Some measure theoretic results in effective descriptive set theory

1975 ◽  
Vol 20 (2) ◽  
pp. 97-110 ◽  
Author(s):  
Jacques Stern
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

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 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.

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.

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