A characterization of jump operators

1988 ◽  
Vol 53 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Howard Becker
Keyword(s):  
Characterization Theorem ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
General Study ◽  
General Topic ◽  
Descriptive Set ◽  
Increasing Function

The topic of this paper is jump operators, a subject which originated with some questions of Martin and a partial answer to them obtained by Steel [18]. The topic of jump operators is a part of the general study of the structure of the Turing degrees, but it is concerned with an aspect of that structure which is different from the usual concerns of classical recursion theory. Specifically, it is concerned with studying functions on the degrees, such as the Turing jump operator, the hyperjump operator, and the sharp operator.Roughly speaking, a jump operator is a definable ≤T-increasing function on the Turing degrees. The purpose of this paper is to characterize the jump operators, in terms of concepts from descriptive set theory. Again roughly speaking, the main theorem states that all jump operators (other than the identity function) are obtained from pointclasses by the same process by which the hyperjump operator is obtained from the pointclass Π11; that is, if Γ is the pointclass, then the operator maps the real x to the universal Γ(x) subset of ω. This characterization theorem has some corollaries, one of which answers a question of Steel [18]. In §1 we give a brief introduction to this general topic, followed by a brief (and still somewhat imprecise) description of the results contained in this paper.

scholarly journals JUMP OPERATIONS FOR BOREL GRAPHS

2018 ◽  
Vol 83 (1) ◽  
pp. 13-28
Author(s):  
ADAM R. DAY ◽  
ANDREW S. MARKS
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Connected Components ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
Borel Equivalence Relations ◽  
Locally Countable ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

AbstractWe investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

Simple r. e. degree structures

1987 ◽  
Vol 52 (1) ◽  
pp. 208-213
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Recursion Theory ◽  
Turing Degrees ◽  
Local Version ◽  
Admissible Set ◽  
Jump Operator ◽  
Admissible Sets ◽  
Closed Sets ◽  
Locally Countable ◽  
To Come ◽  
Admissible Ordinals

Much of recursion theory centers on the structures of different kinds of degrees. Classically there are the Turing degrees and r. e. Turing degrees. More recently, people have studied α-degrees for α an ordinal, and degrees over E-closed sets and admissible sets. In most contexts, deg(0) is the bottom degree and there is a jump operator' such that d' is the largest degree r. e. in d and d' > d. Both the degrees and the r. e. degrees usually have a rich structure, including a relativization to the cone above a given degree.A natural exception to this pattern was discovered by S. Friedman [F], who showed that for certain admissible ordinals β the β-degrees ≥ 0′ are well-ordered, with successor provided by the jump.For r. e. degrees, natural counterexamples are harder to come by. This is because the constructions are priority arguments, which require only mild restrictions on the ground model. For instance, if an admissible set has a well-behaved pair of recursive well-orderings then the priority construction of an intermediate r. e. degree (i.e., 0 < d < 0′) goes through [S]. It is of interest to see just what priority proofs need by building (necessarily pathological) admissible sets with few r. e. degrees.Harrington [C] provides an admissible set with two r. e. degrees, via forcing. A limitation of his example is that it needs ω1 (more accurately, a local version thereof) as a parameter. In this paper, we find locally countable admissible sets, some with three r. e. degrees and some with four.

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.

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.

scholarly journals HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 93-102 ◽  
Author(s):  
MARGARYTA MYRONYUK
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
Gaussian Distributions ◽  
Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

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.

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