Adrian Mathias - The Strength of Mac Lane Set Theory

SAUNDERS MAC LANE has drawn attention many times, particularly in his book {\sl Mathematics: Form and Function}, to the system 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 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; we digress to apply these methods to subsystems of Zermelo set theory; we use non-standard models and forcing to establish two independence results concerning \mac\/; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that Mac Lane set theory proves a weak form of stratified collection, and prove that \mac\/ + $KP$ is a conservative extension of

\mac\/ for stratified sentences; 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 --- that is unprovable in Mac Lane's system, and use it to establish the failure of the full schema of stratified collection in \mac\/; and finally we apply the equiconsistency of \mac\/ with the simple theory of types to demonstrate that an instance of Mathematical Induction is unprovable in Mac Lane's system.

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