Measurable cardinals and the cofinality of the symmetric group

Sy-David Friedman; Lyubomyr Zdomskyy

Displaying similar documents to “Measurable cardinals and the cofinality of the symmetric group”

Strong compactness, measurability, and the class of supercompact cardinals

Arthur W. Apter (2001)

Fundamenta Mathematicae

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We prove two theorems concerning strong compactness, measurability, and the class of supercompact cardinals. We begin by showing, relative to the appropriate hypotheses, that it is consistent non-trivially for every supercompact cardinal to be the limit of (non-supercompact) strongly compact cardinals. We then show, relative to the existence of a non-trivial (proper or improper) class of supercompact cardinals, that it is possible to have a model with the same class of supercompact cardinals...

Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns

Arthur W. Apter (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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We provide upper and lower bounds in consistency strength for the theories “ZF + $\neg A C_{ω}$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal is singular of cofinality ω” and “ZF + $\neg A C_{ω}$ + All successor cardinals except successors of uncountable limit cardinals are regular + Every uncountable limit cardinal is singular + The successor of every uncountable limit cardinal...

A Note on Indestructibility and Strong Compactness

Arthur W. Apter (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is $2^{λ}$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is $2^{δ} = δ ⁺$ supercompact,...

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.

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...

On equivalence relations second order definable over H(κ)

Saharon Shelah, Pauli Vaisanen (2002)

Fundamenta Mathematicae

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Let κ be an uncountable regular cardinal. Call an equivalence relation on functions from κ into 2 second order definable over H(κ) if there exists a second order sentence ϕ and a parameter P ⊆ H(κ) such that functions f and g from κ into 2 are equivalent iff the structure ⟨ H(κ), ∈, P, f, g ⟩ satisfies ϕ. The possible numbers of equivalence classes of second order definable equivalence relations include all the nonzero cardinals at most κ⁺. Additionally, the possibilities are closed...

Stationary reflection and level by level equivalence

Arthur W. Apter (2009)

Colloquium Mathematicae

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We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jónsson cardinal is weakly compact. On the other hand, we force and construct a model for the level by level equivalence between strong compactness...

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 $2^{κ} = θ$ .

Δ₁-Definability of the non-stationary ideal at successor cardinals

Sy-David Friedman, Liuzhen Wu, Lyubomyr Zdomskyy (2015)

Fundamenta Mathematicae

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Assuming V = L, for every successor cardinal κ we construct a GCH and cardinal preserving forcing poset ℙ ∈ L such that in $L^{ℙ}$ the ideal of all non-stationary subsets of κ is Δ₁-definable over H(κ⁺).

Easton functions and supercompactness

Brent Cody, Sy-David Friedman, Radek Honzik (2014)

Fundamenta Mathematicae

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Suppose that κ is λ-supercompact witnessed by an elementary embedding j: V → M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F(α)), and (2) α < β ⇒ F(α) ≤ F(β). We address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while...

The consistency strength of the tree property at the double successor of a measurable cardina

Natasha Dobrinen, Sy-David Friedman (2010)

Fundamenta Mathematicae

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The Main Theorem is the equiconsistency of the following two statements: (1) κ is a measurable cardinal and the tree property holds at κ⁺⁺; (2) κ is a weakly compact hypermeasurable cardinal. From the proof of the Main Theorem, two internal consistency results follow: If there is a weakly compact hypermeasurable cardinal and a measurable cardinal far enough above it, then there is an inner model in which there is a proper class of measurable cardinals, and in which the tree property...

Rectangular square-bracket operation for successor of regular cardinals

Assaf Rinot, Stevo Todorcevic (2013)

Fundamenta Mathematicae

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We give a uniform proof that $λ ⁺ ↛ [λ ⁺; λ ⁺] ²_{λ ⁺}$ holds for every regular cardinal λ.

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...

How many normal measures can $ℵ_{ω + 1}$ carry?

Arthur W. Apter (2006)

Fundamenta Mathematicae

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We show that assuming the consistency of a supercompact cardinal with a measurable cardinal above it, it is possible for $ℵ_{ω + 1}$ to be measurable and to carry exactly τ normal measures, where $τ \geq ℵ_{ω + 2}$ is any regular cardinal. This contrasts with the fact that assuming AD + DC, $ℵ_{ω + 1}$ is measurable and carries exactly three normal measures. Our proof uses the methods of [6], along with a folklore technique and a new method due to James Cummings.

Categoricity of theories in $L_{κ , ω}$ , when κ is a measurable cardinal. Part 2

Saharon Shelah (2001)

Fundamenta Mathematicae

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We continue the work of [2] and prove that for λ successor, a λ-categorical theory T in $L_{κ *, ω}$ is μ-categorical for every μ ≤ λ which is above the $(2^{L S (T)}) ⁺$ -beth cardinal.

On the cardinality of Hausdorff spaces and Pol-Šapirovskii technique

Alejandro Ramírez-Páramo (2005)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we make use of the Pol-Šapirovskii technique to prove three cardinal inequalities. The first two results are due to Fedeli [2] and the third theorem of this paper is a common generalization to: (a) (Arhangel’skii [1]) If $X$ is a $T_{1}$ space such that (i) $L (X) t (X) \leq κ$ , (ii) $ψ (X) \leq 2^{κ}$ , and (iii) for all $A \in {[X]}^{\leq 2^{κ}}$ , $|\bar{A}| \leq 2^{κ}$ , then $| X | \leq 2^{κ}$ ; and (b) (Fedeli [2]) If $X$ is a $T_{2}$ -space then $| X | \leq 2^{aql (X) t (X) ψ_{c} (X)}$ .

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...

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 — $ψ_{c} (X) \leq 2^{d (X)}$ 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 $2^{λ} \leq κ$ . 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 $p \in X$ .

Pcf theory and cardinal invariants of the reals

Lajos Soukup (2011)

Commentationes Mathematicae Universitatis Carolinae

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The additivity spectrum $ADD (ℐ)$ of an ideal $ℐ \subset 𝒫 (I)$ is the set of all regular cardinals $κ$ such that there is an increasing chain ${A_{α} : α < κ} \subset ℐ$ with $⋃_{α < κ} A_{α} \notin ℐ$ . We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $ℐ = ℬ$ or $ℐ = 𝒩$ , where $ℬ$ denotes the $σ$ -ideal generated by the compact subsets of the Baire space $ω^{ω}$ , and $𝒩$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $pcf (A) = A$ , then $ADD (ℐ) = A$ in some c.c.c generic extension...