# 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*for some

_{β}*β*which is at most

*α*. But

*L*does not allow for very large cardinals,

*ω*-Erdös cardinals already cannot exist within

_{1}*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 *ω _{1}*-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.