The super tree property at the successor of a singular

TL;DR

This paper investigates the super tree property and related principles at the successor of a singular cardinal, demonstrating their consistency at certain large and small cardinals, and exploring the extent of compactness in set theory.

Contribution

It proves the super tree property can hold at the successor of a limit of many supercompact cardinals and at leph_{\u221e+1}, and analyzes the levels of ISP at successors of singulars.

Findings

ITP holds at the successor of a limit of leph many supercompact cardinals.

ITP can hold at leph_{\u221e+1} in certain models.

The paper determines which levels of ISP can hold at successors of singular cardinals.

Abstract

For an inaccessible cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $\omega$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{\omega+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining large cardinal-type properties at smaller cardinals.