\(AD_{\mathbb {R}}\) implies that all sets of reals are \(\Theta \) universally Baire

We show that assuming the determinacy of all games on reals, every set of reals is \(\Theta \) universally baire.

1. However, the large cardinal assumption used in [6] is probably optimal.

2. \({\mathbb {R}}\) is the Baire space.

3. This means that there is a set \(U^k\subseteq \omega \times {\mathbb {R}}^k\) such that \(U^k\in \Gamma \) and for every set \(A\in \Gamma \), if \(A\subseteq {\mathbb {R}}^k\) then there is an integer n such that \(x\in A\iff (n, x)\in U^k\).

4. The point of this move is to just avoid discussing \(Env(\Gamma )\). Readers familiar with this notion do not have to make this move.

5. More precisely, “there is an ordinal which is not the surjective image of \({\mathbb {R}}\)”.

6. We will need the freedom to include any \(OD^{M_\Gamma }(B_0)\) set of reals into our condensing sequence.

7. \(\sigma (g)\) is the realization of \(\sigma \).

8. Here \(\vec {A}\) is any sjs for which \(M_\Gamma =L_\gamma (\vec {A}, {\mathbb {R}})\).

9. Recall that \(AD_{{\mathbb {R}}}\) implies that \(\omega _1\) is supercompact.

10. self-well-ordered

11. i.e., the iterates of \({\mathcal {M}}\) are also \(\Sigma \)-premice

12. To see this, first because \(\delta \) is inaccessible, condensation implies that \(\rho ({\mathcal {M}}_n^{\#, \Sigma }({\mathcal {M}}))=\delta \) as otherwise we could take a transitive below \(\delta \) hull \({\mathcal {N}}\) of \({\mathcal {M}}_n^{\#, \Sigma }({\mathcal {M}})\), and by condensation \({\mathcal {N}}\trianglelefteq {\mathcal {M}}\). Next, fix \(f:\delta \rightarrow \delta \) such that \(f\in {\mathcal {M}}_n^{\#, \Sigma }({\mathcal {M}})\). Then following the proof of [8, Theorem on page 115] find an extender E such that \(\pi _E^Q(f)(\mathrm{crit }(E))<strength(E)\) and \(E\cap {\mathcal {M}}\in {\mathcal {M}}\). This E witnesses Woodinness of \(\delta \) with respect to f.

13. \({\mathbb {W}}_\eta ^S\) is the extender algebra of S associated to \(\eta \), a Woodin cardinal of S.

14. \(C^\#\) can be thought as the minimal active mouse over \(C, {\mathbb {R}}\) that is \(\omega _1\)-iterable

15. This could happen if for instance, C is Wadge reducible to C.

Authors and Affiliations

1. Rutgers The State University of New Jersey, New Brunswick, USA

   Grigor Sargsyan

Corresponding author

Correspondence to Grigor Sargsyan.

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