The Determinacy of Long Games

The Determinacy of Long Games

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In this volume the author develops and applies methods for proving, from large cardinals, the determinacy of definable games of countable length on natural numbers. The determinacy is ultimately derived from iteration strategies, connecting games on natural numbers with the specific iteration games that come up in the study of large cardinals. The games considered in this text range in strength, from games of fixed countable length, through games where the length is clocked by natural numbers, to games in which a run is complete when its length is uncountable in an inner model (or a pointclass) relative to the run. More can be done using the methods developed here, reaching determinacy for games of certain length. The book is largely self-contained. Only graduate level knowledge of modern techniques in large cardinals and basic forcing is assumed. Several exercises allow the reader to build on the results in the text, for example connecting them with universally Baire and homogeneously Suslin sets. - Important contribution to one of the main features of current set theory, as initiated and developed by Jensen, Woodin, Steel and others.Itay Neeman. (2) H is j *(W5)-generic/M *; and (3) the sequence (15 | 5 alt; a) belongs to j*(/l)[H]. Claim 7A.1. Let (Z, | 5 alt; o) be a position in G10001(M, r, 0, A) vvliieli is won by player I. Let a#39;U and H witness this. Then oi is precisely equal to j* (6), anbsp;...

Title:The Determinacy of Long Games
Author: Itay Neeman
Publisher:Walter de Gruyter - 2004-01-01

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