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Constructive Set Theory CST Book draft Peter Aczel and Michael Rathjen CST Book Draft 2 August 19, 2010 Contents 1 Introduction 7 2 Intuitionistic Logic 9 2.1 Constructivism . 9 2.2 The Brouwer-Heyting-Kolmogorov interpretation . 11 2.3 Counterexamples . 13 2.4 Natural Deductions . 14 2.5 A Hilbert-style system for intuitionistic logic . 17 2.6 Kripke semantics . 19 2.7 Exercises . 21 3 Some Axiom Systems 23 3.1 Classical Set Theory . 23 3.2 Intuitionistic Set Theory . 24 3.3 Basic Constructive Set Theory . 25 3.4 Elementary Constructive Set Theory . 26 3.5 Constructive Zermelo Fraenkel, CZF ................ 26 3.6 On notations for axiom systems. 27 3.7 Class Notation . 27 3.8 Russell's paradox . 28 4 Basic Set constructions in BCST 31 4.1 Ordered Pairs . 31 4.2 More class notation . 32 4.3 The Union-Replacement Scheme . 35 4.4 Exercises . 37 5 From Function Spaces to Powerset 41 5.1 Subset Collection and Exponentiation . 41 5.2 Appendix: Binary Refinement . 44 5.3 Exercises . 45 3 CST Book Draft 6 The Natural Numbers 47 6.1 Some approaches to the natural numbers . 47 6.1.1 Dedekind's characterization of the natural numbers . 47 6.1.2 The Zermelo and von Neumann natural numbers . 48 6.1.3 Lawv`ere'scharacterization of the natural numbers . 48 6.1.4 The Strong Infinity Axiom . 48 6.1.5 Some possible additional axioms concerning ! . 49 6.2 DP-structures and DP-models . 50 6.3 The von Neumann natural numbers in ECST . 51 6.3.1 The DP-model N! ...................... 52 6.3.2 The Least Number Principle . 53 6.3.3 The Iteration Lemma . 54 6.4 The Natural Numbers in ECST+ .................. 55 6.4.1 The DP-model (N; 0;S) ................... 55 6.4.2 Primitive Recursion . 56 6.4.3 Heyting Arithmetic . 57 6.5 Transitive Closures . 58 6.6 Some Possible Exercises . 59 7 The Continuum 61 7.1 The ordered field of rational numbers . 62 7.2 The pseudo-ordered field of real numbers . 65 7.3 The class R0 of left cuts is a set . 67 8 The Size of Sets 69 8.1 Notions of size . 69 8.2 Appendix: The Pigeonhole principle . 79 8.3 Exercises . 83 9 Foundations of Set Theory 85 9.1 Well-founded relations . 85 9.2 Some consequences of Set Induction . 88 9.3 Transfinite Recursion . 89 9.4 Ordinals . 91 9.5 Appendix: On Bounded Separation . 93 9.5.1 Truth Values . 93 9.5.2 The Infimum Axiom . 95 9.5.3 The Binary Intersection Axiom . 96 9.6 Appendix: Extension by Function Symbols . 97 9.7 Exercises . 99 4 August 19, 2010 CST Book Draft 10 Choice Principles 103 10.1 Diaconescu's result . 103 10.2 Constructive Choice Principles . 105 10.3 The Presentation Axiom . 109 10.4 More Principles that ought to be avoided in CZF . 110 10.5 Appendix: The Axiom of Multiple Choice . 112 11 The Regular Extension Axiom and its Variants 115 11.1 Axioms and variants . 115 12 Inductive Definitions 119 12.1 Inductive Definitions of Classes . 119 12.2 Inductive definitions of Sets . 122 12.3 Tree Proofs . 123 12.4 The Set Compactness Theorem . 126 12.5 Closure Operations on a po-class . 127 13 Coinduction 131 13.1 Coinduction of Classes . 131 13.2 Coinduction of Sets . 133 14 W-Semilattices 135 14.1 Set-generated W-Semilattices . 135 14.2 Set Presentable W-Semilattices . 136 14.3 W-congruences on a W-semilattice . 138 15 General Topology in Constructive Set Theory 141 15.1 Topological and concrete Spaces . 141 15.2 Formal Topologies . 142 15.3 Separation Properties . 145 15.4 The points of a set-generated formal topology . 147 15.5 A generalisation of a result of Giovanni Curi . 150 16 Russian Constructivism 157 17 Brouwer's World 159 17.1 Decidable Bar induction . 160 17.2 Local continuity . 161 17.3 More Continuity Principle . 163 18 Large sets in constructive set theory 165 18.1 Inaccessibility . 165 18.2 Mahloness in constructive set theory . 169 5 August 19, 2010 CST Book Draft 19 Intuitionistic Kripke-Platek Set Theory 173 19.1 Basic principles . 173 19.2 Σ Recursion in IKP . 176 19.3 Inductive Definitions in IKP . 179 20 Anti-Foundation 183 20.1 The anti-foundation axiom . 184 20.1.1 The theory CZFA . 185 20.2 The Labelled Anti-Foundation Axiom . 186 20.3 Systems . 188 20.4 A Solution Lemma version of AFA . 191 20.5 Greatest fixed points of operators . 192 20.6 Generalized systems of equations in an expanded universe . 195 20.7 Streams, coinduction, and corecursion . 197 21 The Interpretation of CZF in Martin-L¨ofType Theory 203 22 The Metamathematics of Constructive Set Theories 205 22.1 ECST .................................205 22.2 The strength of CZF . 205 22.3 Some metamathematical results about REA . 206 22.4 ZF models of REA . 214 22.5 Brouwerian Principles . 215 22.5.1 Choice principles . 216 22.6 Elementary analysis augmented by Brouwerian principles . 217 22.7 Combinatory Algebras . 220 22.7.1 Kleene's Examples of Combinatory Algebras . 223 22.8 Type Structures over Combinatory Algebras . 225 22.9 The set-theoretic universe over Kleene's second model . 228 22.10Predicativism and CZFA . 233 6 August 19, 2010 Chapter 1 Introduction The general topic of Constructive Set Theory originated in the seminal 1975 paper of John Myhill, where a specific axiom system CST was introduced. Constructive Set Theory provides a standard set theoretical framework for the development of constructive mathematics in the style of Errett Bishop1 and is one of several such frameworks for constructive mathematics that have been considered. It is distinctive in that it uses the standard first order language of classical axiomatic set theory 2 and makes no explicit use of specifically constructive ideas. Of course its logic is intuitionistic, but there is no special notion of construction or constructive object. There are just the sets, as in classical set theory. This means that mathematics in constructive set theory can look very much like ordinary classical mathematics. The advantage of this is that the ideas, conventions and practice of the set theoretical presentation of ordinary mathematics can be used also in the set theoretical development of constructive mathematics, provided that a suitable discipline is adhered to. In the first place only the methods of logical reasoning available in intuitionistic logic should be used. In addition only the set theoretical axioms allowed in constructive set theory can be used. With some practice it is not difficult for the constructive mathematician to adhere to this discipline. Of course the constructive mathematician is concerned to know that the axiom system she is being asked to use as a framework for presenting her mathemat- ics makes good constructive sense. What is the constructive notion of set that constructive set theory claims to be about? The first author believes that he has answered this question in a series of three papers on the Type Theoretic Inter- pretation of Constructive Set Theory. These papers are based on taking Martin- L¨of's Constructive Type Theory as the most acceptable foundational framework of ideas that make precise the constructive approach to mathematics. They show 1See Constructive Analysis, by Bishop and Bridges 2Myhill's original paper used some other primitives in CST besides the notion of set. But this was inessential and we prefer to keep to the standard language in the axiom systems that we use. 7 CST Book Draft Introduction how a particular type of the type theory can be used as the type of sets forming a universe of objects to interpret constructive set theory so that by using the Curry-Howard `propositions as types' idea the axioms of constructive set theory get interpreted as provable propositions. Why not present constructive mathematics directly in the type theory? This is an obvious option for the constructive mathematician. It has the drawback that there is no extensive tradition of presenting mathematics in a type theoretic setting. So, many techniques for representing mathematical ideas in a set the- oretical language have to be reconsidered for a type theoretical language. This can be avoided by keeping to the set theoretical language. Surprisingly there is still no extensive presentation of an approach to con- structive mathematics that is based on a completely explicitly described axiom system - neither in constructive set theory, constructive type theory or any other axiom system. One of the aims of these notes is to initiate an account of how constructive mathematics can be developed on the basis of a set theoretical axiom system. At first we will be concerned to prove each basic result relying on as weak an axiom system as possible. But later we will be content to explore the consequences of stronger axiom systems provided that they can still be justified on the basis of the type theoretic interpretation. Because of the open ended nature of constructive type theory we also think of constructive set theory as an open ended discipline in which it may always be possible to consider adding new axioms to any given axiom system. In particular there is current interest in the formulation of stronger and stronger notions of type universes and hierarchies of type universes in type theory. This activity is analogous to the pursuit of ever larger large cardinal principles by classical set theorists. In the context of constructive set theory we are led to consider set theoretical notions of universe. As an example there is the notion of inaccessible set of Rathjen (see [68]).
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