Some Proof-Theoretic Contributions to Theories of Sets

Abstract

Publisher Summary The set and set existence constitute an important part of the research in mathematical logic and the foundations of mathematics. Zermelo–Fraenkel set theory with or without the axiom of choice is the generally accepted formalism and can be considered as the official framework for doing mathematics. Proof theory is concerned with the investigation of the proof possibilities of mathematical systems. One aspect of this research is the ordinal analysis of formal theories. The proof-theoretic ordinal ׀Th׀ of a theory Th is often defined as the least ordinal α such that the consistency of Th can be proved in primitive recursive arithmetic (PRA) plus the scheme of transfinite induction along a standard primitive recursive well-ordering of order-type α. Furthermore, experience shows that the ordinal ׀Th׀ is a good measure for the proof-theoretic strength of Th, so that ׀Th׀ is said to be the proof-theoretic strength of Th.