Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

Abstract

We prove the following result which is due to the third author.

The Consistency Strength of Long Projective Determinacy

“…This will only be applied in X -premice M for some X ∈ P 1 (R) with x 0 ∈ X and as in the proof of Lemma 3.8 "least" refers to the least set in the well order of elements of M definable from x 0 which is given by (•, x 0 , •). In this case, the game G ϕ, is a variant of the Kechris-Solovay game in [11] (see also the game in [21,Lemma 2.3]) adapted as a model game. The winning condition is Π 1 2 , so the game G ϕ, is determined.…”

“…Afterwards, Neeman improved this in [22] even further and showed that for all n, the existence of M n (x) for all reals x implies determinacy of all n (< 2 − Π 1 1 ) sets. Concerning the other direction, Woodin (see [21]) showed that if Σ 1 n+1 games are determined, then M n (x) exists for all reals x, thus establishing a level-by-level characterization of projective determinacy in terms of the existence of inner models with large cardinals. Similar characterizations are known forprojective games of length (see [1] and [3]).…”

“…Further results. The proof of Theorem 1.1 uses game arguments based on techniques from Martin and Steel [16] and [21], tracing back to arguments of H. Friedman [4] and and Solovay [11] as well as inner model theoretic methods based on the unpublished notes [31]. The method of proof of Theorem 1.1 can also be used to show the following two generalizations: Outline.…”

“…In the rest of this section we summarize some facts about premice which we are going to need later. Most of these can be found in [21] and [41] for x-premice for a real x and they straightforwardly generalize to X -premice for arbitrary countable sets of reals X . Fix such a set X ∈ P 1 (R) for the rest of this section.…”

“…This will be used in the proof of Theorem 3.1. For premice over a real x this can be found, for example, in [21,Lemma 2.11]; the argument there generalizes to premice over countable sets of reals. For the reader's convenience we will briefly sketch the main ideas and recall the statements.…”