Shelah has shown that a failure of the PCF conjecture implies the existence of free subsets for internally approachable structures of certain types. This we term the approachable free subset property (AFSP). We will show how to construct scales that refute the AFSP under certain anti large cardinal assumptions.

Scales such as these have been known to exist in inner models. They can also be added through forcing by results due to Pereira and Cummings. Here we will show their existence in V through the use of inner model theory, specifically techniques from the proof of the covering lemma.

We have reasons to think, based on work of Gitik, that our assumptions (concerning the existence of tree-like scales, not failure of AFSP!) are close to optimal. This talk is partially based on joint work with Omer Ben-Neria.

Slides fron the talk