The mouse set conjecture for sets of reals

TL;DR

This paper proves that the Mouse Set Conjecture for sets of reals follows from the more general Mouse Set Conjecture under the assumption of AD++V=L(P(R)), establishing a significant link between these conjectures.

Contribution

The authors demonstrate that the Mouse Set Conjecture for sets of reals is a consequence of the broader Mouse Set Conjecture under certain set-theoretic assumptions.

Findings

Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.

Under AD++V=L(P(R)), the equivalence between definability and membership in an iterable mouse is established.

The result connects two major conjectures in descriptive set theory and inner model theory.

Abstract

Recall that the Mouse Set Conjecture says that under AD++V=L(P(R)), a real is ordinal definable if and only if it belongs to an iterable mouse. The Mouse Set Conjecture for sets of reals says that under the same theory, a set of reals is ordinal definable from a real if and only if it belongs to a mouse over the reals. We prove that the Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.