Kurt Gödel Research Center

2008 seminar talk: Internal Consistency

Talk held by Sy-David Friedman (KGRC) at the KGRC seminar on 2008-10-02.

Abstract

A statement is internally consistent iff it holds in an inner model, assuming the existence of inner models with large cardinals. This notion gives rise to a new type of relative consistency result:

where """ICon""" denotes "internally consistent", "LC" stands for some large cardinal hypothesis and """S""" is a statement of set theory. Internal consistency results demand techniques beyond those used for ordinary consistency results and come in two types. Type 1 results are those where "LC" is taken to be "0# exists". In this case, the methods of generic modification (F-Ondrejovic) and partial master conditions (F-Thompson) are typically used. Type 2 results arise when a statement can be shown by forcing to consistently hold in """V_kappa""" where """kappa""" is a measurable cardinal. In this second case, the methods used are typically generic modification (Woodin) or various tree-forcing methods (F-Thompson for Sacks products, Dobrinen-F for Sacks iterations and F-Zdomskyy for Miller iterations). In this talk I'll discuss internal consistency in the contexts of cardinal exponentiation, global domination, the tree property, embedding complexity and the cofinality of the symmetric group.