# Free subsets in internally approachable models

## TL;DR

This paper investigates properties of internally approachable models and their implications for set theory, focusing on the consistency strength of certain cardinal properties.

## Contribution

The paper provides a direct proof of the consistency strength of the Approachable Bounded Subset Property without relying on PCF scales.

## Key findings

- ABSP requires inner models with measurable cardinals of arbitrarily large Mitchell order below ℵω.
- The result matches the exact consistency strength of ABSP for ascending sequences.
- The proof avoids the use of continuous tree-like scales.

## Abstract

We consider a question of Pereira as to whether the characteristic function of an internally approachable model can lead to free subsets for functions of the model. Pereira isolated the pertinent Approachable Free Subsets Property (AFSP) in his work on the \documentclass[12pt]{minimal}
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\begin{document}$${\text {pcf}}$$\end{document}pcf-conjecture. A recent related property is the Approachable Bounded Subset Property (ABSP) of Ben-Neria and Adolf, and we here directly show it requires modest large cardinals to establish:

Theorem
If ABSP holds for an ascending sequence \documentclass[12pt]{minimal}
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\begin{document}$$ \langle \aleph _{n_{m}} \rangle _{m}$$\end{document}⟨ℵnm⟩m
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\begin{document}$$( n_{m} \in \omega )$$\end{document}(nm∈ω) then there is an inner model with measurables \documentclass[12pt]{minimal}
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\begin{document}$$\kappa < \aleph _{\omega }$$\end{document}κ<ℵω of arbitrarily large Mitchell order below \documentclass[12pt]{minimal}
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\begin{document}$$\aleph _{\omega }$$\end{document}ℵω, that is: \documentclass[12pt]{minimal}
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\begin{document}$$\sup \left\{ \alpha \mid {\exists }\kappa < \aleph _{\omega } o ( \kappa ) \ge \alpha \right\} = \aleph _{\omega }$$\end{document}supα∣∃κ<ℵωo(κ)≥α=ℵω. A result of Adolf and Ben Neria then shows that this conclusion is in fact the exact consistency strength of ABSP for such an ascending sequence. Their result went via the consistency of the non-existence of continuous tree-like scales; the result of this paper is direct and avoids the use of PCF scales.