Turing Invariant Sets and the Perfect Set Property

TL;DR

This paper demonstrates that assuming all Turing invariant sets of reals have the perfect set property leads to all sets of reals having this property, extending to countable analytic equivalence relations.

Contribution

It establishes a logical implication from Turing invariant sets to all sets of reals having the perfect set property, generalizing to analytic equivalence relations.

Findings

All sets of reals have the perfect set property under the assumption.

The result extends to all countable analytic equivalence relations.

The work connects Turing invariance with the structure of sets of reals.

Abstract

We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.