Fine hierarchies and Boolean terms

1995 ◽  
Vol 60 (1) ◽  
pp. 289-317 ◽  
Author(s):  
V. L. Selivanov
Keyword(s):  
Complexity Theory ◽  
Set Theory ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Descriptive Set

AbstractWe consider fine hierarchies in recursion theory, descriptive set theory, logic and complexity theory. The main results state that the sets of values of different Boolean terms coincide with the levels of suitable fine hierarchies. This gives new short descriptions of these hierarchies and shows that collections of sets of values of Boolean terms are almost well ordered by inclusion. For the sake of completeness we mention also some earlier results demonstrating the usefulness of fine hierarchies.

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.

scholarly journals Turing degrees in Polish spaces and decomposability of Borel functions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050021
Author(s):  
Vassilios Gregoriades ◽  
Takayuki Kihara ◽  
Keng Meng Ng
Keyword(s):  
Set Theory ◽  
Polish Space ◽  
Recursion Theory ◽  
Descriptive Set Theory ◽  
Analytic Subset ◽  
Polish Spaces ◽  
Negative Results ◽  
Borel Functions ◽  
Important Open Problem ◽  
Descriptive Set

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture.

Cantor’s Diagonalization Method

2016 ◽  
Vol 2 (3) ◽  
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Set Theory ◽  
Recursion Theory ◽  
Incompleteness Theorem ◽  
Descriptive Set Theory ◽  
Recursively Enumerable ◽  
Diagonalization Method ◽  
Descriptive Set

The set of arithmetic truths is neither recursive, nor recursively enumerable. Mathematician Alexander Kharazishvili explores how powerful the celebrated diagonal method is for general and descriptive set theory, recursion theory, and Gödel’s incompleteness theorem.

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 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
Sign in / Sign up

Export Citation Format

Share Document