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 ©T. 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The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. •*• 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.