DESCRIPTIVE SET THEORY AND UNCOUNTABLE MODEL THEORY

2006 ◽  
pp. 133-145
Author(s):  
MICHAEL C. LASKOWSKI
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Descriptive Set Theory ◽  
Descriptive Set

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
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.

scholarly journals Classification from a Computable Viewpoint

2006 ◽  
Vol 12 (2) ◽  
pp. 191-218 ◽  
Author(s):  
Wesley Calvert ◽  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Body ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Computable Structures ◽  
Structure Theorems ◽  
Sample Structure ◽  
Descriptive Set ◽  
Scott Rank

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

A topological analog to the Rice-Shapiro index theorem

1982 ◽  
Vol 47 (4) ◽  
pp. 824-832 ◽  
Author(s):  
Louise Hay ◽  
Douglas Miller
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Index Theorem ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
The Sixties ◽  
Turing Reducibility ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory ◽  
Theory And Model

Ever since Craig-Beth and Addison-Kleene proved their versions of the Lusin-Suslin theorem, work in model theory and recursion theory has demonstrated the value of classical descriptive set theory as a source of ideas and inspirations. During the sixties in particular, J.W. Addison refined the technique of “conjecture by analogy” and used it to generate a substantial number of results in both model theory and recursion theory (see, e.g., Addison [1], [2], [3]).During the past 15 years, techniques and results from recursion theory and model theory have played an important role in the development of descriptive set theory. (Moschovakis's book [6] is an excellent reference, particularly for the use of recursion-theoretic tools.) The use of “conjecture by analogy” as a means of transferring ideas from model theory and recursion theory to descriptive set theory has developed more slowly. Some notable recent examples of this phenomenon are in Vaught [9], where some results in invariant descriptive set theory reflecting and extending model-theoretic results are obtained and others are left as conjectures (including a version of the well-known conjecture on the number of countable models) and in Hrbacek and Simpson [4], where a notion analogous to that of Turing reducibility is used to study Borel isomorphism types. Moschovakis [6] describes in detail an effective descriptive set theory based in large part on classical recursion theory.

A boundedness lemma for iterations

2001 ◽  
Vol 66 (3) ◽  
pp. 1058-1072
Author(s):  
Greg Hjorth
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Cardinal ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Inner Model ◽  
Direct Limits ◽  
The One ◽  
Image Position ◽  
Descriptive Set

The purpose of this paper is to present a kind of boundedness lemma for direct limits of coarse structural mice, and to indicate some applications to descriptive set theory. For instance, this allows us to show that under large cardinal or determinacy assumptions there is no prewellorder ≤ of length such that for some formula ψ and parameter zif and only ifIt is a peculiar experience to write up a result in this area. Following the work of Martin, Steel, Woodin, and other inner model theory experts, there is an enormous overhang of theorems and ideas, and it only takes one wandering pebble to restart the avalanche. For this reason I have chosen to center the exposition around the one pebble at 1.7 which I believe to be new. The applications discussed in section 2 involve routine modifications of known methods.A detailed introduction to many of the techniques related to using the Martin-Steel inner model theory and Woodin's free extender algebra is given in the course of [1]. Certainly a familiarity with the Martin-Steel papers, [5] and [6], is a prerequisite, as is some knowledge of the free extender algebra. Probably anyone interested in this paper will already know the necessary descriptive set theory, most of which can be found in [4]. Discussion of earlier results in this direction can be found in [3] or [2].

Some applications of coarse inner model theory

1997 ◽  
Vol 62 (2) ◽  
pp. 337-365 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Class ◽  
Set Theory ◽  
Model Theory ◽  
Equivalence Relation ◽  
Equivalence Classes ◽  
Descriptive Set Theory ◽  
Inner Model Theory ◽  
Inner Model ◽  
Descriptive Set

AbstractThe Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. determinacy implies that for every thin equivalence relation there is a real, N, over which every equivalence class is generic—and hence there is a good (N#) wellordering of the equivalence classes. Analogous results are obtained for and quasilinear orderings and determinacy is shown to imply that every prewellorder has rank less than .

scholarly journals Intuitionism and effective descriptive set theory

2018 ◽  
Vol 29 (1) ◽  
pp. 396-428 ◽  
Author(s):  
Joan R. Moschovakis ◽  
Yiannis N. Moschovakis
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory
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