A Solovay-like model at $\aleph_ω$
Alejandro Poveda, Sebastiano Thei
TL;DR
This work constructs a Solovay-type inner model at the first singular cardinal $\aleph_\omega$ by forcing over large cardinals to obtain a model $L(\mathcal{P}(\aleph_\omega))$ where $\aleph_\omega$ is a strong limit and several robust regularity and combinatorial properties hold in a choiceless context. The authors deploy a sophisticated Sigma-Prikry framework based on Merimovich-like extender forcing, together with a commutative system of projections and a constellation mechanism, to force that every $A\subseteq {}^\omega\aleph_\omega$ has the $\aleph_\omega$-PSP, while simultaneously destroying scales, diamonds, and certain square-like principles; SCH fails and AP fails at $\aleph_\omega$, and TP holds at $\aleph_{\omega+1}$. The main theorem delivers the first $\aleph_\omega$-level Solovay model and answers Woodin’s question about SCH and AP at $\aleph_\omega$ in the $\textsf{ZF}+\textsf{DC}_{\aleph_\omega}$ setting. The results bridge Solovay-type regularity phenomena with singular cardinal combinatorics and provide a framework for higher-descriptive-set-theory at singular cardinals using GDST techniques.
Abstract
Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_ω$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_ω))$ satisfies the following properties: (1) Every set $A\subseteq (\aleph_ω)^ω$ has the $\aleph_ω$-PSP. (2) There is no scale at $\aleph_ω$. (3) The Singular Cardinal Hypothesis (SCH) fails at $\aleph_ω$. (4) Shelah's Approachability property (AP) fails at $\aleph_ω$. (5) The Tree Property (TP) holds at $\aleph_{ω+1}$. The above provides the first example of a Solovay-type model at the level of the first singular cardinal, $\aleph_ω$. Our model also answers, in the context of ZF+$\mathrm{DC}_{\aleph_ω}$, a well-known question by Woodin on the relationship between the SCH and the AP at $\aleph_ω$.