The strength of Mac Lane set theory

Abstract

SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and 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 KP + 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 KP ; we show, again using ill-founded models, that KP + V = L proves the consistency of KP ; 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: Département de Mathématiques et Informatique, Université de la Réunion, BP 7151, F 97715 St Denis de la Réunion, Messagerie 9, France outre-mer. Electronic address: ardm@univ-reunion.fr THE STRENGTH OF MAC LANE SET THEORY ii

DOI: 10.1016/S0168-0072(00)00031-2

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Cite this paper

@article{Mathias2001TheSO, title={The strength of Mac Lane set theory}, author={A. R. D. Mathias}, journal={Ann. Pure Appl. Logic}, year={2001}, volume={110}, pages={107-234} }