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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

Citations:

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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