Generalized Silver and Miller measurability

Laguzzi, Giorgio

doi:10.1002/malq.201400020

Cited by 9 publications

(16 citation statements)

References 11 publications

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“…Further background . Note that the notions of “κ‐Miller measurability”, “κ‐Sacks measurability” and “κ‐Silver measurability” considered in and are potential candidates for such properties; however, they are not generated by

< κ

‐closed forcing notions on

κ^{κ}

.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…κ‐Miller measurability can be forced to hold for all sets definable from ordinals and subsets of κ in the Silver model (which requires an inaccessible

λ > κ

); cf. [, Lemma 5.4]. If κ is inaccessible, then κ‐Silver measurability for all sets definable from ordinals and subsets of κ holds in the κ‐Cohen model; cf.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…If κ is inaccessible, then κ‐Silver measurability for all sets definable from ordinals and subsets of κ holds in the κ‐Cohen model; cf. [, Lemma 4.2].…”

Section: The List Of Open Questionsmentioning

confidence: 99%

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Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

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< κ

‐closed forcing notions on

κ^{κ}

.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

“…κ‐Miller measurability can be forced to hold for all sets definable from ordinals and subsets of κ in the Silver model (which requires an inaccessible

λ > κ

); cf. [, Lemma 5.4]. If κ is inaccessible, then κ‐Silver measurability for all sets definable from ordinals and subsets of κ holds in the κ‐Cohen model; cf.…”

Section: The List Of Open Questionsmentioning

confidence: 99%

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Questions on generalised Baire spaces

Khomskii

Laguzzi

Löwe

et al. 2016

Mathematical Logic Qtrly

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“…The Silver model cannot in general be used to separate the various regularity properties: in fact, the results from [Sch,Lag14] show that many of these properties are simultaneously satisfied by Σ 1 1 subsets of κ κ. However, we can use Theorem 1.13 to separate the Hurewicz dichotomy from the κ-perfect set property.…”

Section: Introductionmentioning

confidence: 99%

“…Thus in a Silver model all Σ 1 1 subsets (hence also all Π 1 1 subsets) of κ κ are κ-Sacks measurable. Laguzzi showed in [Lag14] that in fact such sets are also κ-Miller measurable.…”

mentioning

confidence: 99%

The Hurewicz dichotomy for generalized Baire spaces

Luecke

Ros

Schlicht³

2016

Isr. J. Math.

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Abstract. By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space X is covered by a Kσ subset of X if and only if it does not contain a closed-in-X subset homeomorphic to the Baire space ω ω. We consider the analogous statement (which we call Hurewicz dichotomy) for Σ 1 1 subsets of the generalized Baire space κ κ for a given uncountable cardinal κ with κ = κ <κ . We show that the statement that this dichotomy holds at all uncountable regular cardinals is consistent with the axioms of ZFC together with GCH and large cardinal axioms. In contrast, we show that the dichotomy fails at all uncountable regular cardinals after we add a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the κ-perfect set property, the κ-Miller measurability, and the κ-Sacks measurability.

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Special subsets of the generalized Cantor space and generalized Baire space

Korch

Weiss

2020

Mathematical Logic Qtrly

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In this paper, we are interested in parallels to the classical notions of special subsets in R defined in the generalized Cantor and Baire spaces (2 κ and κ κ). We consider generalizations of the well-known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ-sets, γ-sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less-know classes like X-small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver-null sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems.

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Generalized Silver and Miller measurability

Abstract: We present some results about the burgeoning research area concerning set theory of the "κ-reals". We focus on some notions of measurability coming from generalizations of Silver and Miller trees. We present analogies and mostly differences from the classical setting.

Cited by 9 publications

References 11 publications

Questions on generalised Baire spaces

Questions on generalised Baire spaces

The Hurewicz dichotomy for generalized Baire spaces

Special subsets of the generalized Cantor space and generalized Baire space

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