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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, Inÿnity, Power Set, Restricted Separation, Foundation, and Choice, to which system, a orced 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 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 ÿnitely axiomatisable; we study Friedman's strengthening KP P + AC of KP + MAC, and the Forster-Kaye subsystem KF of MAC, and use forcing over ill-founded models to establish independence results concerning MAC and KP P ; we show, again u

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Author

A.R.D. Mathias

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0168-0072)

Published

2001

Language

EN

Field

Computer Science (Physical Sciences)