A Solovay-like model for singular generalized descriptive set theory

TL;DR

This paper constructs a generalized Polish space of singular weight where all subsets have the perfect set property, reducing the large cardinal assumptions needed from I0 to a supercompact cardinal.

Contribution

It introduces a new singular generalized Polish space with the perfect set property, lowering the consistency strength from I0 to a supercompact cardinal.

Findings

Existence of a singular Polish space with the perfect set property.

Reduction of consistency strength from I0 to supercompact cardinal.

Demonstration of the perfect set property for all subsets in the constructed space.

Abstract

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical characteristics, similar to those implied by the Axiom of Determinacy: if $\lambda$ witnesses I0, then there is a topology for $V_{\lambda+1}$ that is completely metrizable and with weight $\lambda$ (i.e., it is a $\lambda$-Polish space), and it turns out that all the subsets of $V_{\lambda+1}$ in $L(V_{\lambda+1})$ have the $\lambda$-Perfect Set Property in such topology. In this paper, we find another generalized Polish space of singular weight $\kappa$ of cofinality $\omega$ such that all its subsets have the $\kappa$-Perfect Set Property, and in doing this, we are lowering the consistency strength of such property from I0 to $\kappa$ $\theta$-supercompact, with $\theta>\kappa$ inaccessible.