ —1—*~ CQ and 0 —>• Co are non-isomorphic (because if they were, Co would be initial, and an inhabited object cannot be initial in a nondegenerate topos). Thus we have an internal functor in £C°P which is not isomorphic to 0, and thus £C°P is nondegenerate. In the case of £-,-,, note that I proved earlier that any -i-i-closed subobject of a sheaf is a sheaf. 1 will always be a -i-i-sheaf, since the unique map ->->A—^1 is always a unique extension of the unique map A —>• 1, and will always be 36 -1-i-closed as a subobject of itself. Thus, if we can prove that 0 is -1-1-closed, we will have at least two non-isomorphic objects in £_,_, (they are non-isomorphic because £ is nondegenerate). To see that 0 is -1-1-closed, we note that ->1 = 0 and -.0 = 1, and thus -.-10 = 0. • Thus the internal functor category and Booleanization constructions will be able to be used for relative consistency proofs. We will see them put to use later. 37 Chapter 3 Algebraic Set Theory In some sense, topos theory captures the categorial analogue of set theory, in that a topos is a category that is "like the category of sets". However, a topos is is no way a model of set theory because the language of set theory requires us to quantify over the class of all sets, while the internal language of a topos only allows us to quantify over a given set by quantifying over the corresponding object of the topos. Thus, topos theory generalizes the notion of a "universe of sets" to the categorical context insofar as a "universe of sets" is a model of higher-order logic. However, if we want to generalize the notion of "universe of sets" insofar as a "universe of sets" is a domain in which set theory takes place, wTe must pursue a different type of category. The study of this sort of category makes up what is known as "Algebraic Set Theory". In this chapter, I will define the basic notions of algebraic set theory that will be used throughout the rest of the thesis. The first section will motivate; the considerations used in defining algebraic set theory, and the second will provide a motivated axiomatic definition. The next section will discuss the notion of "strong category of classes" which will include both elementary toposes and "cosmological categories of classes" (those in which we can quantify over sets) as special cases. I will then move on to discuss the set theory that can be constructed in a cosmological category of classes and the axiomatic set theory 38 that results. The next section will discuss the fact that every strong category of classes comes with a topos embedded within it. Lastly, I will conclude this chapter with a discussion of the methods of constructing new categories of classes from old ones. 3.1 Motivation for the Notion of Category of Classes If we wish to have a Heyting pretopos £ in which the language of set theory can be interpreted, we need to have an object of £ corresponding to the class of all sets. Of course, such an object will be a proper class, and thus the small sets will only form a subcategory of £. Thus the Heyting pretoposes which will form models of set theory will come equipped with the additional structure of a full subcategory of small sets. I will in fact encode the notion of smallness, not by describing a collection of small objects, but by describing a collection of small maps (thought of as those maps which have small fibres). The small objects are then defined to be all those A, for which the map A —>•1 is small. If K is a strongly inaccessible cardinal, we can form a model of Godel- Bernays set theory, by taking the sets to be the elements of VK, and the classes to be the subsets of VK. Since a strongly inaccessible cardinal is regular, VK has cardinality K. Thus, up to isomorphism, the sets of cardinality < K are exactly the subsets of VK, and the sets of cardinality < K are exactly the elements of VK. So, given such a K, we should be able to construct a category of classes whose objects are the sets of cardinality < K, and whose arrows are all set-theoretic functions between these sets. We will call such a category Set 39 3.2 Axiomatic Description of a Category of Classes There are, in the literature, a number of different axiomatizations which all go by the name of "category of classes". Many of the axiom systems are much weaker than ours, as they attempt to incorporate diverse examples (some not even that set-theoretical) within a single type of category. Here, I am more concerned with generalizing set theory than anything else, thus I choose an axiomatization which gets us as close as possible to Godel-Bernays set theory without adopting either the law of excluded middle or the axiom of choice. This is one of the strongest axiomatizations out there (in fact it is exactly the axiomatization that results from considering all the "additional axioms" discussed in [2]), thus I will use the term "strong category of classes" to refer . to it. To define what it means for a pair (£, §) (where £ is a Heyting pretopos and 5" is a collection of arrows of £) to be a "strong category of classes", I will consider the axioms below. Following each axiom, I will motivate it by showing why it holds in the standard models (Set (SubC) The category whose objects are all objects of £ and whose arrows are all maps in S forms a subcategory of £. In particular, this means that every identity map is in S and the composition of two maps in S is again in S. An identity map idx is clearly in SK since all of its fibres are one-point sets. If maps / : A —>- B and g : B —*- C are both in SK, then for any c E C, and b E B g~l{c) and f~l(b) are both sets of cardinality < K. For c G C, ( (Mono) All monos are in S. 40 A map / : A—^B is mono iff for all b G B f~l(b) is either 0 or a singleton. Thus this axiom is true because 0 and one-element sets have small cardinality. This axiom seems trivial, but it will be important because it in fact encodes the set-theoretic axiom of separation - the fact that a subobject of a small object is small. (PIBk) The pullback of a map in S along any map is again in S. Let A along a map g : C—*~A. For each c G C, (,9*(/))~1(c) = f~l{g(c)) by definition of pullback, thus this axiom is satisfied in the standard models because the notion of smallness encoded by SK is a property of the fibres of a map. (Coll) For any two arrows e : Y —*- X and f : X —>• A where e is epi and f is in S, there exists a commutative square: Z ^Y e-^X 9 f N' V B h- *A for which h is epi, g is in S, and the obvious arrow Z—^X x^ B is epi. This axiom captures the set-theoretic axiom of collection (which, classically, is equivalent to replacement) which states that given a small family F of non empty classes, we can find a set which intersects each one of those classes. Suppose, in Set 41 a E A, let Pa denote some such /?. Then, let B be the set of the /Vs and Z their disjoint union with g being the obvious map which sends s E Pa to pa. The fact that h is epi says that we have at least one such Pa for each a E A. g is small because each of the /3a's have cardinality < K. The map Z—^Y is the 1 _1 embedding of each Pa in the disjoint union of the classes e^ (x) for x E / (a), which clearly makes the given square commute. The fact that the unique map into the pullback Z—^X xA B is epi is just a restatement of the fact that, for 1 1 all pa and x E f~ (a), Pa n e~ (x) is nonempty. For the following axioms, we need the notion of a "small object over A". For any object B with an arrow B—*-A, we say that B is "small over Av, when the map B—*-A is small. The next few axioms will be concerned with relations R>-^B x A which are small over A, i.e. such that the second projection R—>-A is a small map. In the standard models we are considering, a relation small over A, R C B x A yields, for each a E A, a subobject of B of cardinality < K given by {b E B : (b, a) E R}. So, if we let V(B) be the set of all small subobjects of B, then there should be a unique corresponding map A—^V(B). (PE) For every object C, there is a power-object V(C) with a relation Ec ^-^C x V(C) small overV(C) such that, for any other relation R >-^- C x X, which is small over X, there is a unique arrow p : X —^V(C) such that the diagram below is a pullback: B, -Gc C xX > C x V(C) In the case of Set 42 R ^ C x X, p is defined by x i-*- {c G C : (c, x) G i?}. The fact that PK(C) will always have cardinality < K when C does uses the fact that K is strongly inaccessible. If we well-order the object C, since K is regular, every subobject of C of cardinality < K is contained in some initial segment of C of cardinality < K. Thus, VK(C) can be constructed as the union, over all (3 < K of V^p), where C@ consists of the first (3 elements of C. Since n is a strong limit, V(Cp) will still have cardinality < K, so the union of K-many such V{Cp) can have cardinality at most K. • It should be noted that the existence of a power-object is a property of (£.S), rather than a structure on £, since power-objects, if they exist, are unique up to isomorphism. We note that the fact that K is a strong limit entails that any set with cardinality < K also has its power-set of cardinality < K. Thus, we want an axiom that says the power-set of a small object will always be small. To formulate such an object, I define, for any C, a relation Cc >-^V(C) x V(C) given in the internal language by ||Q, (3 : \/xec{x e a =^> x G [3)\\. (PS) For any object C, Cc is a small over the second projection to V(C). Informally, this says that, for any small subobject (5 of C, the class of subobjects in V(C) which is contained in (3 is small. In particular, if C itself is small, then the class of all subobjects must be small. This axiom is in fact equivalent to the statement that, in any slice category £/X, the power-object of a small object is small. Again, Set (InfS) £ has a natural numbers object, and this object is small (the unique map N —^ 1 is in S). This axiom holds true in Set Definition 3.2.1. We call a Heyting pretopos £ a "strong category of classes" if it satisfies (SubC), (Mono), (PIBk), (Coll), (PE); (PS), and (InfS). 43 3.3 Cosmological and Insular Categories of Classes It is not hard to see that every elementary topos with natural numbers object 8 can in fact be made into a strong category of classes by taking the collection S of small maps to consist of every arrow in 8. The axioms (SubC), (Mono), (PIBk), (Coll) and (PS) become trivial truths if every arrow is in S, the axiom (PE) holds because an elementary topos has power-objects, and the axiom (InfS) holds because 8 has a natural numbers object. While it is in some sense convenient that elementary toposes are subsumed under our definition of "strong category of classes", this also means that our notion of "strong category of classes" does not always perform its intended purpose - to quantify over the class of all sets. In order to allow.quantification over all sets, we need to assume an additional axiom (how this axiom allows us to quantify over the class of all sets we will see in the next section). (Univ) There is an object U of 8 such that, for every object C of 8, there exists a rnonomorphism C ^^~ U. We call such an object U a "universal object". This axiom clearly holds in the case of Set Definition 3.3.1. We call a strong category of classes which also satisfies (Univ) a "cosmological category of classes". We call a strong category of classes in which all arrows are small an "insular category of classes". This terminology is motivated by the idea that a cosmological category of classes can see the big picture (the universe), whereas an insular category of classes is one whose point of view is restricted to the point of only seeing small objects. It will turn out that every insular category of classes is a topos by theorem 3.5.1, thus I will often times use "elementary topos" and "insular category of classes" interchangeably. 44 The proposition below will imply that no elementary topos is a cosmological category of classes. Proposition 3.3.1. The only strong category of classes which is both insular and cosmological is the degenerate category of classes in which every object is isomorphic. Proof. Let 8 be a category of classes which is both insular and cosmological. By (Univ), there is an embedding V(U) >-^» U. Consider the subobject -i £[/ >—*- V(U) x U defined as the (Heyting) complement of Eu- We can compose this subobject with the map V(U) x U —-—:—->- U x U and pullback along the diagonal U—*-U x U to get a subobject R of U. Since 8 is insular, R is small, so there will be an arrow 1 —^V(U) corresponding to R, and we can consider the composite m o r : 1 —>- U. For those who haven't guessed yet, R is the Russell set. The existence of the composite mor, means we can ask whether or not 1 lh rR~1(m,or). rR,~1(mor) will lead to a contradiction, so we can conclude - So we have (at least) two different sorts of strong categories of classes that we will be considering here. We have will continue to work with elementary toposes (insular categories of classes) as generalizations of the category of sets, and will work with cosmological categories of classes as generalizations of the category of sets and classes. However, since both sorts of categories fall under the single heading of "strong categories of classes", I will make a number of the definitions later in this thesis for the case of "strong categories of classes", thus only needing one definition for notions that will be applied to both the cosmological and insular cases. For example, exponentials with small exponent can be defined in any strong category of classes. If we have objects B and C where B is small, then we can 45 define CB to be the subobject of V[C x B) consisting of all relations which are functional (this can be defined easily in the internal language). Then, any- arrow B x A ^C will correspond to a relation A x B -—-—^C x B x A, for which the projection to A is small (this projection will be just a pullback of the unique map B—*-1). Thus, we will have a unique corresponding arrow A—^V(C x B), and this arrow will in fact factor through the exponential CB. Also, just as in the internal language of a topos, we were able to, given a formula (f>(x,y) with free variables of type X.Y, construct a term {x G X : 3.4 Set Theory in a Cosmological Category of Classes The first thing to note about cosmological categories of classes, is that all of the internal language quantifiers are subsumed by those over U. For example, if we regard C as a subobject of U (and thus ||0|| =—^ C ^-*- U as a subobject of U), the internal language formula Vx6c0(x) will always be equivalent to the r n formula \/x(zu( C (x) =>- 4>(x)). This corresponds to the move from type theory to set theory where quantifiers over types can be seen as restricted quantifiers over the universe of sets. For brevity's sake, we will often omit the G U from quantifiers, thus, in the case of a cosmological category of classes, Vx means The internal language of a cosmological category of classes (S,S), can be, in many ways, seen as a type of "set theory with urelements". It is a sort of set theory because almost all of the structure of (£, S) is encoded in the G relation 46 G >—>U xV(U)>-^U x U. For example UA>-*~U is small" can be encoded by the U formula 3x\/y(y £ x <^=> A(y)), and B>-^U is the power-object of A>-*~U" can be encoded by the formula Vx((Vy(y £ x =>• A(y)) •<=>• B(x)). However, it is unlike ordinary set theory in that extensionality does not necessarily hold. Two elements of U can both be extensionally equivalent in that neither of them contains anything as elements, but they can still be non-equal. These elements will correspond to the part of U that is not in the range of the map V(U) >->- U. In some sense, these elements are not sets, and we call them "urelements". Thus, the basic non-logical symbols necessary to describe the set theory of a category of classes are the relation £ of membership, and the predicate S of set-hood. If we restrict the internal language of a cosmological category of classes to an intuitionistic first-order theory with a single type U, and £ and S as the only non-logical symbols, we arrive at a sort of intuitionistic axiomatic set theory. We will interpret £ as the relation G ^^ U x V{U) >—>• U x U, and S as the subobject V(U)>-^U. Under such an interpretation, in a cosmological category of classes, the following axioms of set theory are always satisfied (to be precise, we take the universal closure of the formulas below, and let 4> be any first-order formula with the appropriate free variables in the axiom schemata of collection and separation). 1. (set-hood) a £ b => S(6) 2. (extensionality) (S(a) A S(6) A Vx(x G a -<=>- x £ b)) =4- a = b 3. (pairing) 3x(S(.r) A Vy(y £x ^^ (y = a\J y = b))) 4. (powerset) S(a) => 3x(S(x) AMy{y £ x <=> (S(y) AVz(z £ y =>• z £ a)))) 5. (union) (S(a) AVy(y £ a ^ S(r/))) => 3x(S(x)AVy(y £x <=> 3z(z £ a Ay £ 6. (infinity) 3x(S(x) A 3y(S(y) AVw-*(w £y) Ay £ x) A Vy{y £ x => S{y) A 3z(S{z) AWw(w £ z <^> (w G y V w = y))) A z £ x)) 47 7. (collection) S(a) A Vy(y G a =>• 3w(f)(y,w)) => 3zS(z) AV9(t/ e a ^> 3w(w G zA 8: (separation) S(a) =>• 32(S(z) A Vx(x G z -<=>• x G a A 0(x))) The fact that the above axioms arc satisfied is proven in [2]. If we want, in addition to the above, a sort of axiom of regularity, we must undertake an additional construction. One of the main results of Joyal and Moerdijk in [15] is that, in any cosmological category of classes (in fact, Joyal and Moerdijk use a slightly weaker axiomatization), it is possible to construction an object V which is the "initial ZF-algebra". The details of this construction are beyond the scope of this thesis, however, the important point to make is that it is in some sense the categorical analogue of the Von Neumann universe construction, in that it creates a well-founded universe out of a not necessarily well-founded one. Given the initial ZF-algebra V, we will have that V(V) ~ V, so that the relation ey ^-^ V x V(V) is in fact a relation on V. This relation ey (with the "set-hood" predicate S interpreted as the subobject V ^^ V) will satisfy all of the above 8 axioms plus the two below. 9. (e-induction) Vx(V?/(y G x =^> 10. (no atoms) VxS(x) The list of axioms that we have above is, in some sense, as much as we can get of classical set theory without assuming excluded middle or the axiom of choice. My definition of "strong category of classes" was in fact chosen to make as many of these axioms true as possible. A number of the axioms that used in the definition of "strong category of classes" which are not used by other authors writing on algebraic set theory have been chosen particularly to yield certain of the set-theoretic axioms above. In a category of classes without (VS), the powerset axiom would not hold, if (Inf) wasn't assumed, the axiom of infinity would not hold, if (Coll) wasn't assumed, then the axiom of collection would not hold, and if (Mono) wasn't assumed, separation wouldn't hold. 48 We can also impose additional conditions on our category of classes that make the axiom of choice or the law of excluded middle hold in our internal set theory. (EM) was in fact defined for an arbitrary Heyting pretopos, and the law of excluded middle holds in the internal set theory of a strong category of classes £ exactly when £ satisfies (EM). (AC) and (IAC) can be defined in similar (although not exactly identical) ways to the way they were defined for elementary toposes. (AC) For any epimorphism g : A —»• B. there is a section f : B —>• A such that g o f = id. (IAC) For any object X, and any small-over-X objects A and B, £/X lh Veej4s(Vae/i3beBa = e(b)) => 3s6j3-4(Va6^e(s(a)) = a). It should be clear that because, in a topos, exponentials in a slice category are constructed using exponentials in the topos, in the case of an insular category of classes, (IAC) agrees with its topos-theoretic definition. In a cosmological, category of classes, if (IAC) is satisfied, then an appropriate version of the axiom of choice will hold in in the internal set theory, replacing the "external" quantifiers over small objects A and B with internal quantifiers over U. 3.5 The Topos of Small Objects I remarked earlier in this chapter (section 3.3), that an elementary topos forms a category of classes if we take all maps to be small. We will see that, just as we can form a category of classes from a topos, we can form a topos from a category of classes. Within a given category of classes, we should be able to construct a topos consisting of all the sets in such a category of classes. Thus, given a category of classes (£,S), we let £$ denote the full subcategory of £ consisting of all objects C for which the unique map C —*-1 is in S. 49 Theorem 3.5.1. If (S,S) is a strong category of classes, then Ss is an ele mentary topos. Moreover, given any formula 4> in the internal language of Ss, the interpretation |||| of The first part of this theorem, that Ss is a topos, is proven in [2], however, I have not seen a statement or proof of the second part, which says that the inclusion functor Ss —*- S is "logical". This could be because this inclusion functor is logical only if (Mono) holds, and most authors writing on algebraic set theory do not wish to assume this axiom. We will first prove an important lemma, which is an easy corollary of a lemma in in [4]. Lemma 3.5.2. If a map fog is in S, then, in particular, g is in S. Proof. Consider the pullback square Pi 9 A -+C g°f Since g o f is small, by (PIBk), so is p This lemma implies, in particular, that any map from a small object is small. I now proceed with the proof of the main theorem. Proof. To prove that Ss is a topos, it is enough to prove that it has a termi nal object, pullbacks, and power-objects. The internal language of a strong category of classes is defined using names of arrows and subobjects, logical op erations coming from the limit and colimit structure, universal and existential 50 quantification, and power-object structure. Thus, to show that interpretations of internal language statements are the same in both cases, we must show that limits, colimits, quantifiers, and power-objects agree. To see that all finite limits exist in Es and agree with those in £, it is enough to show that the limits of objects of Es that exist in £ lie in Es- It is also enough to show this for a terminal object and pullbacks. The terminal object 1 of £ will lie in Es since the identity map 1 —>-1 is always in S. If we have a pullback square: A f-^^B 9* 3 where B, C, and D are all small objects, then by lemma 3.5.2, / is in S. Hence, /* is in S, and since B is a small object, this implies that A must be a small object by (SubC). In a strong category of classes, colimits of small objects can be constructed using limits and power-objects. To see how, see [18], pp. 180-184. Thus, if we prove that limits and power-objects are the same in S, and Es, that will be enough to show that colimits are also the same. Given an arrow / : A —*• B, quantifiers are constructed as right and left adjoints to the pullback functor /* : Sub(B) —^ Sub(A) between subobject lattices. Since pullbacks are the same in E and Es, and since adjoints are unique Tip to isomorphism, to prove that quantifiers are the same, it is enough to show that, given an object A of Es, the subobject lattice of A in Es is the same as it is in E. If we have a subobject X >->- A of A in E, since A is small and monos are small, X will also be a small object, hence will be a subobject of A in Es- Now, suppose X is a subobject of A in Es- We need to show that the map X —>- A remains mono in E. Since a map is mono if and only if its kernel pair is the identity, and since limits in £ and £s agree, monomorphisms 51 in Es are also mono in E, hence the subobject lattices of A in E and Eg agree. So, now we only need to show that the power-objects are the same in £ and Es- Again, it is enough to show that, for any object A in Es, the object V(A) and the relation 6^, constructed in £, are in fact objects in £S) because they will then satisfy the universal property necessary to be a power-object in £. The fact that V(A) will be in Eg follows directly from the axiom (P2). EA is a subobject of A x V(A), which is a product of small objects, hence small by (PIBk). Thus, by (Mono), Gyi is itself a small object. This completes the proof. • 3.6 Constructing new Categories of Classes In order to use "forcing" methods in algebraic set theory, I want a proce dure whereby I can construct, out of a given (strong/cosmological) category of classes, a new (strong/cosmological) category of classes which will satisfy additional axioms. In order to do so, I must prove that the category result ing from such a construction will still be a (strong/cosmological) category of classes. Much of the proofs have already been done for me in [3], [28], [24]. In particular, most of the results that I will need will follow from a theorem of Moerdijk and Van Den Berg's stated in [24] and proven in [26]. Moerdijk and Van Den Berg prove a theorem about the preservation of certain axioms under the construction of internal sheaves over internal Grothendieck sites. Definition 3.6.1. We call an internal category C in a strong category of classes "small" if and only if both CQ and C\ are small objects. Definition 3.6.2. An internal Grothendieck site in a strong category of classes £ is a small internal category C in £ and a topology r given as a subobject of V{C\) x CQ satisfying certain conditions. Using this notion of Grothendieck site, we can define a category Sh^c.r)(£) of internal sheaves on the site, which will be a full subcategory of £C°P. How 52 this definition plays out in general is not important, but what is important is that our internal functor category and Booleanization constructions are special cases of it. If C is a small internal category, let us define tc0 to be the map C0—*~P{C\) (c,id) : which is the transpose of the mono C\ —^Co x C\, and define rt to be the (tc0,id) > topology given by the mono C0 *- f \C\) x Co. Then, the category C P Sh{c^Tt){£) is exactly the category £ ° . Similarly, if we let Cl be the one-object one-arrow internal category CQ — Cj = 1 and let r-,-, denote the subobject of V(l) given by ||Q £ V(l) : -i->a = T||. Then, the Booleanization £_,_, is exactly the category S7i(ci.r^-,)(£)- Definition 3.6.3. // (£. S) is a strong category of classes, and if an arrow o : F—^G in Sh(c.T)(£), when viewed as an internal natural transformation, has object function a0, and a0 is in S, then we call a "pointwise small". Moerdijk and Van Den Berg provide a weaker axiomatization of the notion of "category of classes" than they one that I have provided. I will use "weak category of classes'' to refer to their definition of a "category with small maps". A weak category of classes doesn't necessarily satisfy (P2), (Mono), (PE), (PS), or (InfS) and a weak category of classes satisfying all of these axioms will in fact be a strong category of classes. Moerdijk and Van Den Berg wish to prove that if £ is a weak category of classes then so is Sh(c,r)(£)- However, this result is not valid without further assumptions. To prove that Sh(c,r)(£) is a weak category of classes, it is necessary to construct an associated sheaf functor a : £C°P—*-Sh(c,r)(£)- To construct this functor, additional requirements must be satisfied. Firstly, in addition to their "weak category of classes" axioms, Moerdijk and Van Den Berg require an axiom (IIS) to be satisfied in £, which turns out to be strictly weaker than our axiom (PS), thus it will be automatically satisfied in any strong category of classes. They also require their topology r to have a small basis, which occurs, in particular, if r is small. In a strong 53 category of classes any subobject of V(Ci) x C0 will be small since C\ and CQ are small, so, in the case of a strong category of classes, both these conditions will be satisfied. The more important additional condition that Moerdijk and Van Den Berg require is the axiom (BE). (BE) Every equivalence relation R > m > A x A such that m is in S has a quotient: a coequalizer whose kernel pair is the pair of maps R > A corresponding to m. It turns out, that in the presence of (Mono), (BE) implies that every equiv alence relation has a quotient, which is just (P2). Similarly, (P2) is strictly stronger than (BE) in any category with a distinguished class of maps S. So, in a strong category of classes (which is a Heyting pretopos), all three of Mo erdijk and Van Den Berg's additional conditions will be satisfied, and thus the theorem that they proof will apply. While the final version of the theorem in question and its proof has not yet been published, we can safely assume that the theorem proven in [26] will differ from the one presented below (which is simply a restatement of the theorem stated in [24]) only inconsequentially. Theorem 3.6.1. // (£,S) is a weak category of classes satisfying (IIS) and (BE), and (C,r) is an internal Grothendieck site with a small basis, then Sh(c,T)(£) with the class of pointwise small maps is also a weak category of classes satisfying (IIS) and (BE), and if any of the axioms (Mono), (PE), (PS), (InfS) are satisfied by £. they are also satisfied by S7i(c,T)(£)• The requirement that E be a pretopos, either via (BE) or (P2), seems to be essential to the existence of an associated sheaf functor. In [24], it is noted that (BE) is essential to.the proof of the above theorem. In [3] a different axiomatization of the notion of "category of classes" is presented, and a proof that this notion is preserved under sheaf constructions is given. The authors note that their proof that the associated sheaf functor exists relies crucially on 54 the assumption of axiom (P2). This is, in the end, the reason that I require the category I am dealing with to be a pretopos and not just a Heyting category. Suppose we have a strong category of classes £. In particular, it will be a weak category of classes satisfying (IIS) and (BE), so the above theorem is applicable. This means that, since (Mono), (PE), (PS) and (InfS) are satisfied in £, they are satisfied in S7i(c.T)(£)- I then only need to prove that (P2) is satisfied to show that £ will also be a strong category of classes, and this follows from the fact that (BE) and (Mono) together prove (P2). The definition of the small maps in Sh(c,r)(£) as those that are pointwise small shows that if all maps are small in £ (£ is insular), then Sh(c,r)(£) is also insular. Thus, we get the following corollary of Van Den Berg and Moerdijk's result. Corollary 3.6.2. // (£,S) is a strong category of classes, and (C,r) is an internal Grothendieck site, then S7?,(Cr)(£) with the collection of pointwise small maps is again a strong category of classes. If £ is insular, then so is Sh(c.T){£)- In fact, from this corollary follow two more, which are the results that I will actually use in constructing new categories of classes. Note that since these results are stated for strong categories of classes, they are strict generalizations of the corresponding results for elementary toposes. Corollary 3.6.3. // (£,S) is a strong category of classes, and C is a small internal category, then £C°P with the collection of pointwise small maps is also a strong category of classes. If £ is insular, then so is £G°P. Corollary 3.6.4. // (£, S) is a strong category of classes, then (£-,-,, S n £-,-,) is also a strong category of classes. If £ is insular, then so is'£-,-,. 3.6.1 Preservation of (Univ) So far, we can see that our internal functor category and Booleanization con structions preserve the notion of "strong category of classes". We want also for 55 this construction to preserve the notion of "cosmological category of classes", thus I still must prove the following theorem. Theorem 3.6.5. If £ satisfies (Univ); then so does £-,-,, and if C is a small internal category then £c"v also satisfies (Univ). Proof. To prove the Booleanization part of this theorem, I will need the fol lowing lemma, which was proven in [3] (and will also be proven in [26] when it comes out). Note that the proof of this lemmas was the exact point at which (P2) was needed. Lemma 3.6.6. The inclusion functor £~,^ —*- £ has a left adjoint a. and a preserves finite limits. Note that a, as it preserves finite limits, preserves monomorphisms. This will allow us to define the universal object in £_,_, to just be a(C/), where U satisfies (Univ) in £. This is because, for any sheaf A, we have a monomor- phism A>-^U, which gets mapped by a into a monomorphism BL(A) —*-a(U). Since A is a sheaf, a.(A) ~ A, so we have a monomorphism A—*-a(U), and thus a(f/) satisfies (Univ). Dealing with the internal functor category case is a little more complicated. I noted earlier that the axioms of a strong category of classes are stable under taking slice categories, so S/CQ is also a strong category of classes. Since C\ is a small object in £, it is small in £/CQ, and thus we can take the exponential of the objects U x C0 and C\ over C0. Let Uc* denote this exponential in the slice category £/C0. Informally, the fibre of Uc* over a G C0 consists of all functions from the fibre of C\ over a into U. Now, we can take the pullback of U^1 (as an object over C0) along the map c : C\ —^Co, and can denote the resulting 1 object by U^ xCo C\. We can then define a map dy : f/^ xCo C\ —> UQ* as l the exponential transpose of the composite Uc^ Xp0 C\ XQ0 C\ ——^~Uc Xc0 C\ —^ U x Co, where e is the evaluation map in the slice category. Let U denote the internal functor given by (U^, djj). 56 Now, I will show that, given any other internal functor F = (F0, Fi), there is a monomorphism F—'-^-U. To construct this monomorphism, first note that there is a mono m : F0 ^^ U by (Univ) in 8. Then, consider the composite m FQxCoCi—^F0 >U', and pair it with the map C{-K2) '• F0XCOC\—^C0 to get a map F0 xCo C\ —*~U x C0 over C0. By the universal property of the exponent Uicc \ , thitmos jicjL^yieldos aa maymap uvcoveir uuC0,, ^// Q .: i(j--yF0 U>C() . I will now prove informally in the internal language that the definition of /i0 means that it is mono. If we have x,y E FQ such that HQ{X) = Ho{y), then, in particular, [J®(X) and no(y) lie in the fibre over the same a 6 Co, so that we can evaluate them both at ida. But, /j0(x)(ida) = m(ida(x)) = m(x)1 and ^0{y)(tda) = m{ida(y)) = m(y), and so m{x) = m(y), and, since m is mono, x = y. Now, we just need to see that [AQ is in fact the object function of an internal natural transformation. We must show that the square below commutes. F0 xCo Cx ^—? - t/g xCo d dF du Fn Ho -tfg By exponential transposition, this amounts to showing that the square be low commutes. f-iQXid TCi Fa xc C\ xCo C -^Co XCoCiXcoCa dpxid e(idxo) w X ^0 C0 Cl •+UxC0 {m{dF),c(TV2) The projection to CQ will commute automatically because all the maps are maps over Co- Thus we only need show that the big rectangle below will commute. 57 idxo Ho* id FQ XC0 C\ XCO C\ -*• FQ x c C\ tfgxcoCi dp xid dp 7ri(e) I FQ XCO C\ C7 dp The right-hand square commutes by the definition of fi0 as a certain ex ponential transpose. The left-hand square commutes because F is an internal functor, thus the whole rectangle will commute and /i0 will be the object func tion of an internal natural transformation. • The above results are what allow us to carry out relative consistency proofs in the context of algebraic set theory. They will thus be extremely useful in proving the main results of this thesis. However, before I can carry out any of those proofs, I must first formulate the statements which I will prove the relative consistency of. 58 Chapter 4 Cardinality The Continuum Hypothesis is, in its essence, a hypothesis about the notion of cardinality. It makes a claim about the number of cardinalities occurring between the Natural Numbers and the continuum, and thus cannot be even defined until we know what cardinality is. In the classical set theory of ZFC, the definition of cardinality is well-known. However, in the world of a strong category of classes, it is not altogether clear what is the correct generalization of this definition. Thus, before I can move on to formulating the continuum hypothesis and proving the consistency of its negation, I must lay down what is meant by cardinality. The goal of this chapter is to provide a definition of the notion of cardinality in an arbitrary strong category of classes, and a motivation for this definition. I will open this chapter with a discussion of the Cantor-Bernstein theorem and the connection between cardinal equivalence and cardinal inequalities. I will next discuss two different notions of cardinality, based on monomorphisms and epimorphisms and the differences between them. I will then show how to formulate these two notions of cardinality in both the external language of category theory and the internal language of the strong category of classes in question, and then combine the notions of epi- and mono-cardinality together to yield a unified notion of cardinality. I will then discuss the reasons why 59 cardinality becomes a much simpler notion when we assume (EM) and (IAC), and will end this chapter with a proof of Cantor's theorem, which will justify the notion of cardinality that we have defined. 4,1 The Cantor-Bernstein Theorem In an introductory course on set theory, the definition of cardinality is usually motivated by intuitive descriptions of what it means for two sets to have the same size. For example, Kleene opens Introduction to Metamathematicsvelaied with the following paragraph: A flock of four sheep and a grove; of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. Although the words for numbers have been used to state this truism on the printed page, the relationship to which we refer underlies the concept of cardinal number. Without counting the sheep or the trees, one can pair them with each other, for example by tethering the sheep to the trees, so that each sheep and each tree belongs to exactly one of the pairs. Such a pairing between the members of two collections or 'sets' of objects is called a one-to-one correspondence. In this case and others, cardinality is introduced as an equivalence relation on sets - we say that two sets have the same cardinality if and only if there is a bijection between them. The next step is usually to introduce a "cardinality ordering" on sets by saying that \A\ < \B\ iff there is an injection from A to B, and then to prove the Cantor-Bernstein theorem that if \A\ < \B\ and \B\ < \A\ then \A\ = \B\. The notion of cardinality is thus seen to contain two components: an equivalence relation and a preorder, and the Cantor-Bernstein theorem is seen as the connection between them. One of the obstacles to developing a notion of cardinality in an arbitrary strong category of classes is that the Cantor-Bernstein theorem is not intu- 60 itionistically valid. To see that the Cantor-Bernstein theorem fails in general / consider the two-object category 2 = 0- 1, and consider the presheaf topos Set2 . Let 2 denote the map N —> N given by n 2n, and consider the two presheaves A and B given below. id A N- N B= N. N Consider the natural transformations a and r given by: a : A —*- B = id T : B^A= 2 a and r are both monomorphisms (since they are pointwise mono), but A and B are not isomorphic as presheaves. So, under the traditional definition of cardinality, we have that \A\ < \B\, \B\ < \A\, but \A\ ^ \B\, which contradicts the Cantor-Bernstein theorem. Without the Cantor-Bernstein theorem, there is no longer a connection between the notions of cardinality defined by the traditional definitions of \A\ < \B\ and \A\ = \B\. So, in defining cardinality in an arbitrary strong category of classes, wTe can choose to either use the traditional definition of the equivalence relation \A\ = \B\ (in which case \A\ — \B\ iff A ~ B) and then find a new preorder \A\ < \B\ for which the Cantor-Bernstein theorem holds, or use the traditional definition of the preorder \A\ < \B\, and find a new equivalence relation for which the Cantor-Bernstein theorem holds. It is a general result (valid intuitionistically), that, given any preorder A < B, the definition of A~BasA 61 cardinality as a preorder "set A is smaller than set B", and then it will be clear how to define an appropriate equivalence relation "set A is the same size as set Bv for which the Cantor-Bernstein theorem holds. 4.2 Mono-cardinality and Epi-cardinality In classical set theory, we have two equivalent definitions of cardinality: 1. We say that a set A has cardinality less than or equal to a set B when there is an injection A >-^B. 2. We say that a set B has cardinality greater than or equal to a set A when there is a surjection B—»~ A. These definitions easily generalize to a strong category of classes by replacing "injection" and "surjection" with "mono" and "epi". I will denote the notion of cardinality defined by the first definition by Ce (intended to resemble the relation C of inclusion) and that given by the second definition by )pe (intended to resemble the double arrowhead on the arrow for an epi). The reason for the subscript e is that this inequalities here are defined "externally", rather than in the internal language of a given strong category of classes. The classical result that A Ce B implies B )pe A (provided that A is nonempty) requires the law of the excluded middle: the standard surjection from B to A is defined piecewise on im(^4) and its complement. Such a piecewise definition requires excluded middle because otherwise there is no guarantee that im(A) even has a complement. The result that B ^e A implies A\Ze B requires the axiom of choice. In fact, finding such an injection from A to B is equivalent to choosing an element from each fibre of the surjection onto A. So, in an arbitrary strong category of classes, without (IAC) or (EM), we have no guarantee that these two definitions of cardinal inequality are equivalent. In fact, we can provide an example of a category of classes in which these two definitions are not equivalent. Consider the topos Set2 given above, and 62 consider the presheaves A and B defined above. There is a monomorphism B ^^ A given above, so B Ele A. However, given the arrows a, j3 : B I A given by N ^-^N N ^^N a — P n^[n/2\ nt-*[(n+l)/2\ there is no arrow cj> : A —>• B for which a o ^ j3 o <\>. This means that there is no epi A —>- B, and thus it is not the case that A )pe B. Similarly, if we consider the presheaf C given by C= "-L"/2J N ; N there is an obvious epimorphism A —»- C, but if we consider natural trans formations 5,7 : A I C given by § = ni—>2n ro—>[n/2J ™ — n>—»2n+l n^[n/2J there is no ^ : C —>- /I for which tp o 7 ^ ip o 5. This means that /I ^e C, but it is not the case that C Ce A. So, in this topos, neither of these two cardinal inequalities implies the other. 4.3 Internal and External language In order to define the inequalities C and =^ precisely in an arbitrary strong category of classes, I will reformulate these inequalities in the internal language of the category in question. 63 Definition 4.3.1. If A and B are small objects of a strong category of classes, we define A £1 B to be the internal language formula 3jeBAVXj?/e/i(/(x) = f(y) =>x = y), and A =4 B to be the formula 3f€AB\'yeA3xeB(f(x) = y). The statement that £ lh A C B does not quite mean that there is a monomorphism A—*~B; it in fact means, more generally, that there is some in- habited object X, and a map A x X ^B such that < /. 7Ti >: A x X—^B x X is mono. Clearly, if this object X has a global element 1 —^X, then such an / induces a monomorphism from A—^B. Thus, in a strong category of classes £ in which every inhabited object has a global element (the topos Set2 is such an example), £ lh A C B iff there is a mono from A to B. Similarly, the statement that £ lh A =4 B means that there is an X and / as above such that < /> TI"I >: B x X —>• A x X is epi, and if every inhabited object has a global element, then £ lh B =$ A iff there is an epi from A to B. Using C and = Definition 4.3.2. For A and B small objects of a strong category of classes, we define A \Z B to be the internal language statement (A C B) A —>{B C A), and A ^ B to be (A =$ B) A ->(B ^ A). The negation in these statements is in the internal language, meaning that A C B does not just say that there are no monos from B to A, but that there are not even any "partial monos" (a "partial mono" would be a map / : X xA—^B for X a nonzero subobject of f with < /, -nx >: A x X —>• B x X mono). While I will use the internal language cardinal inequalities C, =^. c, -< when ever possible, I may sometimes still have need of the external statements about the existence of monos and epis. For example, if A and B are objects that are not known to be small in a category of classes, we have no guarantee that the exponential AB will exist. However, we can still define an external category- theoretic notion of cardinal inequality between them. 64 Definition 4.3.3. For arbitrary objects A and B of a strong category of classes, we say that A Qe B iff there is a mono from A to B, A \ze B iff there is a mono from A to B but not from B to A, and we define A =4e B and A - It should be noted that, in any category of classes in which every inhabited object has a global element, the internal and external weak cardinal inequalities (but not necessarily the strict ones) agree. But in general we have distinct internal and external methods of talking about cardinality in a category of classes £. It should be noted that if A Ce B then clearly A Q B, and if A =4e B then A = 4.4 Combining Epi- and Mono- Cardinality Neither ^ nor C will on their own suffice to fully define cardinality in a strong category of classes. CI is a convenient definition of cardinality since we want subobjects of a given object to be smaller than the object itself. However, the intuitive definition of "countability" of an object A involves the existence of an enumeration function e : N —»• A, thus =^ is the notion of cardinality that we use to define count ability. It is very much possible to combine the two notions of cardinality together, by saying that B is smaller than A exactly when there is an epimorphic partial function from A to B. Definition 4.4.1. Given A and B small objects of a strong category of classes. 65 we define A < B to be the internal language formula 3/ep(^xS)Vfc)b'GB((3a6^((a, b) G / A (a, b') G /)) =>• 6 = b') A Vj,e_B3agyi(a, and we define A < B to be the internal language formula A < B A —>B < A. Given arbitrary objects A and B, we define A Q and ^ are preorders on the set of small objects of a strong category of classes £ because they are reflexive and transitive. Reflexivity follows from the fact that the identity map is mono and epi, and transitivity follows from the fact that monos and epis both compose to yield new monos and epis respectively. We may worry at first that the relation < that we have defined will not be a preorder. However, as with C and =^, the identity map is an epimorphic partial map, and the composite (as partial maps) of two epimorphic partial maps will again be an epimorphic partial map. If we have objects A B and C, and subobjects S of A and R of B and epimorphisms / : S —>- B and g : R —>- C, consider the diagram below: ABC where the bottom diamond is a pullback. Then, by the preservation of monos and epis under pullback, S' will be a subobject of A with an epimorphism S' —^ C, which will be the composite of the two given epimorphic partial maps. Thus A >e B and B >e C implies A >e C. This entire argument can be reformulated in the internal language of a strong category of classes £ to yield the implication that A > B A B > C => A > C. 66 It should be clear that, in any strong category of classes 8, A C B =>- A < B and A =4 B ==>• A < B. To prove the first informally in the internal language, take the subobject S of B to be.A>-*-B given by A C .£?, and the epimorphism yl —> yl to be the identity. To prove the second, take the subobject S to be B itself, and the epimorphism to be the epi given by A =4 B. 4.5 Relationship with Excluded Middle and the Axiom of Choice We know that, in classical set theory, the preorders A C B, A =4 B and A < B agree on nonempty objects. In fact, the agreement between these three relations holds in any strong category of classes which satisfies (IAC) (and hence (EM) - see [13] for a proof that (EM) follows from (IAC)). In the Boolean case, the preorder A < B reduces to A =4 B (on non-empty objects). In addition, in any strong category of classes satisfying (IAC), the preorder A < B reduces to A IZ B. Thus, in a strong category of classes satisfying (IAC), all three preorders coincide. This result is captured in the two theorems that follow, which are based upon arguments present in [13] and [18]. Theorem 4.5.1. In any strong category of classes 8 satisfying (EM), if A and B B are small objects, we have that 8 \\- A =4 B A < B A 3/£A (/ = f). Proof. We will prove this theorem informally in the internal language of 8. We have already argued above that in any strong category of classes 8, A =4 B =^> A < B. Also, 3feAB(f = f) follows from the fact that, in the case of A =4 B, there is an (internal) epi from B —>• A. B So, now suppose that A < B A 3/6J4 (/ = /)• Let S be the subobject of B and / the epi S —^ A witnessing A < B. By (EM) S has a complement ->S such that B ~ S + ->S. So, to define a map B —3- A, it suffices to define two maps S —^ A and ~>S —>• A. Since AB is inhabited, we can find a map g : B —5- A, and restrict it to -^S to get a map —>S —>• A. We let the map 67 S —>- A be /. Then, map B —*- A given by the coproduct will be epi since / is epi, and we are done. • Theorem 4.5.2. In any strong category of classes £ satisfying .(IAC); if A and B are small objects, 8 lh A C. B <==> A < B. Proof. Again, we have argued above that if A C B then A < B. To see the converse, suppose that A < B. We will argue informally in the internal language. There is thus a subobject S of B and an epi / : 5—>A. By (IAC), / has a section s : A —>• S. Since / o s is the identity, s must be mono, so the composite of s with the inclusion i : S—*~ B is again mono, and we have a monomorphism A—*~B. • 4.6 Cantor's Theorem The notion of cardinality would be uninteresting if all infinite sets had the same cardinality. So, one of the first things to be done in any study of cardinality is to prove some strict cardinal inequalities between infinite sets. The simplest classical example of this is Cantor's theorem, which states that the cardinality of any set X is strictly less than that of its power set V(X). Cantor's original proof of the theorem uses the famous "diagonal argument" which constructs, given a surjection X —^VX, a new subset of X which is not in the range of this surjection, yielding a contradiction. We will see that this proof can be easily generalized to the non-classical case. Theorem 4.6.1. If A is any small object of a strong category of classes £, then£\\- A Proof First, to show that A < V(A), it is enough to show that A Ce V(A) by constructing a mono A—^V(A). Intuitively, this mono will be the "singleton" function a i—> {a}. We can define it by considering the "equality" relation A >—^->- A x A, and taking the corresponding map A :—*-.V(A), given by 68 the universal property of the power-object. We can show directly that { —}A defined as such is monic by noting that, for any / : X —*- A, we have both squares (and hence the big rectangle) pullbacks in the diagram: X A eA AxX + Ax A AxP(A) idAxf idAx{-}A Thus, if we have two functions f,g:X I A for which {-^0/= { — }A o g1 then the bottom row of the above diagram will be the same for both these functions, and then we have that < /, idx > and < g, idx > are both pullbacks of the same map. So, there is an isomorphism a of X for which < /, idx > oa =< g, idx >, but then idx ° a = idx, so a is the identity and / = g. This implies that { — }A is monic, and so we have a mono witnessing E lh A C.e V(A). To show the strict inequality, we wish to show that, internally, there are no epimorphic partial functions from A to V(A). I will do so via an informal argument in the internal language. Note that while proof by contradiction is used in this proof, it is only used to prove negative statements, thus the proof is in fact intuitionistically valid. First, assume we have a subobject S of A with an epimorphism / : S—>V{A). Define a subobject R, of S by {s E S : s> ^ f(s)}. Since A is small, S and R will both be small, so R can be seen as a term of type V(A). Thus, since / is epi, there is a r 6 S such that f(r) = R. We will first prove by contradiction that r $ R. Suppose r E R. Then, by definition of R, r E~ f(r). But, since f(r) = R, this is a contraction. So, r E1 R. But then, since f(r) = i?, we have that r E1 /(?")• But thus, by the definition of R, r E R. WThich is again a contradiction. This contradicts the assumption that / is epi. So, we have A-jtV{A). a This result and its proof have consequences for versions of Cantor's theorem 69 for =^ and tZ. The first corollary follows from the fact that C and = Corollary 4.6.2. For any small object A of a strong category of classes, we have A^V(A) and A^V{A). Corollary 4.6.3. For any small object A of a strong category of classes, we have A C. V{A). 4.6.1 A counterexample While it is now clear that A )fi V(A) in general, and that A C V(A), there seems to be no general way to prove that, for an inhabited object A, A =<; V(A) - there is no canonical epi from V(A) to A. We can, in fact, construct a counterexample. Let V be the category: ^c -< •% >- % Then, consider the topos Setv°P, and the presheaf A: {o}c—-{o,i}-—>{i} Then, if we let So denote the 6-element set of subpresheaves of {0]c—>{0,1}, and S\ the 6-element set of subpresheaves of {1}°—>- {0,1}, then the presheaf V(A) is: S0^-V({0A})~^Si / -f Suppose now that we have an inhabited presheaf E = E0 —^ £"0,1 ^—-— Ei and a map a: 70 S0 x E0 P({0,l})xE0, S1 x Ex CTQ O"0.1 {0} {0,1} {1} I will show that the map (a, ir2) into the product Ax E cannot be epi, thus showing that, in this topos, it is not the case that V(A) )p A. I will do this by s n e using corollary 2.5.4 and showing that (co;i,7r2) i °t Pi- Since E is inhabited, E0 is nonempty, and so we have some e0 G E0. For any s G V({0,1}), there is an s0 G So such that s is the image of sQ. Since 0"o(sOjeo) = 0, we must have cro,i(s;/o(eo)) = 0 for any such s G 5. Now, e consider the element (1, /o( o)) of {0, l}xEoti. If it lies in the range of (cr0)1,7T2), then 1 must be the image under <70,i of some (s, /o(eo))- But this is impossible by the above argument. Thus, the range of (<7o,i, TT2) doesn't contain (1, /o(eo)), and thus (oo.i, TT2) cannot be epi. This yields a counterexample to the statement V(A) V A. 71 Chapter 5 The Continuum Hypothesis In its classical form, the Continuum Hypothesis states that there are no sets which lie intermediate in cardinality between the natural numbers and the con tinuum. In the last chapter I discussed how, without the classical assumptions of (EM) and (AC), the notion of cardinality becomes much more complicated. Similarly, I will note here that the Continuum Hypothesis itself is a lot more complicated, even if I use < as the only notion of cardinality under consider ation. This complication arises from the fact that the notion of "continuum'' becomes a lot more complicated without classical assumptions. The goal of this chapter is to define the continuum hypothesis and its nega tion in an arbitrary strong category of classes. I will open this chapter with a discussion of the constructions needed in order to quantify over all sets of a certain cardinality. I will then proceed to discuss what the continuum hy pothesis and its negation will look like once we fix a certain object R to stand for "the continuum". I will then examine a number of different candidates for R - both the object 7>(N) and the different real numbers objects. I will then briefly discuss the strategies that one might use for proving the consistency of the continuum hypothesis. The next two chapters of this thesis will then be focused on the proof of the consistency of its negation. 72 5.1 Subquotients and PERs The Continuum Hypothesis involves a quantification over all sets which are smaller than the continuum (which I will for now not define and simply denote by R). Classically, one only needs to quantify over the subsets of R. However, in the case of an arbitrary strong category of classes, the notion of "smaller than it"' is more complicated. Given an object A in a strong category of classes, there are two different ways of making from A an object smaller than A. We can take a subobject S>—>A, or a quotient A—^Q. Subobjects of A correspond to objects that are smaller than A under the Ce ordering, and quotients of A correspond to object that are smaller under the =4e ordering. The question remains as to what to call an object which is smaller than A under the Definition 5.1.1. We say that an object B is a "subquotient" of A if B is a quotient of a subobject of A, that is if B In order to quantify over all subquotients of R, we will need a single object that corresponds to all these subquotients. We have, for any small object A, an object V(A) of subobjects of A. We would like to, similarly, create an object SQ(A) which will be the "object of all subquotients". Every map X—^SQ(A) should correspond to a "subquotient-over-X of A". A "subquotient-over-X of A" is just a subobject 5 of X x A, together with a map S > B such that the pair (TTI, f) '• S—*-X x B is epi. This object SQ(A) should have some universal subquotient-over- 73 X x B - »- A^ ;: k S *AA v X xA^^SQ(A) xA In the case of Set, SQ(A) is {a £ P(P(4)) : VseaS + 0AVSiT6aSnT = 0}. Then AA is {a £ A. a E SQ{A) : 3Seaa £ S}, and XA is {S £ V{A)1a £ SQ(A) : S £ a}, and the epi A^ to A^ sends (a, a) to (S, a), where a £ f S £ a. Given a subquotient-over-X S —»• B , and x £ X, let Sx denote {a £ A : (a, x) £ S}, and fx : Sx —^ B be the map a *-> /(a. x). The the map 1 g : X —>- «SQ(v4) sends each x G AT to the set {/^" (6) : b £ B} of fibres of fx over X (which will all be subsets of A), and the map h : X x B —>- A^ will 1 send (x, 6) to the pair (/aT (6), g(x))- Just as, in Set, we can define SQ(A) to be a certain subset of the double power set of A consisting of all partitions of A into disjoint nonempty classes (that don't necessarily cover all of A), we can define SQ(A) in an arbitrary strong category of classes using a similar internal language formula. Definition 5.1.2. For a small object A in a strong category of classes, we define SQ(A) to be the subobject ofV(P(A)) given by \\a £ P(P(A)) : Vsea^aeAa £SA Vs,T6Q((3a6yl(a G S A a G T)) => S = T)\\ However, there is a simpler way of defining the object of subquotients. Because we are working in a pretopos, epimorphisms e : S^^B are in bijection with equivalence relations R>~^S x S. So, every quotient B of a subobject S of A could be equally well viewed as an equivalence relation R>~^S x S^-^A x A. • An equivalence relation R on a subobject S of A, is what is called a "partial 74 \ equivalence relation" or "PER" on A. A relation R on A x A is a PER if and only if it is symmetric and transitive. We can then recover S from R by defining S — \\a £ A : R(a,a)\\. Definition 5.1.3. For any small object A in a strong category of classes, we define the object V£TZ(A) to be the subobject ofV(A x A) given by \\r £ P(AxA) : Va>66A((a, b) £ r => (6, a) £ r-)AVaAcgyi((a, b) £ r/\{b, c) £ r => (a, c) £ r Theorem 5.1.1. For ant/ smal/ object A in a strong category of classes, the objects SQ(A) and VSTZ(A) are isomorphic. Proof I will construct, informally in the internal language, maps / : SQ{A)—>-V£lZ(A) and g : V£7Z(A) —>• SQ(A) which will be mutually inverse. / associates to any a £ SQ(A) the relation r in V(A x A) given by r(a, b) -<=> 3Sep(A)(S £ a A a, £ S Ab £ 5). r will be symmetric directly from the definition, and will be transitive because if there is a set S in a containing a and b and a set T contain ing b and c, then, since b is in both S and T, S1 = T is a set containing a and c. Now, to construct g : V£7Z(A)—*-SQ(A), let a symmetric, transitive relation r be given, and define a to be {.S £ V(A) : 3aeA(r(a, a) AV(,g^(6 £ 5* <^=> r(a, &)))}• Each such a consists of nonempty sets S since each will contain the a which is assumed to satisfy r(a,a). Each will consist of disjoint sets, since if S and T both contain the same element 6, we have a? and as such that r(as, b), r(aT, b), c £ S <^=> r(as,c), and c £ T -<=>• r(ar,c). By symmetry and transitivity of r, this implies that c £ 5 if and only if c £ T. This shows that / and g are maps between the correct pair of objects. It is an easy verification to show that f(g(r)) = r and g(f(a)) = a. • When quantifying over subquotients of an object A, I will often quantify over V£1Z(A) rather than SQ(A), since V£7Z(A) is defined as a subobject of V(A x A), which is in some sense less complex than the double power-object V(V(A)). V£TZ(A) is also important in other contexts in category theory, not 75 directly related to the study of cardinality. For example, in [22], Scott defines a category whose objects are partial equivalence relations, and uses this category as a Cartesian closed category appropriate for interpreting the A-calculus of theoretical computer science. The general result that, in an appropriate con text, categories of partial equivalence relations form Cartesian closed categories is discussed in [5]. It is interesting to note that Scott introduces PERs as an alternative to subsets, just as we have done. 5.2 Formulating (CH) and (NCH) If we want to formulate the continuum hypothesis in a strong category of classes S, we will need some small object R to correspond to the "set of real numbers". Given such an R, the Continuum Hypothesis will be the statement that there is no other object X such that N < X < R. The negation of the Continuum Hypothesis will state that there exists such an X. Thus both the Continuum Hypothesis and its negation involve quantification over all such objects X - quantification over subquotients of R. We will use the object V£7Z(R), defined in the last section, to do this. Since X will be viewed as a term of type V£TZ(R), we need a way of defining cardinal inequalities between terms of type V£TZ(R), not just between small objects. Let us work in Set for a moment. Suppose we have two partial equivalence relations a and j3 on objects R and S respectively, and let A and D denote their respective quotients. Let dom(o;) = {a G R : a(a,a)} and dom(/?) = {b G S : a(6, b)}, and, given, a in dom(a), we let a denote the equivalence class of a mod a, and similarly define b. If we have a surjective partial function / between A and B, this will yield a relation 0 on R x S such that a G dom(a) A /(a) = b. Similarly, given a relation 76 an a G dom(a) with (a, b). Then we can see that / will be a surjective partial function. So, I make the definition below. Definition 5.2.1. For any terms a,/3 of type V£TZ(R) C V(R x R), we define a < /? to be the internal language statement: 3fev(RxR)( V6efl((&, b) E p => 3a6H((a, b) E / A (a, a) e a)) A Voi,a2,6i,626fl«ai: &i> e / A (a2, b2) £ f A {alt a2) G a =^> (61, 62) G /?)) PFe oiso define a < (3 accordingly. By the above definition, terms and not just objects can be compared using cardinal inequalities. Note here that N and R, as they will both be subquotients of R, can be regarded as terms of type V£TZ(R). I then define the continuum hypothesis (CHR) and its negation (NCH#) for any small object R with N (NCH^) 3aeV£n{R)N (CHR) Va6KR(fl)Q Note that (NCH#) is not, strictly speaking, the negation of (CHR). The reason for this is, that, in the non-classical setting, there are ways that (CHR) can fail, other than there being a cardinality between, N and V(R). (CHR) is actually a very strong statement because it implies that there is no subquotient of R which is incomparable to N under <. The fact that < is not necessarily a linear order makes it much easier for (CHR) to fail. Thus, I will not just prove that strong categories categories of classes in which (CHR) fails, but that there are models in which it fails because (NCH#) holds. 5.3 Different choices for R It was mentioned earlier that it is not entirely clear what object of a strong category of classes S should take the place of R in the formulation of (NCH.R). 77 Since I proved in the previous chapter that N < 7>(N), we might think that V(N) should take the place of K. This allows us to formulate (NCHp(N)). In fact Cohen's "forcing" proof of the independence of the continuum hypoth esis generalizes easily to construct a topos (or category of classes) in which (NCH-p(pj)) is satisfied (see chapter 6). However, such a proof will turn out to be too weak. The reason is that there are many strong categories of classes in which there is an object of intermediate cardinality between N and 'P(N), but this object itself has some claim to still being "the continuum". We don't want it to be the continuum itself that negates the continuum hypothesis. To see a concrete example, consider the topos Set73^, where 'P(R) is given some well-ordering, making it order-isomorphic to an ordinal a, and thus mak ing it into a category with arrows given by the order relation. The natural numbers object in this category is the constant presheaf AN with value N. To explicitly describe the object T-^AN) in this topos. I need to first make some set-theoretic definitions. For every ordinal i < a, let «j denote the set {j G a : j > i} U {*}, and, for i < j let TTJJ denote the map «j —>• ctj which sends everything > i and < j to j, and leaves * and everything else in a fixed. Proposition 5.3.1. The object P(AN) in the topos Set7'^ is the presheaf which sends each i £ a to a^ and the arrow i —>- j to 7rN-. Proof. It is well known that, in a presheaf topos, the power-object V(A) of a presheaf A is the functor which associates, to any object X the set of all subfunctors of the functor y{X) x A, where y is the Yoneda functor. For the case of Set7>^R-), X will be an ordinal i < a, and y(i) will be the functor which sends j > i to the one-element set and j < i to the emptyset. So y(i) x N is the functor which takes constant value N on {j e a : j > i} and is the empty set elsewhere. A subfunctor of such a functor is simply a chain {Sj : j > i} of nested subsets of N, where Sj C Sk for j < k. Such a chain of nested subsets can alternately be given as a function S : N —*- cti where each n 6 N is sent to k if Sk is the minimal Sj containing n, or is sent to * if n is not in any of the 78 Sj's. If i < j, the map which restricts a subfunctor of y(i) x N to y(j) x N, will forget about all of the Sk's for k < j. Thus, if Sk for k < j was the minimal one containing n before restriction, Sj will be the minimal one after restriction. This shows why trfj is the restriction map af —*~af- D Now, we may consider the presheaf AIR which sends every object to M and every arrow to the identity. We have an inclusion R —>• 2^ ~ 2N which sends each real to the lower half of the Dedekind cut defining it, which becomes a monomorphism between constant presheaves AM—>• A(2N). We also have, for each i, an inclusion ji : 2>—^a^ which sends 0 to * and 1 to i. It can be easily seen that these /j's are natural in i, and thus, together, they define a mono /N : A(2N) -^V(AN). So, we definitely have that AM C P(AN), and thus AM < P(AN). I will now show that there is no cardinal inequality in the other direction. To see this, note that, since every inhabited object in Set17 has a global element (this is because a has an initial object), it is enough to show that there is no subobject of AM which maps epimorphically onto V(N). A subobject of AM is simply given by a nested chain of subsets Si of M such that Si C Sj when i < j. If we had an epi from such a subfunctor to V(N), it would have to be pointwise epi (by corollary 2.5.4), in particular, we would have a surjection from So to (a U {*})N. But, by Cantor's theorem for sets, since a ~ V(M), there can be no such epi, so we have that AM < P(AN). It is also easy to show that AN < AM So, this object AM, negates (NCHp(N)). However, this object AM is in some sense the object that corresponds the "real numbers" in Setp(R) (we '11 see how in the next section). We don't want the continuum hypothesis to be negated by the real numbers - there should have to be something smaller required to negate it. Thus, V(N) is not the best choice for R in (NCHH). 79 5.3.1 Real Numbers as Dedekind Cuts In an arbitrary strong category of classes 8, the rational numbers object Q can be easily defined as a certain subset of N x N. The order relation < on Q can also be defined easily using the order relation on N (see [18] for details). Moreover, in an arbitrary strong category of classes, the objects N and Q are in fact isomorphic - the isomorphism between them can be defined recursively using the same definition as the bijection between them in Set. This implies that the order relation on Q is "decidable" - Q lh VXj3/eQX < y\J x = y\/ x > y. Using the rationals, we can then define the real numbers. One way of doing so is with the definition of the real numbers as Dedekind cuts. Definition 5.3.1. We say that a term L of type V(Q) is an "open lower section" if it satisfies the following 1. BqeQ(q e L) 2- Vq,reQ(q 3- Vg€Q(g Gl^> 3r6Q(r 6 L A q < r)) and we say that a term U is an "open upper section" if it satisfies the same con ditions with < replaced by >. We let Q( and Qu denote the subobjects ofV(Q) consisting of all open lower sections and all open upper sections respectively. A Dedekind cut will then consist of a pair (L,U) such that L is an open lower section and U is an open upper section, so that L and U are disjoint but almost meet. Classically, there are many ways of formulating this notion of "L and U almost meet". In the case of an arbitrary topos, two different ways of formulating this yield two different definitions of the real numbers. Definition 5.3.2. The Dedekind (also sometimes called "Dedekind-Tierney") real numbers object R^ is defined as the subobject of Q; x Qu given by \\L, U : VxeoHa: E LAx e U)A\/x 80 The MacNeille (also sometimes called "Dedekind-MacNeille") real numbers object Mm is defined as the subobject of Qi x Q„ given by \\L,U : V^Q-^X G LAX G U) AVx < y e Q((->x G L => yeU)A(^yeU => x e L))\\. It should be clear that the Dedekind reals are a subobject of the MacNeille reals, and in many strong categories of classes, these two objects coincide. In the case of the topos Set^M', the Dedekind real numbers object and the MacNeille real numbers object are both the presheaf AM discussed above. Also, any pair (L, U) can be recovered from just the lower section L by the construction U = {x G Q : 3y 5.3.2 Real Numbers as Cauchy Sequences There is also another sort of real numbers that can be defined in a strong category of classes. Instead of defining the real numbers as a subobject of Dedekind cuts of V(Q) x P(Q), we can also define them as'the subquotient of QN corresponding to equivalence classes' of Cauchy sequences. We let CS(Q) denote the subobject of QN consisting of Cauchy sequences - the subobject defined in the internal language by. N \\s G Q : Vee(Q(e > 0 =3- 3neN\/ij€N(i > n/\j > n =>• s — Sj < eAsj-sl < e))|| We can then define an equivalence relation = on QN by setting = >-^-QN x QN to be the subobject: N \\s. t G Q : VeeQ(e > 0 =» 3nGNVieN(z > n => s4 - U < e A tt - s, < e)) || 81 We can hen restrict (pullback) this equivalence relation to an equivalence rela tion on CS(Q). We can then take the quotient (coequalizer) of this equivalence relation to be the Cauchy real numbers object Rc. The real numbers objects we have considered above are not unrelated. If we have a Cauchy sequence r G Rc, we can define; a corresponding Dcdekind cut as {q e Q : 3„GN3eeQVmeN(m > n =$• q < rm - e)}. This yields a monomorphism from Rc to R Similarly, we noted above that that R^ C Rm and RTO C T-^N). This ordering of our candidates for R enables us to order our versions of the negation of the continuum hypothesis. (NCHp(pj)) is the weakest, followed by (NCHRm), (NCHKd), and (NCHKc) is the strongest. 5.4 The Consistency of the Continuum Hy pothesis The consistency of (CH) with ZFC was known decades before the consistency of (NCH) with ZFC. So, we might wonder if, in the case we are dealing with, the consistency of (CH) with the theory of elementary toposes with natural numbers objects, or the theory of cosmological categories of classes, can be shown easily. In, [27] a proof is given (this proof is due to unpublished work of Joyal) of the existence of a nondegenerate topos in which the Baire space NN is a subquotient of the naturals. Since Q ~ M, our definition of Rc shows it to be a subquotient of the Baire space. Thus, in this topos in question, Rc will in fact be a subquotient of N, and (CHMC) will in fact hold trivially. Of course, such an example is, in some sense, a degenerate example, since Cantor's theorem doesn't hold for the Cauchy reals in such a case. However, it is not entirely clear what additional condition we can put on an elementary topos to rule out such examples. Maybe the more ambitious goal would be to prove the relative consistency of (CH-p(j^) with topos theory, since we know that Cantor's theorem holds for V(N). As 82 far as I know, this has not yet been done. In the case of a cosmological category of classes, the relative consistency of the continuum hypothesis is in fact just a corollary of the corresponding result for ZFC. If we are given a cosmological category of classes £, we can take its Booleanization £-,_, which will still be a cosmological category of classes. We know from section 3.4 that the universal object U in such a category of classes will in fact model a weak version of intuitionistic set theory, and that Joyal and Moerdijk's construction of the initial ZF-algebra will allow us to build a model of a stronger intuitionistic set theory. In the Boolean case, this stronger set theory is just ZF. Thus, we have constructed a model V of ZF, and thus shown ZF to be consistent. We can use this model V to define a constructive universe L within V which will then model ZFC + (CH). We can then define a new cosmological category of classes £' which will have as objects the sets and classes given as predicates on L in the language of ZFC, and as arrows the predicates on L x L which define functional relations between the given predicates. It can been seen then that £' will model (CUp^)). In fact, in this way, consistency results for ZFC will translate directly into consistency results for cosmological categories of classes, since, by variations on the above argument, the existence of a nondegenerate cosmological category of classes will imply the consistency of ZFC and vice versa. However, it is useful to have a method of proving consistency which doesn't rely upon set- theoretic methods, and only relies upon the category-theoretic constructions of taking internal functor categories and Booleanizations. The argument for the consistency of the negation of the continuum hypothesis in the remainder of this thesis is such an argument. I also hope that the above "constructible sets" argument can be rephrased in categorical terms, although doing so is beyond the scope of this thesis. 83 Chapter 6 The Consistency of (NCHpm\) I am now ready to present the first of my "forcing"-style consistency proofs. In this chapter, I will examine Cohen's original consistency proof, and present three generalizations. Cohen's proof dealt with the relative consistency of (NCHP(H)) with standard classical set theory ZFC. I will eventually show that (NCHp(pj)) is consistent with the theory of strong categories of classes, and we will accomplish this in three successive steps of generalization. The first section of this chapter will present a summary of Cohen's proof and the key ideas present there, that I wish to carry over to our category-theoretic proofs. The next section will present a summary of the Lawvere-Tierney proof and how it adapts Cohen's ideas to a category-theoretic context. I will then make a short digression to discuss why the Lawvere-Tierney proof works, us ing the language of classifying toposes. The next section will generalize the Lawvere-Tierney proof to the case in which neither (IAC) nor (EM) is as sumed to hold in the base topos (the fact that this generalization can be done is noted in [9]. although I have yet to find it worked out in the literature). I will then conclude the chapter with a generalization of the relative consistency theorem to the case of a strong category of classes, which turns out to be an easy consequence of the work that will have already been done. 84 6.1 Cohen's Proof Cohen's groundbreaking innovation in set theory was the method of "forcing" that he used to prove that, if ZFC is consistent, then ZFC + (NCH) is also consistent. In short the method involves starting with a given model V of set theory, and constructing a new model V of set theory in which (NCH) is satisfied. For brevity's sake, I will not provide a complete rendition of the proof here, but will comment briefly on how the new model V is constructed. A more or less complete proof is given in [12] The first step in the construction of V is to construct a Boolean algebra B and a "Boolean-valued" model of set theory VB. The formulas in this case will not simply be satisfied or unsatisfied in the model, but will take truth values in the Boolean algebra B. The "elements" of VB will be "i?-valued sets of elements of VB". Since a set a is determined by saying, for each j3 G VB, the truth value of (3 G a, we can realize elements of VB as functions / : VB —^ B. Of course, this definition as it is is circular, but can be reformulated so that it is recursive, (see [12] for a rigourous definition). For elements a and f3 of VB, we can assign to the formula (3 G a the truth-value \\0 G a\\= a(f3). We can then define a semantics on VB in a natural way, such that every formula <\> has a truth-value ||0|| in B. The next step is to define a notion of "I?-generic collection" (I use "collec-. tion" here to denote any extensionally-determined object, whereas "set" will refer to specifically those collections that appear in our model V of ZFC). Definition 6.1.1. Given a poset P with maximal element 1 such that every p.q G P which are bounded below have a meet pf)q, we say that a subcollection G of P is P-generic if and only if it satisfies the following four conditions: 1. leG 3- if p < q and p G G then q G G 3. if p.q G G then 3rec(r < p A r < q) (we might as well take r = pil q) 85 4- if D is a subset of P such that \/pep3q£pq < p A q G D (we say that D is "dense"), then 3P£GP £ D. Since a Boolean algebra B is, in particular, a poset with maximal element, this definition defines a notion of 5-generic collection. Metatheoretic consid erations will tell us that there will exist a generic collection of elements of B, but it will not necessarily be a set. We can then "force" G to appear in our model of set theory (to be a set) by considering a new model of set theory V[G\. For any formula 86 to P. Since G is P-generic, it will be P-generic, and hence G will be P-generic. The existence of a mono from A —>• V(N) will then follow from the following lemma: Lemma 6.1.1. If, in any model ofZFC, there exists a set G which is P-generic, then there exists an injection A—*-V(N). Proof. For aGi.nGN, define pa>n to be the condition with domain {(a.n)} which maps (a,n) to 1. The injection g : A—^V(N) is given by a M {n G N : pa,n £ G}. g will turn out to be a monomorphism because G is generic. Suppose a and b are distinct elements of A. Then, given any condition p, there is some n such that neither (a, n) nor (b. n) is in the domain of p. Thus, we have a stronger condition q which sends (a,n) to 0 and (b,n) to 1. Thus, the set D = {p : 3neNp(a.n) = 0 Ap(b,n) = 1} is dense, and thus it intersects G, since it is generic. Let p be in D n G, and let n be such that p(a, n) = 0 and p(b, n) = 1. We can see that the other conditions on a generic set G entail that Pa,n & G but pb,n G G, and so n G g(b), but n £ g(a) , and thus g(a) and g(b) are distinct. • Lastly, in order to show that V(N) has cardinality at least H^, we must show that A has cardinality at least K„. To prove this we will need the a fact about P known as the "countable chain condition". We say that a poset P satisfies the "countable chain condition" if every set Q C P such that for all p,q G Q, there is no r G P such that r < q and r < p is at most countable. Then, we have the following lemma. Lemma 6.1.2. If P is a poset satisfying the countable chain condition, and G a P-generic subset of P, then if \A\ < \B\ in V, then \A\ < \B\ in V[G] One of the consequences of the above theorem is that if a set S has cardi nality H,3, then S has cardinality at least K^. This entails that, in V[G], the cardinality of V(N) can be made as big as we want it to be. 87 6.2 Boolean Toposes with Choice Cohen's proof constructs, from any given universe of sets, a new model a set theory in which the negation of the continuum hypothesis is satisfied. Thus, if we wish to generalize the results to topos theory, we need a method of construct ing, from a given elementary topos, a new topos satisfying (NCH). fn [23], Lawvere and Tierney presented such a construction for the the closest topos- theoretic analogue of ZFC: the case in which the given topos satisfies both (EM) and (IAC) (Lawvere and Tierney actually assumed two-valuedness as well, but their proof is valid if we only assume (IAC), which in turn implies (EM)). Thus, the theorem they prove is the following: Theorem 6.2.1. If there exists a nondegenerate Boolean topos with choice £, then there exists another nondegenerate Boolean topos with choice £' satisfying. (NCHP(N)). In Cohen's proof, we started with a large set A, and constructed from it a poset P, a Boolean algebra B, and a "i?-generic" subset G of B. This subset G allows us to construct a model V[G] of ZFC and (NCH). If B is a Boolean algebra, and / : VB—*-B is a B-valued set which is only non-zero on ordinary sets, then category-theoretically, / can be regarded as the functor from the category Bop to Set which sends p G B to {a : p < f(a)}. Thus, we will replace the "I?-valued sets" construction with an "internal functor category" construction. In fact, the internal functor category construction we will use will be that for Cohen's poset P of forcing conditions. In Cohen's proof, P was the poset of finite approximations of subsets of A x N. Thus, to define an analogous internal poset P, we must first define what it means for a subset of A x N to be finite. For any object A of a topos £, we define Fin(A) to be the subset of V(A) given in the internal language by \\a G V(A) : 3n£N3jeNAVaeA(a G a =>- f(a) < n)A\/ien{i < w => 3!ae^(a G aAf(a) = i))\\. I then define PQ to be the subobject \\a.p G Fin(A x N) : Va6yixN-.(a G a A a G p)\\ of Fin{A x N) x Fm(A x N). 88 We can think of a as corresponding to the subset of A x N on which the finite approximation is 0 and (5 as the subset on which the finite approximation is 1. We then define a partial order P\ as the subobject of PQ X P0 given by \\{a,P), {y,S) G P0 : Vo6ylxN(a G7^aGa)A(aG<5=»aG /?)||. The fact that this pair of objects, with the obvious domain, codomain and composition maps, forms an internal category in £ can be easily verified. Given this internal category P, we now consider the internal functor cat egory £P°P. In Cohen's construction, an injection was constructed from A to V(N), thus making the cardinality of V(N) larger than A. Similarly, in the topos-theoretic case, we will construct a monomorphism g from an object corre sponding to A to the object P(N) in £P°P (this monomorphism will correspond directly with the injection g given in the proof of theorem 6.1.1). The "object corresponding to A1 will be the "diagonal of A' A{A), which is given by the % P P discrete fibration A x P1 ° i A x P0 in £ ° . An arrow A(^)—>P(N) can idxd alternatively be given by a subobject of A(A) x N. We also note that the nat ural numbers object in £p° is just the diagonal A(N) of the natural numbers object in £. So, we want a subobject G, in £p°v, of A(A) x A(N), which is just a subobject G0 in £ of A x N x P0 that is appropriately functorial (i.e. for (l,n,p) G A x N x Pi, if (l,n,c(p)) G G0, then (l.n,d(p)) G G0). In fact this subobject Go is defined in a way completely analogously to Cohen's case. We : n a define Go in the internal language by ||/ G A, n G N, (ap, j3p) G Po (^ ) £ p\\- We can then prove the following: Lemma 6.2.2. The subobject GQ given above in fact yields a subpresheaf of A(^4) x A(N), and in £v°v. the corresponding map g : A (A)—^P(N) is mono. So, in the topos £P"P, we have at least completed the first step of negating the continuum hypothesis. However, proving that £P°P satisfies (NCHp(N)) will not prove our theorem as £v°v is not even Boolean. Thus, we must move to the Booleanization (£pop)^. We have an "associated sheaf" func tor a : £P°P —5- (£P°P)_^, and we will apply it to g to get a map a(g) : 89 a(A(A))—*~a.(V(N)). Since a preserves monomorphisms, a(g) will also be mono, and thus we will have a(A(J3)) < a(P(N)) in (£pop)^. In fact, the ob ject a(P(N)) is exactly the power set of the natural numbers object in (£P°P)^. So, in the topos £P°P, we have put a(A(B)) below V(N) where we want it. pop Now, to see that (£ )^ satisfies (NCEfp(N)), we will construct an object A' such that N < A' < a(A(^)) in (£P°P)^. Suppose there is some number n and objects A{ for each i < n with A{ < Aj for each i < j, N < A0 and An < A. In Cohen's case, this chain of A's would become a chain of ASs, and similarly in the topos-theoretic case, this chain of Ai's becomes a chain of a(A(ylj))'s. It will follow from the following theorem that we will have the cardinal inequalities N = a(A(N)) < a(A(A0)) < a(A(yl1)) < ... < po a(A(^n)) < a(A(A)) in (£ ")_. It will then suffice to take n = 2, A0 = N, A1 = V(N), A = A2 = V(V(N)). Theorem 6.2.3. For any objects A, B of £. if A < B in 8, then a(A(.4)) < a(A(Z?)) m (£P°P)^. This theorem is proven in [18]. Just as, in Cohen's case, the preservation of cardinal inequalities depended on the Boolean algebra satisfying a "countable chain condition", the proof here depends on the fact that P satisfies an "internal countable chain condition" in £. Since the internal poset P was constructed in a way analogous to the poset in Cohen's proof, we expect this internal countable chain condition to be satisfied in general. However, this proof will fail to generalize it that it depends crucially on the fact that the topos £ satisfies (EM) and (IAC). In fact, the proof uses the fact that A C B <=> A < B <£=> A -< B, among other things. Thus, we shouldn't expect this proof to generalize to the non-Boolean case. So, we can see how Lawvere and Tierney proved that (£p )_,_, in fact satis fies (NC£Lp(N)). This topos will also satisfy (EM) since it is a topos of double negation sheaves. However, to complete the proof of the theorem, we need one more lemma. 90 Lemma 6.2.4. If the topos £ satisfies (EM) and (IAC), and P is an internal poset in £, then the topos (£pop)_,-, satisfies (IAC). For a proof of this lemma in the case £ = Set see [18]. Again, this lemma, like the previous theorem, since it depends on (EM) and (IAC), will not directly generalize to the case of an arbitrary elementary topos. However, it will not be needed, as we will not need to prove that (IAC) is satisfied. 6.3 Connection with Classifying Toposes One may wonder at first what the connection is between this proof and Cohen's proof using a P-generic set G. The connection is that the topos £' = (Setp P)-,-, is in fact the "classifying topos" for the theory (over £) of a P-generic set. Given a first-order theory T (more precisely, an "internal theory" in a topos £), if T is axiomatically simple enough to be what is known as "geometric", there exists a topos £[T] called the "classifying topos" for T, which contains within it a "universal" (the word "generic" is often used in the literature, but I will not use it here, since I use it above with a different meaning) model of T. E[T] will be nondegenerate exactly when T is consistent. The importance of classifying toposes for this proof can be seen in the following theorem. Theorem 6.3.1. If P is an internal poset in a Boolean topos with choice £ which satisfies the conditions of definition 6.1.1, and Tp is the theory of a P'-generic object, then the classifying topos £[Tp] is in fact the topos £' = (Setp°P)_. The "universal model" of Tp will correspond to the generic set G in V[G], and both can be used in analogous ways to construct an appropriate monomor- phism. For a proof of this this theorem see [27]. For more on Classifying Toposes see [13] or [14]. 91 The fact that the £' in question proves the consistency of (NCHp(tj)) relies upon two facts about E'. The fact that £' is the classifying topos for a TP entails that there will be a monomorphism from a(A(^4)) —>P(N) in £'. The additional fact that £' can be constructed using internal functor category and Booleanization constructions is what we need to show that £' is nondegenerate, and that cardinal inequalities will be preserved. 6.4 Generalization to an Arbitrary Topos Let us suppose that we want to prove that (NCHp(^)) is relatively consistent with the theory of toposes with natural numbers object or, equivalently, with higher-order intuitionistic logic. Supposing that topos theory is consistent, we will have a nondegenerate topos with natural numbers object £. We then wish to construct, from this topos £, a new topos £' which will satisfy (NCH^) and will still be nondegenerate. £' will in fact be the topos £P°P, where P is almost the same internal poset defined above in the Lawvere-Tierney proof. However, I won't define P using exactly the same internal-language formula as above, because the notion of finiteness used above is not the correct one to use in an arbitrary topos. In an arbitrary topos, the most mathematically fruitful notion of finiteness is the notion of Kuratowski-finiteness or K-finiteness. For any object A, the power-object V(A) forms a U-semilattice. Definition 6.4.1. We define K(A) to be the sub-U-semilattice of V(A) gen erated by the subobject A > ^V(A). We say that A is K-finite if the arrow 1 {aeAT} > V{A) factors through K{A). We note that K{A) is the subobject of V(A) consisting of all Kuratowski- finite subobjects, in that if a subobject S >—^ A corresponds to a map Xs '• 1 —?- V(A) which factors through K(A), then S is K-finite. For more on Kuratowski-finiteness, see [13]. 92 One of the consequences of the notion of Kuratowski-finiteness is that there is a map f: K(N) -^N such that for all a G K(N), TO) 0 a. Since K(N) is a sub-U semilattice generated by { —} it suffices to define | on 0, {n} and to define it on a U (5 when it is already defined on a and /?. We define j(0) = 0, f({n}) = Sn and j(a U /?) = max(|(a), T(/?))- Intuitively, 1(a) = max(a) + 1. I will use f in the proof that follows. So, given any inhabited object A of £, we can define P as the set of all pairs of disjoint sets in K(A x N) x K(A x N), with the same ordering as above. We can also define the subset G0 of A x N x PQ using the same internal language definition as above, and the first thing we will need to prove is the following theorem: Lemma 6.4.1. If £ is any topos, and. A, P and G0 are as given, then G0 is in fact a subpresheaf of A(A) x A(N). and in £P°P, the corresponding map g : A (.4)—>P(N) is mono. Proof. Let < denote the partial order Pi on P0. We first need to prove that Go is a subpresheaf, that is that £ II- \/aeA^nen^peP0^geQo({p < Q A (a, n, q) 6 G0) => (a,n,p) G Go). I will prove it informally in the internal language. Suppose p < q. Let (otp,Pp), (aq, f3q) denote the pairs of K-finite subsets of AxM corresponding to p and q respectively. Since p < q, aq C ap. Also, since (a, n, q) G G0, (a, n) 6 ag, so (a, n) G ap and thus (a, n,p) € Go- It can be easily verified ([18]) that the object V(N) in £P°P is the internal functor with object of objects P(N)0 being given by the subobject of V(N x Po) x P0 given below. \\a E V(NxP0),p e P0 : V„6NVqePo((n,g) e a ^ (q < pAVrePo(r < q =>• (n, r) G Then, if we let / be the map A x P0—^V(N x P0) which sends (l.p) to {n. q : (/,7T2) : (a, n) G aq A q < p}, then it can be seen that the map A x PQ ^ V(N x -Po) x Po in feet factors through P(N)o as the "object function" go of g. Since to prove that g is mono, it is enough to prove that go is mono, it will also suffice 93 to prove that / is mono. So, suppose we have (a,p), (b,q) such that f((a,p)) — f({b,q)). Since the image of a K-finite set is K-finite and the union of a finite number of 7r a K-finite sets is K-finite (see [13]), we can take n =T( i( p U (3P U ag U (3q)) in N. Let r be the condition given by ar — ap U {(n.a)}, f3r = j3p. Then, clearly (n,r) G f((a,p)), so (n.r) £ f((b,q)). But, then this means that (n,b) G ar, so, since n £ iri(ap), this means that (n,b) = (n,a), so a = b. Also, (n,r) £ f((k,q)) entails that r < q, so (5q C /3p and ag C ar. Since ar = ap U {(n, a}}, and since (n, a) &" aq by choice of n, we have that aq C ap. So, p < q. Now, defining s by as = «9 U {(n,/)} and /?s = /3g, the same argument shows that q < p, so (/,/?) = (k, q), and thus / is mono. D 6.4.1 Preservation of Cardinal Inequalities I have now proved that, in Ev°v, A(>1) E ^(N), and thus A(A) < V(N). In order for this internal functor category construction to actually provide us with a way of negating the continuum hypothesis, we need to know that A(A) will be as big, in £P°P, as A was in £. In other words, we need to know that a chain N < A0 < ... < An < A will be sent to a chain N = A(N) < A(A0) < ... < A(An) < A(A). We need to know that the functor A preserves strict cardinal inequalities. We need not only that that if A < B then A(A) < A(B), but also that if A <£ B then A(A) % A{B). Lemma 6.4.2. If A and B are objects of an elementary topos £ then if A < B in £, A(A) < A(B) m £P°P, and if A t B then A{A) £ A(B). This is simply an internal version of a familiar lemma about natural trans formations between diagonal functors. This proof is only nontrivial because internal functors (discrete fibrations) are a little bit more unwieldy than func- -tors. Proof. First, suppose A < B. Then, there is an inhabited object X, a subobject S of X x B and a map S *- A such that the map S :—*- A x X is epi. 94 2 2 Since X is inhabited, the maps X x P0 — —^ P0 and X x P1 — —^ P1 are epi, so the internal functor A(X) is inhabited as an object of £P°P. Then we will have a subobject A(S) of A(B) x A(X), and a map A(S) A(/) > A{A) given by A(/)0 = (/, TT3) : 5 x P0 C 5 x X x P0 —->- A x P0. The map (/i ^2) xid : S x P0—*-^4 x 1 x P0 is epi since (/, 7r2) is, thus the corresponding arrow A{S) Let X0 denote the "object of objects" of X. The object of objects of A(B) x X is B x XQ, the object of objects of S is a subobject So of B x XQ, and the map P P f of £ ° has object function a map So —^ Ax P0. Let /' denote the map 7i"i o /0. Then, we wish to show that SQ • >- A x X0 is epi. This is exactly the arrow of £ corresponding to the map 5 : *~ A(A) x X, so since the second is epi, so is the first, by corollary 2.5.4. Thus X0 is an object satisfying the criteria at the beginning of this paragraph, and thus, since A ^ B, X0 ~ 0. Since the object 0 is initial (and stable under pullback), any internal presheaf with object of objects isomorphic to 0 must in fact be initial in X, and thus X ~ 0 in SP°P. * D I now still need to show that the poset in question P is in fact inhabited, so that the internal functor category Sp v will be nondegenerate. The map PQ —^ 1 can be seen to be epi since it has a section 1 —>• P0 given by the pair a = 0, j3 = 0 of K-finite subsets of A x N. So, it follows from this and the above two lemmas that: Theorem 6.4.3. // £ is any nondegenerate topos. and A is any object of £ such that A > V(N) (e.g. take A = V(V(N))), and if P is as given, then the topos £v°v is nondegenerate and satisfies (NCH/p(pn). 95 6.5 Generalization to Algebraic Set Theory The further generalization of this theorem to the case in which £ is not neces sarily an elementary topos, but an arbitrary strong category of classes, will turn out to be routine because I defined the notion of "strong category of classes", such that every strong category of classes will have a elementary topos embed ded in it. If £ is a category of classes, the full subcategory £3 of small objects is an elementary topos. We want start with an arbitrary strong category of classes 8, and construct from it a new category of classes £' such that 8' sat isfies (NCH-p(N)). By theorem 3.5.1, £' satisfies (NCHp(N)) as a category of classes exactly when (£')s does as a topos. And we do know how to construct, from the elementary topos 8s, a new topos (8s)' satisfying (NCRV(N)). The question then is whether we can construct a category of classes £' which has (£')s = (£s)'- I wiH use fne following theorem. Theorem 6.5.1. For any internal category C in a strong category of classes 8. c P C P for which the objects C0 and C\ are both small, we have that (£ ° )s = (£s) ° Proof. Clearly, an internal functor in (£s)c°P is also an internal functor in £G°P (since pullbacks in £ and £s coincide). Also, since arrows between internal functors F and G are defined using arrows in £, (£s)C°P is a full subcategory of £C°P. Thus, I only need to prove that (£c"P)s and (8s)c°P have the same objects. Suppose that F is an internal functor in (£c°P)s- Then F is a small object C P of £ ° , so the map F0 —>• C0 is small. Since F\ —>• C\ is a pullback of this map, it is also small. Thus, since CQ and C\ are both small objects of £, the composites F0 —>• Co —>• 1 and Fj —*- C\ —>• 1 are both small, so F0 and F1 are both objects in £s- Thus, the discrete fibration corresponding to F can be realized in (£s)c°P• Now, suppose F is an internal functor in (£s)C°P • Then, the objects FQ and F\ are both small. Since, in a strong category of classes, any map between small objects is small, the map FQ—^CQ is small, so F is small as an object of 96 £C°F, and thus is in (£c°P)s- O This theorem allows us to directly transfer our earlier result from topos theory to algebraic set theory. Suppose we are given a small object A and small objects N < A0 < A\ < ... < An < A. For example, we can take A = Pl(N). Then, we can construct P as above in the internal language. Since the object A is small, the internal poset P will be a subset of a small set and will thus also be small. Therefore, theorem 6.5.1 will be applicable. We we will have A(N) < A(A0) < A(AX) < ... < A{An) < A(A) < V(N) in (£s)G°P by the results of section 6.4, so by theorem 6.5.1, we have that c A(N) < A{A0) < A(yli) < ... < A(An) < A{A) < V(N) in (£ °")s and hence in (£C°P). This then gives us a way of constructing from any given category of classes £, a new category of classes £' satisfying (NCH-p(N)). This yields the main theorem of this chapter. Theorem 6.5.2. // there exists a nondegenerate strong category of classes £, then there exists another nondegenerate strong category of classes £' of the form £p° satisfying (NCHp(^). If £ is insular (a topos), then so is £', and if £ is cosmological than so is £'. This theorem says exactly that (NCHp(N)) is relatively consistent with both topos theory and the theory of cosmological categories of classes. 6.5.1 Remarks on the Generality of this Construction This method of generalizing consistency proofs from topos theory to algebraic set theory works because of the nature of my axiomatization of the notion of "strong category of classes", and because of the nature of the axiom (NCH) that I show to be consistent. It is important to note that the above theorem relies heavily on the fact that within every strong category of classes is em bedded an elementary topos with natural numbers object. Many of the weaker axiomatizations of the notion of "category of classes" presented in the litera- 97 ture (e.g. [4], [24], [2], ...) do not have such a strong correspondence with topos theory. Secondly, it is important that the axiom (in this case (NCH)) that we wish to show to be consistent is definable with reference only to small objects, because we want it to hold in £$ exactly when it holds in £. This is what allows us to deduce that (NCH) holds in £ from the fact that it holds in £$• Theorem 3.5.1 says that any formula of the internal language of the topos of small objects holds in £$ iff it holds in £, however, this need not hold of formulas which contain quantifiers over objects which may or may not be small. In fact such formulas may fail to be expressible in the internal language of £3. For example, suppose we wanted to construct a consistency proof for the generalized continuum hypothesis. (GCH) can be formulated in the internal language of a strong category of classes using quantification over the universe U. However, (GCH) cannot be formulated internally in an elementary topos as a single axiom since it involves quantification over an object which is not small. However, we can do with (GCH) what we did with (IAC), and reformulate it as an axiom schema which implicitly quantifies over all objects of a topos £ (for each object A we would have an internal language formula which says that there is no object B with A < B < V{A)). We can let (GCH') denote this axiom schema, but then we cannot conclude that a category of classes will satisfy (GCH) just because its full subcategory of small objects satisfies (GCH'). This is because (GCH) uses internal quantification over U where (GCH') uses external quantification over small objects, and these two notions of quantification do not in general agree. While there are definitely cases to which the method above does not gener alize, we will see in the next chapter that the argument above does generalize to the case where we consider (NCHRm) rather than (NCHp(N)). 98 Chapter 7 The Consistency of (NCH^) We saw earlier that (NCHp(pj)) is, in some sense, "too easy" to negate, because there are toposes in which the real numbers object itself negates the continuum hypothesis. So, we should not be satisfied with simply proving the consistency of (NCHp(fjj), but should at least prove the consistency of the next stronger version of the negation of the continuum hypothesis: (NCHfm). In order to do so, the method used in the previous chapter will have to be slightly refined and modified. In this chapter I will present a proof of the consistency of (NCHfm). I will first restrict myself to dealing with the case where £ is a topos. I will begin with a section motivating the definition of the poset P and the map g that I will use, and will prove that in £P°P, the appropriate cardinal inequality will hold. It will turn out that the map g will only be mono if the large object A we are considering satisfies a special property. The next section will deal with the construction of an object A which will both be as large as we want it to be and will satisfy this special property of being natural-distinguishable. I will then discuss why the proof I have presented only shows the consistency of (NCHRm) and not (NCHKd). The last section of this chapter will deal with the generalization of my result from topos theory to an arbitrary strong category of classes. 99 7.1 The Poset P and the map g I will first prove the consistency of (NCHjRm) with topos theory, by construct ing a topos in which (NCHRm) holds. One method that comes to mind as a way of generalizing the proof of the consistency of (NCHfp^)) is to undertake a classifying topos construction similar to that remarked upon in section 6.3. In the case that £ satisfies (EM) and (IAC), a geometric theory T^ can be formulated which says that "there is a monomorphism a(A(^4)) —^Mm", and one can consider the classifying topos £"[T^]. This classifying topos will in fact have the required monomorphism in it, however, we have no a priori guarantee that a(A(^4)) will be large in £[TA] or that £[TA] will be nondegenerate. To prove either of these, we would need a concrete construction of £[T^], such as one using internal functor category and Booleanization constructions. Such a concrete construction is not easy to come by. There is a general construction that will yield a classifying topos for any theory, however, this construction does not even, in general, preserve nondegeneracy. Thus, we must look for a different classifying topos to construct. We saw in Theorem 6.3.1 that, given an internal poset P, the classifying topos for the theory of a P-generic object is just (£pop)^. So, we wish to find an internal poset P for which a P-generic set yields a monomorphism from A—^lm as an analogue of Theorem 6.1.1. 7.1.1 Definitions of P and g To construct P, we recall that, in the case of (NCHp(N)), P was the poset of approximations to a map A—*-V(N). Thus, in this case P will be the poset of approximations to a map A—^M^. A real number x determines a Dedekind cut of rationals ({q E Q : q < x}. {q E Q : q > x}), thus an approximation to a real number x is a determination, for each rational q, whether it is definitely < x, definitely > x or too close to x to tell. Thus, we can think of approximations to real numbers as intervals [I. u] containing all the rationals which are "too close 100 to tell". In order to allow our poset P to have a maximal element, we will only consider subintervals [/, u] of [—1,1]. P0 will then be the set of all functions in A (Q x Q) which assign, to each a G A, a pair — 1 < la < ua < 1. Definition 7.1.1. In a topos £, we let [—1,1] denote the subobject ofQ given by \\x G Q : -1 < x < 1||. For f a term of type ([-1,1] x [-1,1])A, we let fi denote TVi(f) and fu denote 7r2(/). Given an object A, we define a poset P byPo = 11/ e ([-1,1] x [-1,1])^ : VaeAfi(a) < /u(a)||, andP1^^P0 x P0 as \\p,q G P0 : Vieyl(j9;(a) > qt{a) Apu(a) < qu(a))\\. We can see that p < q iff, for every a G A, the interval [pi(a), qi(a)] is a subset of the interval [qi(a),qu(a)]. The fact that P then forms an internal poset can be easily checked. Now, suppose we have a P-generic object G. Then, we wish to define a map which sends every set in A to a Dedekind cut. We can define it by a H-> (La, Ua), where La = {x e Q : 3p€G x < pi(a)} and Ua = {x G Q : 3peG x > pu(a)}- The fact that such a map determines a Dedekind cut for each a is the same as the analogous proof below for the more general case of a map g : A(A)—^Km. This map will also be a mono, provided that we assume (EM), for similar reasons to those used in the proof of theorem 6.1.1. So, this will imply that we P P will have a mono a(A(A)) —^ffim — Rd in (£ ° )-,^. However, as before, if we don't assume (IAC) in £, we will have no way of guaranteeing that a(A(A)) will remain large. In the case of (NCHp(pj)), I eliminated dependency on (EM) and (IAC) P P p P by using the topos £ ° rather than (£ ° )-,-n. So, in the case of (NCHRm), I will use the poset P described above, and simply take the presheaf topos Ev°v. P P My goal is to construct a mono g : A(A) —5- Km in this topos £ ° which is analogous to the map a 1—> (La, Ua) given above. Since Km is a subobject of Qi x Qu, which is in turn a subobject of V(Q) x V(Q), I will first construct g as a map into V(Q) x P(Q), and then show that it factors through M.m. First note that a map g : A(A) —*~V(Q) x V(Q) is given by a pair of maps gl.gu : A{A) —*-V(Q). Maps A(^4) —^V(Q) are in correspondence 101 with subobjects of A(A) x A(Q) in £v°v, which are in correspondence with the subobjects of A x Q x F0 in £ which turn out to be subpresheaves. So, I will l u l define g and g as subobjects G0, G$ of A x Q x P0. The object G Q will be given, in the internal language, by \\a G A, x G Q,p G PQ : x < p;(a)||, and GQ by ||a G A, x G Q,p G Po : fl > P«(a)|| To see that there is a corresponding subpresheaf Gz (respectively, G") of A(A) x A(Q), we must show that if p < q l and (a, x, q) G G 0 (respectively, GQ) then (a, x,p) G G0 (respectively, GQ). This holds because if p < q, qi{a) < pi(a) and qu{a) > pu(a), and thus if x < qi(a), x < pi(a), and if x > qu(a), x > pu(a). This thus defines the map g. 7.1.2 g Factors Through Rm The next step is to prove the following theorem: Theorem 7.1.1. For any topos £ and A an object of £, ifP and g are defined as above, g : A(A)—*~V(Q) x V(Q) factors through Mm. Proof. I will begin the proof by showing that g factors through Qt x Qu. If I can show that gl factors through Q;, the proof that gw factors through Q„ l will be completely analogous. Thus, I must show that A(y4) Ih 3X6A(Q)^ G g , l l l A(A) Ih Vx,yeA(Q)(x Next, I need to show that A(A) Ih Vx,yeA(®)(x < yAy G g => x G g). Let 102 through Gl when composed with the map sending (x,y) to x. Since maps in pop £ into A(A)x This completes the argument that g factors through Qj x Qu. Now, I need to show that g, which we now know factors through Q; x Qu, l u actually factors through Mm. First, I show that A(A) lh VxeQ-i(x G g Ax G g ). To show this, I show that Gl and Gu have empty intersection as subobjects of A x Q x P0. This holds true because, for any p G PQ, we cannot have both x < pi(a) and x > pu(a) since pi(a) < pu(a). l u u Lastly, I will show that A(A) lh Vx<2/eQ((->.r G g => y G g ) A (~^y G g => 1 l x G g )). By symmetry, it suffices to show A{A) lh Vx Vg Vg qi{a0), and, in particular, x > piciQ. If we had x < pM(a0), we could 103 {Pu[a x) define q < p by qi(a) = m&x(pi(a), pu(a) - °> ), qu(a) = pu(a). But, for this g, since pu(a0) - fc4at*l = ^(f|)±E) > x > pj(ao)> ft(ao) = M^ which is bigger than x, and so we have a contradiction. Again, since the order on Q is decidable, this proof by contradiction shows that x > p«(a0). Thus, \\a,x,y,p : Vg 7.1.3 g is Mono The next step is to show that the map g is in fact mono. It is enough to prove that gl : A(A)—*~V(Q) is mono, for which it is enough to show that the l corresponding map g 0 in £ is mono. In £, "P(Q)o is a subobject of V(QXPQ)XP0, l and the composite of g 0 with this inclusion is the map A x P0 >-V(Q x P0) x P0 given in the internal language of £ by (a.p) i—> ({x, q : q < p A x < qi(a)},p). Clearly, if f((a,p)) = /((&, q)), thenp = q. So, we need to show, that if we have a,b E A and p E PQ with {.x, r : r < p A x < ri(a)} = {.x, r : r < p A x < r/(6)}, then a — b. If, we have distinct a and b for which n(a) = ri(b) for every rG?o, then it is clear that no such a proof can go through. In an arbitrary topos, there may be objects A for which there are such a and b. So, in order for g to be mono, we have to impose a further condition on the object A. Definition 7.1.2. An object A of a topos £ is natural-distinguishable if £ lh Va,bG^(V/eNA./(a) = /(&)) =>• a = b. Informally, A is natural-distinguishable if, for every a, b in A, if every func tion / : A —>- N agrees on a and b, then a and 6 are identical. Clearly, every object A C N is natural-distinguishable, however, since we want to be able to make A "large", such an A will not do. Every A of the form Nx is natural- distinguishable. This is because we have a map X —> pjN which sends each x x to the "evaluation at x" map ex : N —5- N. Thus if we have a and b such that yfeNAf(a) = f(b), then we will have VxeXex(a) = ex(b), so VX£Xa(x) = b(x), 104 so a = b. We will see later that this will allow us to construct large natural- distinguishable sets A, so I will now proceed by proving that g is mono in the case that A is natural-distinguishable. Lemma 7.1.2. If A is natural-distinguishable, f, and hence g, is mono. Proof. I will work informally in the internal language of £. I wish to prove that if we have a, b,p with {x,r : r < pAx < ri(a)} = {x,r : r < pAx < ri(b)}, then a = b. First note that, if we want to show that two terms x, y of type Q are equal, it is enough to show that the sets {z 6 Q : z < x} and {z G Q : z < y} are equal. To prove it, remember that the order on Q is decidable. If we have x, y such that {z e Q : z < x} = {z E Q : z < y}, then since x ^ {z G Q : z < x}, we can't have x < y, and similarly, can't have y < x, so must have x = y. Since {x.r : r < p A x < T;(a)} = {x,r : r < p A x < n(6)}, we have {x : x < Pi(a)} = {x : x < Pi(b)}. So, by the above pi(a) = pi(b). Also, Pl{a)+ Ma if we define r to be the condition in PQ given by a ^ [ 2 \pu(a)}, then r < p, so we have {x : x < p'(a)+p"(a)} = {x : x < p,(fe)7"(fe)}, and thus Pi(a) + pu(a) = Pi(b) +pu(b), and since we already know that pi{a) = pi(b), we must havethat pu(a) =pu(b). Now, to show that a = b, I will show that for every / : A—^N, f(a) = f(b). Let such an / be given. Let rf be the condition in P0 given by a i—> [pi(a) + P"f(aT+2: >P"(a)]- -^or eveiT /) r^ < P> so f°r every /, we have {x : x < p (a) r{(a)} = {x : x < r[(6)}. So, r/(a) = rf (6), and thus p,(a) + "^'2 = P((^) + /TM+2 • Thus, because p;(a) = Pz(6) and pu(a) = pu{b), we have that f(a) + 2 = /(6) + 2, and thus /(a) = f(b). So, provided that A is natural- dintinguishable, a = b. D 105 7.2 Construction of a Natural-Distinguishable object A If, in our topos £ we are able to construct a natural-distinguishable object A and objects Ai with N < A0 < A\ < ... < An < A, then, by the above lemmas and lemma 6.4.2, we will have that, in £p°v, there are n cardinalities between N and RTO. So, to complete the proof of the consistency of (NCHRm), we need to be able to construct such a natural-distinguishable object A. I noted above that any object of the form Nx will be natural-distinguishable. If, in general, we always had a monomorphism X —>- Nx or an epimorphism Nx —^ X, we would be set, as we could just find a large set X (say Vn(N)), and then take A = Nx. In the topos Set, given an element x G X, we can take Xx £ ^X which maps y E X to 1 if y = x and 0 otherwise. However, if the topos in question is not Boolean, such a piecewise definition will not be legitimate. The goal of this chapter is to construct, from a given nondegenerate topos £, a new topos £' satisfying (NCHKm). However the topos £' need not be of the form £p° . Because when a topos £ is nondegenerate, so is its Booleanization £-,-,, we might as well first construct £^, construct the object A and the internal poset P within £_,_,, and then take £' to be (£-,^)p"P. Then, having N < AQ < Ai < ... < An < A in £_,_, will be enough to guarantee N' < A{A0) < ...< A(/l1) < A in £'. So, to complete the proof of the consistency of (NCHRm) we will let each Ai be the object ^(N), and let A be the object N7'71"1^. This A is clearly n 1 natural-distinguishable. Then, since An = V (N) = IF"" ™, to show that An E A, it is enough to construct a mono Q —>• N. Lemma 7.2.1. In any Boolean topos there is a mono fi—^N. Proof. It is proven in [13] that N ~ 1 + N, and thus N ~ 1 + 1 + N. But, in a Boolean topos, 1 + 1 ~ 0, (see [18] for a proof), and so N ~ fl + N, and the coproduct injection Q —>- N will be a monomorphism. • 106 From the three lemmas above and Theorem 6.4.2 the following theorem then obviously follows. Theorem 7.2.2. From any nondegenerate topos £ we can construct a nonde- P P generate topos of the form (£^) " satisfying (NCHRm). 7.3 The Case of Rd One might think, a priori, that the poset of approximations to a function into Rd would allow us to prove the consistency of (NCHud). However, in the proof above, I only proved the consistency of (NCHRm). The reason for this is that the map g : A(A) —*-V(Q') that I constructed factors through Rm, but not through Rd. x To see that g doesn't factor through M^, I will show that A(A) \f Vx.ye —*-Q x P0, and P0 ^A x P0 be the maps corresponding to x,y, a, then the subobject \\x £ gl(a)Vy £ #u(a)|| corresponds to the subobject of P0 in £ given by \\p £ P0 : ^(rzofa)) < PI(OQ(J>)) V ni(yQ(p)) > pu(a0(p))\\. When x is the map that sends everything to 0 and y sends everything to 1, this subobject will now be \\p £ P0 : 0 < pi{a0(p)) V 1 > Pi(a0(p))\\. The p which sends everything in A to the interval [0,1] will not be in this subobject, because it satisfies 0 = Pi(a0(p)), and 1 = pu(o-o(p)), so this subobject is not all of PQ. Thus, this shows that g does not factor through the Dedekind reals R. A proof of the consistency of (NCHRJ may still be possible. One would have to think of a different poset P and/or a different map g : A(A)—*~V(Q') such that g would actually factor through Mdl not just Mm. However, I have not yet found such a poset. There are other posets P for which the existence of 107 a P-generic set entails the existence of a mono from A(A) —^K^. In the cases which I have tried, the map g that we can construct will still only factor through Mm, not M 7.4 Generalization to Algebraic Set Theory The last thing I will do with (NCHnm) is to show that it is relatively consistent with the theory of a strong category of classes. Given a strong category of classes £, I construct a new category of classes £' satisfying (NCHRm). Again, we do so using the result for toposes and theorem 6.5.1. First, given the strong category of classes £, we can construct its Booleanization £-,-,. We can then define A, P and g accordingly in the topos (£-,-,)s, and consider (£_,_,)pop. The pop results of this chapter will show that ((£-,_,)s) satisfies (NCHHm), and thus pop by theorem 6.5.1, (£_) will satisfy (NCHRm). So, the main result of this chapter is the following theorem, which says that (NCHiRm) is relatively consistent with both topos theory and the theory of cosmological categories of classes. Theorem 7.4.1. If there exists a nondegenerate strong category of classes £, then there exists another nondegenerate strong category of classes £' of the P P form (£^_,) ° satisfying (NCHRm). If £ is insular, then so is £', and if £ is cosmological than so is £'. 108 Chapter 8 Consequences and Corollaries One of the. virtues of the constructions made in the past two chapters is that they are mainly internal functor category, or "presheaf" constructions. Presheaf constructions have an advantage over more general sheaf construc tions (such as those used by Lawvere and Tierney), in that they preserve much more of the structure of a strong category of classes. This preservation of struc ture will allow us to deduce a number of consequences from these constructions which do not follow from Cohen's proof. 8.1 The Diagonal Functor One of the reasons that presheaf constructions preserve so much structure is that the diagonal functor A : £—^£c°p preserves much of the logical structure of the internal language. Given a small internal category C = C\ t Co, c and an object X of £, we define A(X) to be given by the discrete fibration dycid C-y x X ^Cp x X, as was done before. Then, to any subobject S^~^X in cxid £. wc can define a corresponding subobject A(S)>—*-A(X) in £C°P. This allows us to define, for any formula (p(x) in the internal language of £, a corresponding formula A((f)(x)) in the internal language of £C°P via A( 109 The fact that A preserves the logical structure of Theorem 8.1.1. If A(->0) = ->A(0), A{3xeX(p{x, y)) = 3x€A(x)A(0(a;,y)), and A(VxeX \/xe&(x)A((j)(x.y)), where equality between formulas is defined extensionally to mean equality of interpretations. Proof. Since the internal language is defined using the Heyting pretopos struc ture of finite limits, colimits, and universal quantifier (sups and images can be defined using colimits). it is enough to show that this structure is preserved by the functor A. For limits, we see that if the first square below is a pullback in 8, then the second and third squares are also pullbacks, and since limits in 8C°P are constructed using limits in 8, the corresponding square in 8C°P - the image of the first square in 8 under A - will be a pullback. Also, A(l) is the terminal object C in 8C°P, so A will preserve all finite limits. s f A —+ B Ax C0 -^B x C0 AxC\ ^ B x C\ h gxid hxid gxid • • < C ^D CxC0^DxC0 CxCi-*Dxd j jxid ]Xid For coproducts, we note first that it was proven in theorem 2.5.3 that the functor U : 8C°P—>E/CQ creates colimits, so A(0) is the initial object of 8C°P. Since, in a Heyting pretopos, coproducts are stable under pullback, products distribute over coproducts (A + B) x CQ ~ (A x Co) + (B x C0), and similarly for C\. Then, the creation of coproducts by U constructs A(A) + A(B) exactly as A(^4 + B). Thus A preserves coproducts. HO To see that A preserves coequalizers of equivalence relations, note that if R IA—»-B is an equivalence relation and its coequalizer, then stability under pullback implies that R x Co > A x C0 —^ B x C0 and the corresponding diagram for C\ are also coequalizer diagrams, which implies that the diagram A(R) > A(A) —»- A(B) is a coequalizer diagram in £C°P by the creation of colimits by U. The last thing to show is that universal quantification is preserved, that is if S^^-A is a subobject in £ and / : A—>B is given, then A(WfS) is the same C P subobject of A(B) as VA(/)A(5I). If we specialize the definition of V/ in £ ° given in theorem 2.5.3, we get that \/^f)A(S) is {a,x)eAxCo {(cAB(b,p)=yAdAB(b,p) = (A/)0(a,x)) => AS( an Noting that CA(B) d <^A(B) don't affect B, and that A(/)0 doesn't affect Co, and A(5) doesn't depend on C0, we get the equivalent formulation ||(6, y) E B x C0 : Vp€c1VaG^VxeCo((c(p) = y A d(p) =iA /(a) = 6) => 5(a)) || Which can be seen to be just A of ||6 G B : Vaej4(/(a) = b =>• 5(a))||, so thus VA(/)A(5) = A(V/5), and A preserves the universal quantifier. • While this tells us that A preserves the logic of the internal language we have, as of yet, no guarantee that A will actually preserve any non-trivial formulas at all, since we don't yet know of any atomic formulas (other than T and _L) that A must preserve. The atomic formulas preserved by A will be equalities, as by the following lemma. Lemma 8.1.2. If £ is a Heyting pretopos, and =A denotes the equality predi cate on Ax A, then A(=^) is the same map as —A(A)- Proof. We know that =A is given by the diagonal map A —>• A x A. Taking products with Co and C\ will preserve the diagonal-ness of this map, thus A(=A) will be the diagonal map A{A) —^ A(A) x A(A). • 111 In a strong category of classes, we will also have certain canonical term- forming operators preserved by A. The natural numbers object N with its defining arrows 0 : 1 —^N and S : N—^N will be preserved by A. Lemma 8.1.3. If £ is a strong category of classes, N its natural numbers object, and C a small internal category, then A(N) with the arrows A(0) : A(l) —*- A(N) and A(S) : A(N) —^ A(N), is the natural numbers object in £c°v. Proof. Given an object A of £C°P, and arrows a : 1 —*- A, r : A—>• A, an arrow a : A(N) —*- A makes the first diagram below commute iff its object map ao makes the second diagram commute, which occurs iff its exponential transpose a' makes the third diagram commute. A(l)—U-A(N)—-tA(N) Co >NxC0^NxC0 a0 a0 Co A A CTO TO 0 -°—*N ?- N The fact that N is a natural numbers object in £ means that there is a unique a' making the last diagram commute, so there is a unique a making the first diagram commute. We now only need to show that a is in fact an internal natural transfor mation, and to do that we need to show that the two squares below both commute. 112 N x C0 -^ C0 Nxd'^NxCo QO eta A0 Co A Ao a0 d,A In the case of each square, both routes around the square correspond by exponential transposition to a map / making a certain diagram of the form below commute. -—^N —^N X X f Thus, the universal property of N says that these maps must be the same and therefore implies that the diagrams above commute. • The above two lemmas will imply that A preserves the first-order arithmetic of a strong category of classes. For example, we can define, recursively, the maps +, x :Nx N—5~N, and because A will preserve the diagrams used to define + and x, the maps A(+) and A(x) will be the unique maps in £C°P making the corresponding diagrams commute. Thus, if 113 Lemma 8.1.4. If A and B are small objects in a strong category of classes, and C is a small internal category, then the exponential in £C°P A(A)A^ is in fact just A(AB), and the evaluation map e : A(A)A^ x A(B) —>- A(A) is just A of the evaluation map e : AB x B —>• A. Proof. We wish to show that, given any object X of £C°P and a map a : X x A(B)—*-A(A), there exists a unique map a' : X—^A(AB) such that the composite X x A{B) a'xid > A{AB) x A(B) A(e) > A{A) is just a. In other words, given a, we wish to show there is a unique a'Q such that the diagram below commutes, and the horizontal composite of the rectangular part of the diagram yields a. B d xCo X0x B —°-^+ CxxA x B -^^ dxA dx^id dxidxid dxid " ,xid B idxe XQx B ^CQxA x B -^^ CQxA The fact that the bottom-left triangle above commutes, means that a'Q is of B the form (XQ, f) for some / : XQ —^A , thus finding a unique a'Q is equivalent to finding a unique / such that the diagram below commutes and the composite of the bottom row is 7r2(ao) f(ir2)xid B Ci-XCo X0 X B- *- A X B A dx x id id • fxid . n XQ x B — ^AB x B A The universal property of AB says that there is a unique map / such that 114 e(f x id) = 7T2(Q;O)- SO, we just need to show that the diagram above in fact commutes for this /. Since a is an internal natural transformation, the big rectangle commutes. Since the right-hand square obviously commutes, we have that e(/(7T2) x id) = e(f(dx) x id), but that means that f(iT2) and f(dx) are both the exponential transpose of the same map, and thus they must be equal. So, the unique / we have found does yield an internal natural transformation a'0, and thus a'0 always exists and is unique. • Imagine we had a formula Definition 8.1.1. Given a formula 4> such that all the types in d> are generated from a finite list of small types (small objects of £) 1, 0, N, X\, X2,..., Xn via products, coproducts, and exponentiation, and such that (f> contains only only variables; +, x, 0, S, =, and evaluation maps e; A, V7 ->, and =>; and quanti fiers over types generated from the above finite list, then we let From the previous three lemmas follows the following theorem. Theorem 8.1.5. If (p is any formula satisfying the criteria of the above defi nition, and C is an inhabited internal category in £, then \\A(d>)\\ = A(||0||) = an C P n ||0A||; d thus, if (f) has no free variables, £ ° lh 0A iff £ ^ 4>- ^ the case that the list of types X\... Xn is empty, 0A = The fact that C is inhabited is used to conclude, from the fact that ||0A|| is 1 in £C°P, that ||0|| is 1 in £C°P, since 1 in £C°P is just C itself. We can compare this theorem to lemma 6.4.2 about the preservation of cardinal inequalities. A < B is in fact not a formula of the form discussed 115 above, since it quantifies over V(A x B), and not just AB or BA. However, A C B and A =• Sh(c,T){£) which will preserve a lot of the structure of £, but it won't preserve nearly as much structure as does A on its own. We will see that the corollaries stated in this chapter use the fact that A preserves the universal quantifier and exponentials, and thus the same corollaries would not hold if I replaced my presheaf constructions with more general sheaf constructions. 8.2 The Construction for (NCHP(N)) In chapter 6, I provided a way of constructing, from a given strong category of classes, a new strong category of classes satisfying (NCHp(fj)). This construc tion was in fact just a presheaf construction. Starting with the strong category of classes £, I constructed a new strong category of classes of the form £P°P. 116 If £ is a topos, so is £P°P, and if £ is a cosmological category of classes, so is £P"P. Because £P°P was constructed from £ using only an internal functor category construction, there will in fact be many other properties of £ which will be preserved when moving to £V°P, and thus my construction allows us to negate the continuum hypothesis while still satisfying such properties. Theorem 8.2.1. If $ is an internal language formula with no free variables which is defined as for theorem 8.1.5 from an empty list of types Xly..., Xn, then, given any strong category of classes £ satisfying $; we can construct a new strong category of classes £', still satisfying $, but which also satisfies (NCHp(pj)). If £ is a topos then so is £' and if £ is cosmological then so is £'. This theorem follows directly from theorem 8.1.5 and the results of chapter 6. In terms of consistency it has the following corollary. Corollary 8.2.2. //$ as above is consistent with the axioms of topos theory, then so is $ + (NCH^)), and if<& is consistent with the theory of cosmolog ical categories of classes, then so is $ + (NClTp^)). 8.2.1 Church's Thesis There is at least one instance of this corollary which is mathematically inter esting. In a strong category of classes, the universal property of the natural numbers object allows us to define functions recursively, and thus every recur sive function N—^N is realized as an arrow in any strong category of classes £. It is in fact possible to construct a topos £ in which all functions from N—^N are in fact recursive. It is a well-known fact of recursion theory (see [16]), that there is a first- order formula p(n,x,y) in the language (0, S, +, x,=) which says, informally, "the n-th recursive function takes value y at rr". For every recursive function /, there is an n such that f(x) = y if and only if p(n, x, y), and, similarly, if / is such that there is an n such that f(x) = y iff p(n, x. y), then / is recursive. This will allow us to define Church's thesis. 117 The classical version of Church's thesis says that every function N —*- N which is computable (in the sense that we, as thinking beings, can compute it) is in fact recursive. We only require the computable functions to be recur sive because the law of the excluded middle allows us to construct a function which is not recursive (but is not intuitively computable) (again, see [16]). Intuitionistically, we can do better. There is nothing in intuitionism that re quires the existence of a non-computable function, and thus we can ask for a much stronger version of Church's thesis to hold. This motivates the following definition. (CT) V/GNN3„eNVX;y6N(/(x) = y <^^ p(n,x,y)) It can be seen from the definition of Church's thesis, that (CT) is a formula of the type specified in theorem 8.1.5, thus it will be preserved by presheaf constructions. This gives the following corollary of our relative consistency proof. It should be noted that, due to Martin Hyland's ([10]) construction of the effective topos, we in fact know that (CT) is consistent with topos theory, and work is being done on generalizing this "realizability" construction to algebraic set theory (see [25]). For more on the effective topos, see [11]. Corollary 8.2.3. If (CT) is consistent with topos theory or the theory of a cosmological category of classes, then the conjunction of (CT) and (NCH^pj)) is also consistent with the same theory. This result is something that follows from my proofs which does not follow from the results of Cohen, Lawvere and Tierney. As (CT) contradicts (EM), it definitely does not hold in any universe of sets, or in a Boolean topos with choice. Thus the construction of a new universe of sets, or Boolean topos with choice, from an old one, will have no bearing on the relative consistency of the Continuum Hypothesis with (CT). However, in the case of my proof, the base category £ could in fact be a strong category of classes satisfying (CT), in which case, we can prove the consistency of (CT) and (NCHp(n)). So, our result here does go beyond the previously existing consistency proofs. 118 8.2.2 The Cauchy Reals One thing to notice from the above discussion of (CT) is that, in any strong category of classes satisfying (CT), the Baire space NN is actually a subquotient N of N, since every function in N has an n defining it. But, Kc was defined as a subquotient of QN ~ NN, and thus any strong category of classes satisfying (CT) has Rc < N. In particular, this means that (CT) implies (CHR,.), and is thus inconsistent with (NCHmc). The above considerations imply that there is no way of constructing, from a given strong category of classes £, a new strong category of classes £P°P satisfying (NCHRc). If there was, such a construction would preserve (CT), and would result in inconsistency. To see a more direct reason for why this is true, note that Rc was defined as the quotient of a partial equivalence relation = on QN, where the internal language formula defining = was in fact of the form specified in 8.1.5. This means that the PER defining the Cauchy real numbers in £ is in fact sent to the PER defining the Cauchy real numbers in £P°P via the functor A. Since A(NN) = NN, and A(R) = =, it follows that A preserves the quotient as well: A(KC) = Rc. Since A preserves the Cauchy real numbers object, we have no hope of finding a large object A in £ such that C P A(A) < Kc in £ ° unless A itself is already < Mc in £. Whether £ is in fact our "base category of classes" or its Booleanization, we cannot force (NCHKr) to hold in £C°P unless it already holds in £. Thus, if we want to prove the relative consistency of (NCHRc), we will have to use a different method than the presheaf-based method used here. 8.3 The Construction for (CHEJ The construction I used in chapter 7 to prove the consistency of (CHKm) was a little bit more complicated than the presheaf construction used in chapter 6. Given a topos £, I didn't immediately construct an internal functor category £C°P, but first moved to the Booleanization £_,-,- Since we can't expect prop- 119 erties of £ such as (CT) to be preserved by Booleanization constructions, we have no guarantee that (£_,-, )po;P will satisfy any of the same properties that £ does. However, there are still conclusions about (£^)pop which can be drawn from the fact that it is constructed as an internal functor category in £_,-,. We P P recall, that I proved that, in (£^) ° there is a chain of objects N' < A(^40) < l vn N ... < A(An) < A(A) < Mm where A{ = V (N), and A = N ~^ l Since we know that, in an strong category of classes, Mc < W4 < V(N), we have that, in £_,_,, Rc < Ai < ... < An < A. By the arguments above, we know that, in pop (£_) , A(KC) = Rc, and thus Kc < A(4i) < ... < A(An) < A{A) < Rm. This yields the following theorem. Theorem 8.3.1. Given a nondegenerate strong category of classes £, and given a natural number n, we can construct a new nondegenerate strong category of classes £' such that, in £, there are n — 1 objects A2.A3,.. .An such that Ec < A2 < A3 < ... < An < Mm (in other words, there are n — 1 cardinalities lying between the Cauchy and MacNeille reals). If £ is a topos or a cosmological category of classes, then so is £'. This theorem is rather surprising. Classically, Rc and Mm are isomorphic, and one might expect them to lie close together in most strong categories of classes. However, it is clear from the above theorem, that if we want to make the cardinality of Mm large, we can do so while keeping the cardinality of Mc small. It would be even more interesting to provide a method of constructing a topos in which Md and Rm are far apart in cardinality, as they are much more similar in definition than Rc and Mm. However, this endeavour would go beyond the scope of this thesis. It is also important to note that theorem 8.3.1 does not in any way follow from the results of Cohen, Lawvere, or Tierney. The Cohen and Lawvere- Tierney constructions both result in a new model (of ZFC or topos theory) in which the objects (or sets) Mc and Rm are isomorphic. This is because the 120 models in question satisfy (IAC). Thus these constructions are in no way able to force cardinalities to come between M.c and Rm. However, we may still ask if we can do better than this. It would be nice to be able to prove the consistency of (CT) + (NCHRm) in addition to that of (CT) + (NCHp(n)). If we had a pure presheaf construction which resulted in a strong category of classes satisfying (NCHrem) we would in fact achieve this goal, since (CT) is preserved under presheaf constructions. But, in the proof given in chapter 7, we had to Booleanize before applying a presheaf construction in order to ensure that we would have a large natural-distinguishable object A. There are other choices for an internal poset P which seem like they might lead to an internal functor category satisfying (NCHRm). For example as "finite approximations to a Dedekind cut" we could take a pair (Lf, Uf) of disjoint finite subsets, of A x Q such that £ lh Vae^Va:eQVy6Q(a, x) G L A (a, y) G U => x < y. However, in order to make the map g mono in this case, we will still need a sort of "distinguishability" property on A - that there are enough conditions in P such that, given any condition p in Po and a, b in A, if every stronger condition than p agrees at a and b, then a = b. In the case of the proof for (NCHp(fj)), a stronger condition differing at a and b was easy to produce, since we only had to add a single element to a finite set. However, in the case we are considering here, we have to add an element to the set such that the property (a, x) G L A (a,y) G U ==>• x < y is still satisfied. If, for given a, the set of all x such that (a. x) G L had a determinate maximum or sup, we'd be fine. However, for an arbitrary pair (a,.x), the formula (a,x) G L does not necessarily take the value T or 1, making there be no such determinate maximum. Making sure that a stronger condition q still satisfies this property is not easy unless we assume something like Booleanness. I have made many attempts such as that above to try to circumvent the requirement of natural-distinguishability. However, none of them have worked. It could be that my attempts are in vain because there is some strong category of classes £ such that the object Km is limited in size in every internal functor 121 category 8 . However, I have not succeeded at finding such a category either. It could be that there is some formula 8.4 Conclusion My goal in this thesis was to provide a proof of the consistency of the intuition- istic continuum hypothesis. I wished to conduct such a proof in the framework of both topos theory and algebraic set theory, which can be unified by the the ory of "strong categories of classes". In an intuitionistic framework the phrase "the- continuum hypothesis" is misleading, as there are many classically equiv alent ways of formulating (CH). I chose two weaker versions of the negation of the continuum hypothesis (NCH^(N)) and (NCHRm), and showed that each was relatively consistent with both topos theory and the theory of cosmolog- ical categories of classes. These relative consistency theorems on their own, while they use slightly different methods than the existing relative consistency proofs, don't actually tell us much more than the results obtained by Cohen, Lawvere, and Tierney. However, the method of taking categories of presheaves rather than sheaves used in these proofs has important corollaries which lead to relative consistency results not yet known. We now know that (NCHp(N) is relatively consistent with Church's thesis in either of our frameworks, and that the existence of many cardinalities between Rc and K.m is consistent with both topos theory and algebraic set theory. 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