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 emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Laneʹs system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Laneʹs 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 Friedmanʹs strengthening View the MathML source of View the MathML source, and the Forster–Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and View the MathML source; we show, again using ill-founded models, that View the MathML source proves the consistency of View the MathML source; 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 View the MathML source is a conservative extension of MAC for stratified sentences, from which we deduce that View the MathML source proves a strong stratified version of View the MathML source; we analyse the known equiconsistency of MAC with the simple theory of types and give Lakeʹs proof that an instance of Mathematical Induction is unprovable in Mac Laneʹs 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.
Keywords :
Forster–Kaye set theory , Conservative extension , Simple theory of types , Failure of collection , Failure of induction , Mac Lane set theory , Kripke–Platek set theory , Axiom H spectacles , Mostowskiיs principle , Constructibility , Power-admissible set , Forcing over non-standard models , Stratifiable formula
