Strong tree Properties for two successive cardinals

TL;DR

This paper demonstrates that the existence of two supercompact cardinals implies the consistency of models where certain strong tree properties hold at two successive cardinals, extending the understanding of large cardinal implications.

Contribution

It proves the consistency of models with strong tree properties at two successive cardinals assuming two supercompact cardinals.

Findings

Models with strong tree properties at  and  are consistent.

The result extends the known implications of supercompact cardinals.

Simultaneous tree properties at  and  are achievable under large cardinal assumptions.

Abstract

An inaccessible cardinal $\kappa$ is supercompact when $(\kappa, \lambda)$-ITP holds for all $\lambda\geq \kappa.$ We prove that if there is a model of $\ZFC$ with two supercompact cardinals, then there is a model of \ZFC where simultaneously $(\aleph_2, \mu)$-ITP and $(\aleph_3, \mu')$-ITP hold, for all $\mu\geq \aleph_2$ and $\mu'\geq \aleph_3.$