Displaying similar documents to “Clones on regular cardinals”

A partition property of cardinal numbers

N. H. Williams

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CONTENTSIntroduction....................................................................................... 5§ 1. Notation and definitions......................................................... 5§ 2. Negative relations.................................................................... 9§ 3. The Ramification Lemma ..................................................... 10§ 4. The main theorem................................................................... 13§ 5. A result for cardinals...

Supercompactness and failures of GCH

Sy-David Friedman, Radek Honzik (2012)

Fundamenta Mathematicae

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Let κ < λ be regular cardinals. We say that an embedding j: V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V. Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ⁺⁺-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ⁺⁺-tall measurable cardinal κ. Now more generally, suppose that κ ≤ λ are regular and one wishes the GCH to fail at λ with κ being λ-supercompact. Silver’s...

Supercompactness and partial level by level equivalence between strong compactness and strongness

Arthur W. Apter (2004)

Fundamenta Mathematicae

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We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if δ + α is regular, δ is δ + α strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are δ + α strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function φ reflects an infinite cardinal κ , if given a topological space X with φ ( X ) ≥ κ , there exists Y ∈ [ X ] ≤ κ with φ ( Y ) ≥ κ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

L-like Combinatorial Principles and Level by Level Equivalence

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: (1) ♢ δ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ₀ such that for every δ ∈ S, ◻ δ holds and δ carries a gap 1 morass. (3) A weak version of ◻ δ holds for every...

The number of L ∞ κ -equivalent nonisomorphic models for κ weakly compact

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

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For a cardinal κ and a model M of cardinality κ let No(M) denote the number of nonisomorphic models of cardinality κ which are L ∞ , κ -equivalent to M. We prove that for κ a weakly compact cardinal, the question of the possible values of No(M) for models M of cardinality κ is equivalent to the question of the possible numbers of equivalence classes of equivalence relations which are Σ¹₁-definable over V κ . By [SV] it is possible to have a generic extension where the possible numbers of equivalence...

Initially κ -compact spaces for large κ

Stavros Christodoulou (1999)

Commentationes Mathematicae Universitatis Carolinae

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This work presents some cardinal inequalities in which appears the closed pseudo-character, ψ c , of a space. Using one of them — ψ c ( X ) ≤ 2 d ( X ) for T 2 spaces — we improve, from T 3 to T 2 spaces, the well-known result that initially κ -compact T 3 spaces are λ -bounded for all cardinals λ such that 2 λ ≤ κ . And then, using an idea of A. Dow, we prove that initially κ -compact T 2 spaces are in fact compact for κ = 2 F ( X ) , 2 s ( X ) , 2 t ( X ) , 2 χ ( X ) , 2 ψ c ( X ) or κ = max { τ + , τ < τ } , where τ > t ( p , X ) for all p ∈ X .

On simple partitions of [ κ ] κ

David Asperó (2003)

Fundamenta Mathematicae

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For every uncountable regular cardinal κ, every κ-Borel partition of the space of all members of [ κ ] κ whose enumerating function does not have fixed points has a homogeneous club.

The tree property at the double successor of a measurable cardinal κ with 2 κ large

Sy-David Friedman, Ajdin Halilović (2013)

Fundamenta Mathematicae

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Assuming the existence of a λ⁺-hypermeasurable cardinal κ, where λ is the first weakly compact cardinal above κ, we prove that, in some forcing extension, κ is still measurable, κ⁺⁺ has the tree property and 2 κ = κ ⁺ ⁺ ⁺ . If the assumption is strengthened to the existence of a θ -hypermeasurable cardinal (for an arbitrary cardinal θ > λ of cofinality greater than κ) then the proof can be generalized to get 2 κ = θ .

Uncountable cardinals have the same monadic ∀₁¹ positive theory over large sets

Athanassios Tzouvaras (2004)

Fundamenta Mathematicae

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We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)ϕ(X) and (∃X)ϕ(X), for ϕ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ,∈), (λ,∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures ( 2 κ , [ 2 κ ] > κ , < ) , ( 2 λ , [ 2 λ ] > λ , < ) are...

Interpolation of κ -compactness and PCF

István Juhász, Zoltán Szentmiklóssy (2009)

Commentationes Mathematicae Universitatis Carolinae

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We call a topological space κ -compact if every subset of size κ has a complete accumulation point in it. Let Φ ( μ , κ , λ ) denote the following statement: μ < κ < λ = cf ( λ ) and there is { S ξ : ξ < λ } ⊂ [ κ ] μ such that | { ξ : | S ξ ∩ A | = μ } | < λ whenever A ∈ [ κ ] < κ . We show that if Φ ( μ , κ , λ ) holds and the space X is both μ -compact and λ -compact then X is κ -compact as well. Moreover, from PCF theory we deduce Φ ( cf ( κ ) , κ , κ + ) for every singular cardinal κ . As a corollary we get that a linearly Lindelöf and ℵ ω -compact space is uncountably compact, that is κ -compact for all uncountable cardinals...

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of P κ ( X ) , where κ ⊆ X. Suppose μ < κ, μ < μ = μ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in [ H ( λ ) ] μ ⁺ which are internally club but not internally approachable.

A MAD Q-set

Arnold W. Miller (2003)

Fundamenta Mathematicae

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A MAD (maximal almost disjoint) family is an infinite subset of the infinite subsets of ω = 0,1,2,... such that any two elements of intersect in a finite set and every infinite subset of ω meets some element of in an infinite set. A Q-set is an uncountable set of reals such that every subset is a relative G δ -set. It is shown that it is relatively consistent with ZFC that there exists a MAD family which is also a Q-set in the topology it inherits as a subset of P ( ω ) = 2 ω .

Ordinal remainders of classical ψ-spaces

Alan Dow, Jerry E. Vaughan (2012)

Fundamenta Mathematicae

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Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain T α : α < λ of infinite subsets of ω, there exists ℳ ⊂ [ ω ] ω , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain T α : α < λ , hence a ψ-space with...