Combinatorics in Higher Solovay Models

Abstract

We construe the singular-cardinal analogue of the classical Solovay model. Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\mathcal{P}(\hat{I}º))$ that are typically implied by Woodin's axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $\hat{I}{º}^{+}$-sequences of distinct members of $\mathcal{P}(\hat{I}º)$ and that Shelah's approachability property $\AP_κ$ fails. We prove that every set in our inner model satisfies a singular analogue of the complete Ramsey property and that the partition relation $\hat{I}º\stackrel{\mathrm{OD}}{\to }(\ddot{I}‰{)}^{\ddot{I}}{‰}_{V_{\hat{I}}¼}$ holds for all $\hat{I}¼<\hat{I}º$.