Download The determinacy of long games by Itay Neeman PDF
By Itay Neeman
Neeman stocks his study into proofs of determinacy for video games of countable size, for readers with an information of simple set conception, and a few familiarity with Silver indiscernibles and extenders and new release trees-all at a graduate point. He covers the elemental parts, video games of mounted countable size, video games of constantly coded size, pullbacks, while either gamers lose, alongside a unmarried department, and video games that extend neighborhood cardinals. He appends a quick exposition of extenders, widely used extensions, and iterability, yet warns that it could possibly serve greater as a booster or reminder than as an preliminary publicity.
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Extra resources for The determinacy of long games
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Mirroring G∗ , let H ∗ be the played as follows: Again I and II alternate playing natural numbers xn , creating together the real x = xn | n < ω . But now they play auxiliary moves in the game B[x]. The first player to violate these rules loses. Infinite plays here are won by I. I II x0 b0−I b0−II b1−I x1 b1−II x2 ... 5. The game H ∗ . G∗ ∈ M is an open game for player I, hence determined in M. Similarly H ∗ is open for II, and hence determined in M. 1. Case 1. If I wins G∗ in M. We intend to argue that I wins Gω (C) (in V).
1. Fix σ ∗ ∈ M, a winning strategy for II in G∗ . Fix similarly τ ∗ ∈ M, a winning strategy for I in H ∗ . ˙ δ, and X. Let τgen be the Let σgen be the generic strategies map associated to A, ˙ mirrored generic strategies map associated to B, δ, and X. Working inside M[g] we construct x, a, and b so that (1) x = xn | n < ω is a real; (2) a, x is a run of G∗ played according to both σ ∗ and σgen [x]; (3) b, x is a run of H ∗ played according to both τ ∗ and τgen [x].
IE (κ) is at least as large as λ by Fact 2 in Appendix A, since E is λ-strong. So projλ (iE (u)) in the definition makes sense. 9. Let M and N be models of ZFC∗ . Suppose that M κ + 1 = N κ + 1. Let u be a type in M with dom(u) = κ. ” N Then E can be applied to N and StretchE λ (u) (as computed in M) is equal to projλ(iE (u)). Proof sketch. The ultrapower embeddings iEN and iEM agree on subsets of M κ, and u is a subset of M κ. 10. A (κ, n)-type u is called elastic just in case that u− is defined and u contains the following two formulae: 1A The auxiliary games map 19 • “δ is an inaccessible cardinal”.