Definable tree property for uncountable regular cardinals

TL;DR

This paper constructs a ZFC model using large cardinals to prove the definable tree property holds for all uncountable regular cardinals, answering a longstanding open question.

Contribution

It establishes the first ZFC model where the definable tree property is valid for all uncountable regular cardinals, using supercompact and measurable cardinals.

Findings

Definable tree property holds for all uncountable regular cardinals in the constructed model.

Utilizes supercompact and measurable cardinals to achieve the result.

Answers an open question in set theory about the definable tree property.

Abstract

The primary goal of this paper is to establish a model of $ZFC$ wherein the definable tree property is affirmed for all uncountable regular cardinals. This endeavor commences with the utilization of both a supercompact cardinal and a measurable cardinal that exceeds it. Subsequently, we construct a $ZFC$ model. Within this model, we demonstrate that the definable tree property holds for all uncountable regular cardinals. Thereby we respond to an inquiry raised in [1].