Definable tree property for successors of cardinals

TL;DR

This paper explores the consistency strength of the definable tree property at successors of regular and singular cardinals, linking it to large cardinal assumptions and providing new consistency results.

Contribution

It establishes the exact large cardinal assumptions needed for the definable tree property at successors of regular and singular cardinals, improving understanding of its consistency strength.

Findings

Definable tree property at successors of all regular cardinals requires many small large cardinals.

At successors of singular cardinals, the property is consistent with the existence of a supercompact and a measurable cardinal.

The consistency strength for regular and singular cases is precisely characterized.

Abstract

It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.