Generalisations of stationarity, closed and unboundedness, and of Jensen's □

Brickhill, Hazel; Welch, Philip D.

doi:10.1016/j.apal.2023.103272

Cited by 2 publications

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References 23 publications

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“…In order to prove the characterizations of limit points of sets in the spaces (P κ X, τ ξ ) (Theorem 3.16(1)), we introduce new iterated forms of two-cardinal stationarity and two-cardinal pairwise simultaneous stationary reflection, which we refer to as ξ-strong stationarity and ξ-s-strong stationarity (see Definition 3.7). Let us note that the notions of ξ-strong stationarity and ξ-s-strong stationarity introduced here are natural generalizations of notions previously studied in the cardinal context by Bagaria, Magidor and Sakai [4], Bagaria [2] and by Brickhill and Welch [8], as well as those previously studied in the two-cardinal context by Sakai [26], by Torres [28], as well as by Benhamou and the third author [7].…”

Section: Introductionmentioning

confidence: 75%

“…It is conceivable that some two-cardinal (κ)-like principle could be used to address Questions 5.3. For example, (κ) implies that every stationary subset of κ can be partitioned into two disjoint stationary sets that do not simultaneously reflect (see [20,Theorem 2.1] as well as [14,Theorem 7.1] and [8,Theorem 3.50] for generalizations). Question 5.4.…”

Section: Questions and Ideasmentioning

confidence: 99%

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Higher indescribability and derived topologies

Cody

2023

J. Math. Log.

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We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of [Formula: see text]-indescribability and [Formula: see text]-indescribability of a cardinal [Formula: see text] for all [Formula: see text]. In this context, universal [Formula: see text] formulas exist, there is a normal ideal associated to [Formula: see text]-indescribability and the notions of [Formula: see text]-indescribability yield a strict hierarchy below a subtle cardinal. Additionally, given a regular cardinal [Formula: see text], we introduce a diagonal version of Cantor’s derivative operator and use it to extend Bagaria’s [Derived topologies on ordinals and stationary reflection, Trans. Amer. Math. Soc. 371(3) (2019) 1981–2002] sequence [Formula: see text] of derived topologies on [Formula: see text] to [Formula: see text]. Finally, we prove that for all [Formula: see text], if there is a stationary set of [Formula: see text] that have a high enough degree of indescribability, then there are stationarily many [Formula: see text] that are nonisolated points in the space [Formula: see text].

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Section: Introductionmentioning

confidence: 75%

Section: Questions and Ideasmentioning

confidence: 99%

Higher indescribability and derived topologies

Cody

2023

J. Math. Log.

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Two-Cardinal Derived Topologies, Indescribability and Ramseyness

CODY,

LAMBIE-HANSON,

ZHANG

2024

J. symb. log.

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We introduce a natural two-cardinal version of Bagaria’s sequence of derived topologies on ordinals. We prove that for our sequence of two-cardinal derived topologies, limit points of sets can be characterized in terms of a new iterated form of pairwise simultaneous reflection of certain kinds of stationary sets, the first few instances of which are often equivalent to notions related to strong stationarity, which has been studied previously in the context of strongly normal ideals. The non-discreteness of these two-cardinal derived topologies can be obtained from certain two-cardinal indescribability hypotheses, which follow from local instances of supercompactness. Additionally, we answer several questions posed by the first author, Holy and White on the relationship between Ramseyness and indescribability in both the cardinal context and in the two-cardinal context.

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Generalisations of stationarity, closed and unboundedness, and of Jensen's □

Cited by 2 publications

References 23 publications

Higher indescribability and derived topologies

Higher indescribability and derived topologies

Two-Cardinal Derived Topologies, Indescribability and Ramseyness

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