Operational set theory and small 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...

Large Cardinals and Small Dowker Spaces

We prove that, if there is a model of set-theory which contains no first countable, locally compact, scattered Dowker spaces, then there is an inner model which contains a measurable cardinal. A Hausdorff space is normal if, for every pair of disjoint closed sets C and D, there is a pair of disjoint open sets, U containing C and V containing D. A (normal) space is binormal if its product with t...

Large cardinal properties of small cardinals

Applications of Large Cardinals to Graph Theory