Strong Tree Properties for Small Cardinals

TL;DR

This paper demonstrates that the existence of infinitely many supercompact cardinals implies a strong tree property for small cardinals in a ZFC model, extending the known large cardinal consequences.

Contribution

It proves that from infinitely many supercompact cardinals, a model can be constructed where small cardinals satisfy the (aleph_n, mu)-ITP for all n>1 and relevant mu, establishing a new link between large cardinals and small cardinal properties.

Findings

Models with infinitely many supercompact cardinals imply (aleph_n, mu)-ITP for small cardinals.

The result extends the known implications of supercompactness to small cardinal tree properties.

The paper constructs models satisfying strong tree properties for all small cardinals above a certain size.

Abstract

An inaccessible cardinal kappa is supercompact when (kappa, lambda)-ITP holds for all lambda greater than or equal to kappa. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where for every natural number n greater than 1 and for every ordinal mu greater than or equal to aleph_n, we have (aleph_n, mu)-ITP.