## The strength of Mac Lane set theory.(English)Zbl 1002.03045

In this paper the author investigates various systems of set theory which are related with Mac Lane’s system (MAC). This system is naturally related to systems derived from topos-theoretic notions. It is shown that the consistency strength of MAC is not increased by adding the Kripke-Platek set theory (KP) or the Axiom of Constructibility.
Essential in his investigations is the so-called Axiom H, which asserts the existence of “universal” transitive sets. This axiom and its consequences are studied in detail. As a byproduct, the author gives a new proof that Z is not finitely axiomatisable. He studies Friedman’s strengthening of KP+MAC and the Forster-Kaye subsystem KF of MAC. He shows that there is a variant $$\text{KP}^{\mathcal P}$$ of KP such that $$\text{KP}^{\mathcal P}+V=L$$ proves the consistency of $$\text{KP}^{\mathcal P}$$. This is done using forcing over ill-founded models. The author analyses the known equiconsistency of MAC with the simple theory of types and shows a weak form of Stratified Collection. The paper closes with some philosophical remarks.

### MathOverflow Questions:

Strength of Borel determinacy

### MSC:

 3e+70 Nonclassical and second-order set theories 3e+30 Axiomatics of classical set theory and its fragments
Full Text:

### References:

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