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.

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

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.

scholarly journals RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

2018 ◽  
Vol 83 (2) ◽  
pp. 766-789 ◽  
Author(s):  
MERLIN CARL ◽  
PHILIPP SCHLICHT
Keyword(s):  
Set Theory ◽  
Positive Lebesgue Measure ◽  
Descriptive Set Theory ◽  
Infinite Time ◽  
Admissible Sets ◽  
Random Forcing ◽  
Image Position ◽  
Descriptive Set ◽  
Share Information ◽  
Effective Descriptive Set Theory

AbstractWe study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines (ITTMs) introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as${\rm{\Pi }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets.

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.

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