HANDMADE DENSITY SETS

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.

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RANDOMNESS VIA INFINITE COMPUTATION AND EFFECTIVE DESCRIPTIVE SET THEORY

Journal of Symbolic Logic ◽

10.1017/jsl.2018.3 ◽

2018 ◽

Vol 83 (2) ◽

pp. 766-789 ◽

Cited By ~ 1

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.

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COMPUTABLE POLISH GROUP ACTIONS

Journal of Symbolic Logic ◽

10.1017/jsl.2017.68 ◽

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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A boundedness lemma for iterations

Journal of Symbolic Logic ◽

10.2307/2695092 ◽

2001 ◽

Vol 66 (3) ◽

pp. 1058-1072

Author(s):

Greg Hjorth

Keyword(s):

Set Theory ◽

Model Theory ◽

Large Cardinal ◽

Descriptive Set Theory ◽

Inner Model Theory ◽

Inner Model ◽

Direct Limits ◽

The One ◽

Image Position ◽

Descriptive Set

The purpose of this paper is to present a kind of boundedness lemma for direct limits of coarse structural mice, and to indicate some applications to descriptive set theory. For instance, this allows us to show that under large cardinal or determinacy assumptions there is no prewellorder ≤ of length such that for some formula ψ and parameter zif and only ifIt is a peculiar experience to write up a result in this area. Following the work of Martin, Steel, Woodin, and other inner model theory experts, there is an enormous overhang of theorems and ideas, and it only takes one wandering pebble to restart the avalanche. For this reason I have chosen to center the exposition around the one pebble at 1.7 which I believe to be new. The applications discussed in section 2 involve routine modifications of known methods.A detailed introduction to many of the techniques related to using the Martin-Steel inner model theory and Woodin's free extender algebra is given in the course of [1]. Certainly a familiarity with the Martin-Steel papers, [5] and [6], is a prerequisite, as is some knowledge of the free extender algebra. Probably anyone interested in this paper will already know the necessary descriptive set theory, most of which can be found in [4]. Discussion of earlier results in this direction can be found in [3] or [2].

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Two Applications of Inner Model Theory to the Study of Sets

Bulletin of Symbolic Logic ◽

10.2307/421049 ◽

1996 ◽

Vol 2 (1) ◽

pp. 94-107 ◽

Cited By ~ 3

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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