Realism, nonstandard set theory, and large cardinals

Operational set theory and small large cardinals

Article history: Received 5 December 2006 Revised 21 April 2008 Available online 11 April 2009 A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of...

Interpretability logics and large cardinals in set theory

This paper is a first exploration how the modal logic of interpretability can play a role in the study of large cardinals in set theory. First we reason that such a role is likely to be fruitful. Next, for two particular readings of the binary modality we study its repercussions for the corresponding logic. The first reading of A B is “Inside any Vκ where A holds we can find a Vκ′ where B holds...

Axiom of Choice in nonstandard set theory

We verify that the best-known nonstandard set theories: IST, BST, and HST, with the Axiom of Choice deleted, are conservative extensions of ZF + Boolean Prime Ideal Theorem. 2010 Mathematics Subject Classification 26E35 (primary); 03E25, 03E70, 03H05 (secondary)

A nonstandard set theory in the ∈-language

Wedemonstrate that a comprehensive nonstandard set theory can be developed in the standard ∈-language. As an illustration, a nonstandard Law of Large Numbers is obtained.

Applications of Large Cardinals to Graph Theory