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 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

HANDMADE DENSITY SETS

2017 ◽  
Vol 82 (1) ◽  
pp. 208-223 ◽  
Author(s):  
GEMMA CAROTENUTO
Keyword(s):  
Set Theory ◽  
Real Line ◽  
Borel Measure ◽  
Point Of View ◽  
Topological Complexity ◽  
Descriptive Set Theory ◽  
Locally Finite ◽  
Finite Borel Measure ◽  
Image Position ◽  
Descriptive Set

AbstractGiven a metric space (X , d), equipped with a locally finite Borel measure, a measurable set $A \subseteq X$ is a density set if the points where A has density 1 are exactly the points of A. We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools—and from the point of view—of descriptive set theory. In this context a density set is always in $\Pi _3^0$. We single out a family of true $\Pi _3^0$ density sets, an example of true $\Sigma _2^0$ density set and finally one of true $\Pi _2^0$ density set.

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.

COMPUTABLE POLISH GROUP ACTIONS

2018 ◽  
Vol 83 (2) ◽  
pp. 443-460
Author(s):  
ALEXANDER MELNIKOV ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Set Theory ◽  
Group Actions ◽  
General Setting ◽  
Structure Theory ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Sufficient Condition ◽  
Computable Structure Theory ◽  
Image Position ◽  
Descriptive Set

AbstractUsing methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

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