A null ideal for inaccessibles

“…In the recent years the theory of the generalized Cantor and Baire spaces was extensively developed (cf., e.g. [5–8, 10, 27, 28, 31, 43, 44] and many others). An important part of the research in this subject is an attempt to transfer the results in set theory of the real line to 2 κ and

κ^{κ}

(the list of open questions can be found in [22]).…”

Section: Introductionmentioning

confidence: 99%

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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“…We will be interested in a natural generalization of such properties to the spaces 2 κ and κ κ for uncountable (and in this paper: always inaccessible) cardinals κ. This area of research has progressed quickly in recent years; [KL + 16] collected many questions inspired by workshops on generalized reals, and several recent results can be found in [BB + 18], [FL17], [She17], [CS19].…”

mentioning

confidence: 99%

“…A construction of a κ + -c.c. κ κ -bounding forcing notion using a different diamond is given in [FL17] but it implies 2 κ = κ + , therefore that setup does not allow us to investigate cardinal characteristics.…”

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confidence: 99%

The higher Cichoń diagram

2021

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For a strongly inacessible cardinal κ, we investigate the relationships between the following ideals: (1) the ideal of meager sets in the <κ-box product topology, (2) the ideal of "null" sets in the sense of [She17], (3) the ideal of nowhere stationary subsets of a (naturally defined) stationary set S κ pr ⊆ κ. In particular, we analyze the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable. While some results from the classical case (κ = ω) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null) ≤ non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant. We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with <κ-support (and a strong "Knaster" version of κ +-c.c.) and one iteration with ≤κ-support (and a version of κ-properness).

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A null ideal for inaccessibles

Cited by 6 publications

References 3 publications

Questions on generalised Baire spaces

Questions on generalised Baire spaces

Special subsets of the generalized Cantor space and generalized Baire space

The higher Cichoń diagram

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