Kurt Gödel Research Center

2010 seminar talk: Condensation, Large Cardinals and the consistency strength of PFA

Talk held by Peter Holy (KGRC) at the KGRC seminar on 2010-11-25.

Abstract

Gödel's constructible universe L satisfies the strongest possible form of condensation: if M is an elementary submodel of any L_α, then M is isomorphic to L_β for some β which is at most α. But L does not allow for very large cardinals, ω₁-Erdös cardinals already cannot exist within L. We will generalize and then weaken Gödel's condensation principle to obtain new condensation principles (= fragments of condensation) and investigate whether those principles are consistent with the existence of (very) large cardinals. We will introduce (and deal with) the following principles:

Strong Condensation: strong, inconsistent with ω₁-Erdös cardinals

Stationary Condensation: consistent with ω-superstrong cardinals but pretty weak

Local Club Condensation: pretty strong, consistent with ω-superstrong cardinals

Acceptability: incomparable to the above, weak, consistent with ω-superstrong cardinals

A very interesting open question is whether Local Club Condensation and Acceptability are (simultaneously) consistent with the existence of an ω-superstrong cardinal. If this question has a positive answer, we would probably be able to prove the following:

Conjecture: Let S(κ) denote any large cardinal property of κ consistency-wise weaker than supercompactness. It is then consistent that there exists κ such that S(κ) holds but no proper forcing extension satisfies PFA.

Since any reasonable way to obtain a model of PFA seems to be starting with a model with large cardinals and to then obtain PFA in a proper forcing extension, this would be a strong hint towards the consistency strength of PFA actually being that of a supercompact cardinal.

This is joint work with Sy Friedman.