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The Strength of Mac Lane Set Theory The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that KF proves a weak form of Stratified Collection, and that MAC + KP is a conservative extension of MAC for stratified sentences, from which we deduce that MAC proves a strong stratified version of KP; we analyse the known equiconsistency of MAC with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion | namely that there exists an infinite set of infinite sets, no two of which have the same cardinal | and use it to establish the failure of the full schema of Stratified Collection in Z; and we determine the point of failure of various other schemata in MAC. The paper closes with some philosophical remarks. Mathematics Subject Classification (2000): Primary: 03A30, 03B15, 03B30, 03C30, 03E35, 03E40, 03E45, 03H05. Secondary: 03A05, 03B70, 03C62, 03C70, 18A15. key words and phrases: Mac Lane set theory, Kripke{Platek set theory, Axiom H spectacles, Mostowski's principle, constructibility, forcing over non-standard models, power-admissible set, Forster{Kaye set theory, stratifiable formula, conservative extension, simple theory of types, failure of collection, failure of induction. Postal address: D´epartement de Math´ematiques et Informatique, Universit´e de la R´eunion, BP 7151, F 97715 St Denis de la R´eunion, Messagerie 9, France outre-mer. Electronic address: [email protected] THE STRENGTH OF MAC LANE SET THEORY ii Table of Contents Section Paragraph Page Page 0 Introduction 4 1 Adding Foundation and Transitive Containment 10 2 Adding Axiom H 14 3 Examination of Axiom H 20 4 Adding KPL to M 24 The simulation of G¨odel's L in the system M by L-strings along well-orderings 24 On elementary submodels 26 G¨odel's operations 28 Condensation for L-strings 30 The construction of G¨odel's L in the system M + H by transitised factored L-strings 31 The construction of G¨odel's L in the system M + H: a direct argument 31 5 The increasing strength of subsystems of Z 33 Proof of Axiom H in KLZ0 33 Σ_ k hulls 34 The minimality axiom 36 A weak fine-structural lemma 37 Consistency proofs for fragments of KLMZ 38 Proving Σk Separation in L 41 A strong fine-structural lemma 41 6 Two independence results and a consistency proof 45 The Takahashi hierarchy 45 Takahashi's reductions 48 Equivalence of KPP and KPs(P) 50 Development of KPP and KPR 52 Well-founded parts of ill-founded models 54 A transitive model of KPP + AC in which every ordinal is recursive 57 A transitive model of KPP + AC + 8κ9κ+ with a long well-ordering of the continuum 58 7 Stratifiable formulæ 60 An embedding elementary for stratifiable formulæ 61 Stratifiable schemata provable in MAC 62 Coret's argument in KF and MAC 62 8 The simple theory of types 64 The system TST 64 The interpretation of TST in KF 65 A direct limit of types: the method of Forster and Kaye 65 Kemeny's Princeton Thesis of 1949 72 Completing the circles 75 9 Limitations of MAC and Z 76 Failure of stratifiable Π1 Collection in Z 76 P Failure of ∆0 and Π1 Replacement in Z 77 Failure of Induction and other schemata in MAC 78 An algebraic illustration 80 Peroration 81 10 Envoi 82 Further reading 84 Acknowledgments 85 Bibliography 86 iii A. R. D. MATHIAS Chart of the set-theoretic systems considered y n dn mt n x Ext ? fx; yg x x r y P(x) Sep Foun TCo ! Choice Repl Coll V=L ON=@! Ax H RAS S0 X X X SX X 6 S1 X X X X X X P P 8 KF X X X X [X] X a-∆0[a-∆0 ] [a-∆0 ] P P M0 X X X X [X] X ∆0[∆0 ] [a-∆0 ] P P 1 M1 X X X X [X] X ∆0[∆0 ] set X [a-∆0 ] P P 8 KFI X X X X [X] X a-∆0[a-∆0 ] AxInf [a-∆0 ] P M X X X X [X] X ∆0[∆0 ] set X X P 10 ZBQC X X X X [X] X ∆0[∆0 ] set X X AC P M X X X X [X] X ∆0[∆0 , a-Σ1] set X X X [oa-Σ1] 10 MOST X X X X [X] X Σ1 set[Σ1; Π1] X X X [Σ1] ∆0[s-Σ1] [X] KP X X X X [X] ∆0[∆1] Π1 [X] ∆0[Σ1] KPI X X X X [X] ∆0[∆1] Π1 [X] X ∆0[Σ1] 4 KPL X X X X [X] ∆0[∆1] Π1 [X] X [X] ∆0[Σ1] X 6 KPR X X X X [X] [X] ∆0[∆1] Π1 [X] X ∆0[Σ1] X P P P P P 6 KP X X X X [X] X ∆0[∆1 ] Π1 [X] X ∆0 [Σ1 ] [X] 5 Zk X X X X [X] X Σk set X 5 KZk X X X X [X] X Σk Π1[Σk; Πk] [X] X ∆0[(s)-Σ1] [k > 1] 5 KLZk X X X X [X] X Σk Π1[Σk; Πk] [X] X [X] ∆0[s-Σ1] X [k > 0] 5 KLMZk X X X X [X] X Σk Π1[Σk; Πk] [X] X [X] ∆0[s-Σ1] X X [k > 2] 7 Z X X X X [X] X full set X [strat] ZC X X X X [X] X full set X X [strat] 5 KLMZ X X X X [X] X full Π1[full] [X] X [X] [strat] ∆0[s-Σ1] X X [X] ZF X X X X [X] X full set[full] [X] X [full] X [•] [X] [X] ZFC X X X X [X] X full set[full] [X] X X [full] X [•] [X] [X] Notes 1. Unexplained terms are defined in the text of the paper. Most systems are introduced in section 0. Under x is given the number of a later section where the given system is discussed. 2. The entries [X] and [•] mean respectively that the statement is derivable or refutable in the given system; not all facts of this nature appear in the chart. 3. In M0, TCo implies the existence of the transitive closure of a set. 4. The equivalence AC () WO of the Axiom of Choice and the Well-ordering Principle is provable in M0, using the power set axiom and the existence of Cartesian products. 5. Save for KFI, those systems with an axiom of infinity take it as asserting the existence of the infinite von Neumann ordinal !. AxInf is the statement that there is a Dedekind infinite set. 6. s-Σ1 collection is strong Σ1 collection; it and Axiom H are provable in KZk for k > 1, and in KLZk for k > 0. Σ1 collection is provable in KZ0; indeed in KP. 7. a-Σ1 means stratified Σ1; o-Σ1 means strong Σ1; oa-Σ1 collection means strong stratified Σ1 collection. 8. RAS is the assertion that 8ζ fx j %(x) < ζg 2 V , where %(x) is the set-theoretical rank of x. 9. Quine's New Foundations, NF, is mentioned briefly in sections 0 and 7, and two versions|TSTI, TST; TKTI, TKT|of the simple theory of types with and without infinity are discussed in section 8. THE STRENGTH OF MAC LANE SET THEORY 4 0: Introduction We describe here the scope and results of this paper in the semi-formal style, customary among logicians, that makes no visual distinction between levels of language; at the end of this introduction we shall summarise certain more rigorous conventions that will be used, with occasional relaxations, in later sections. Our underlying logic is classical throughout the paper. We suppose that set theory is formalised with two primitive notions, 2 and =, a class-forming operator f j : : : g, and apart from the usual quantifiers 8 and 9, restricted quantifiers 8x :2 y and 9x :2 y , (in which x and y are distinct variables) for which the following axioms are provided.
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