The axiom of determinacy and the modern development of descriptive set theory

V. G. Kanovei

doi:10.1007/bf01092890

The axiom of determinacy and the modern development of descriptive set theory

Journal of Soviet Mathematics ◽

10.1007/bf01092890 ◽

1988 ◽

Vol 40 (3) ◽

pp. 257-287 ◽

Cited By ~ 1

Author(s):

V. G. Kanovei

Keyword(s):

Set Theory ◽

Descriptive Set Theory ◽

Axiom Of Determinacy ◽

Modern Development ◽

Descriptive Set

Download Full-text

Optimal Proofs of Determinacy

Bulletin of Symbolic Logic ◽

10.2307/421159 ◽

1995 ◽

Vol 1 (3) ◽

pp. 327-339 ◽

Cited By ~ 20

Author(s):

Itay Neeman

Keyword(s):

Set Theory ◽

Large Cardinals ◽

The Other ◽

Descriptive Set Theory ◽

Transfer Theorem ◽

Axiom Of Determinacy ◽

Structural Connection ◽

Projective Hierarchy ◽

Descriptive Set

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(ℝ).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(ℝ) ⇒ HODL(ℝ) ⊨ 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.

Download Full-text

Two Applications of Inner Model Theory to the Study of Sets

Bulletin of Symbolic Logic ◽

10.2307/421049 ◽

1996 ◽

Vol 2 (1) ◽

pp. 94-107 ◽

Cited By ~ 3

Author(s):

Greg Hjorth

Keyword(s):

Los Angeles ◽

Set Theory ◽

Model Theory ◽

Large Cardinals ◽

Descriptive Set Theory ◽

Inner Model Theory ◽

Regular Cardinals ◽

Inner Model ◽

Definable Sets ◽

Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

Download Full-text