Determinacy for games ending at the first admissible relative to the play.(English)Zbl 1105.03053

For $$C$$ a set of real sequences, each indexed by a countable ordinal, the game $$G_{\text{adm}}(C)$$ has two players (I and II) who at ordinal stage $$\xi$$ alternately pick an $$\omega$$-sequence of natural numbers, i.e., a real $$y_\xi$$. The players continue until they reach the first ordinal $$\alpha$$ for which $$L_{\alpha}[y_{\xi} \mid \xi<\alpha]$$ is admissible. The game then ends; I wins if $$\langle y_\xi \mid \xi<\alpha\rangle\in C$$, otherwise II wins.
The author develops methods for proving the determinacy of $$G_{\text{adm}}(C)$$ for appropriately defined $$C$$ and for optimal large cardinal assumptions. These methods result from the combining of the author’s machinery for proofs of determinacy of long games [I. Neeman, The determinacy of long games. Berlin: Walter de Gruyter (2004; Zbl 1076.03032)] with his rank games [I. Neeman, “Unraveling $$\Pi_{1}^{1}$$ sets, revisited”, Isr. J. Math. 152, 181–203 (2006)]. The large cardinal assumptions depend on $$C$$’s complexity, but they always involve a cardinal $$\kappa$$ that is a limit of Woodin cardinals and has Mitchell order $$\kappa^{++}$$.
The heart of the article is a proof that $$G_{\text{adm}}$$ is reducible to an iteration game on models with sufficient large cardinals. In the paper’s final section the author uses his reduction to obtain several optimal results about the determinacy of $$G_{\text{adm}}$$.

MSC:

 03E60 Determinacy principles 91A44 Games involving topology, set theory, or logic

Zbl 1076.03032
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References:

 [1] DOI: 10.1016/S0168-0072(00)00022-1 · Zbl 0979.03040 [2] DOI: 10.1090/pspum/042/791065 [3] DOI: 10.1007/BFb0071698 [4] DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035 [5] The determinacy of long games 7 (2004) · Zbl 1076.03032 [6] DOI: 10.1016/0168-0072(93)90037-E · Zbl 0805.03043 [7] DOI: 10.1016/0003-4843(82)90017-1 · Zbl 0573.03026 [8] Israel Journal of Mathematics [9] Inner models and large cardinals 5 (2002)
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