Optimal proofs of determinacy

Bulletin of Symbolic Logic 1 (3):327-339 (1995)
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Abstract

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L.The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL ⇒ HODL ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals.

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Citations of this work

Long games and σ-projective sets.Juan P. Aguilera, Sandra Müller & Philipp Schlicht - 2021 - Annals of Pure and Applied Logic 172 (4):102939.
Forcing the [math]-separation property.Stefan Hoffelner - 2022 - Journal of Mathematical Logic 22 (2).
Optimal proofs of determinacy II.Itay Neeman - 2002 - Journal of Mathematical Logic 2 (2):227-258.

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References found in this work

Inner models with many Woodin cardinals.J. R. Steel - 1993 - Annals of Pure and Applied Logic 65 (2):185-209.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.
Analytic determinacy and 0#. [REVIEW]Leo Harrington - 1978 - Journal of Symbolic Logic 43 (4):685 - 693.

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