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

Sy-David Friedman; Liuzhen Wu; Lyubomyr Zdomskyy

Displaying similar documents to “Δ₁-Definability of the non-stationary ideal at successor cardinals”

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.

Reflection implies the SCH

Saharon Shelah (2008)

Fundamenta Mathematicae

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We prove that, e.g., if μ > cf(μ) = ℵ₀ and $μ > 2^{ℵ ₀}$ and every stationary family of countable subsets of μ⁺ reflects in some subset of μ⁺ of cardinality ℵ₁, then the SCH for μ⁺ holds (moreover, for μ⁺, any scale for μ⁺ has a bad stationary set of cofinality ℵ₁). This answers a question of Foreman and Todorčević who get such a conclusion from the simultaneous reflection of four stationary sets.

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...

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...

Splitting stationary sets in $_{κ} λ$ for λ with small cofinality

Toshimichi Usuba (2009)

Fundamenta Mathematicae

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For a regular uncountable cardinal κ and a cardinal λ with cf(λ) < κ < λ, we investigate the consistency strength of the existence of a stationary set in $_{κ} λ$ which cannot be split into λ⁺ many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in $_{κ} λ$ such that every stationary subset of S can be split into λ⁺ many pairwise disjoint stationary subsets. ...

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...

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...

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...

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...

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 $2^{κ} = θ$ .

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...

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 $p \in X$ .

Some properties of stationary sets

C. A. Di Prisco, W. Marek

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CONTENTSIntroduction..................................................................51. Derivative of a stationary set...................................72. Stationary degrees ...............................................133. Subsets of $P_{ϰ} (λ)$ ..............................................194. Stationary subsets of $P_{ϰ} (λ)$ .............................255. Superstationary substes of $P_{ϰ} (λ)$ ....................326. End-stationary subsets of $P_{ϰ} (λ)$ ......................34References................................................................37 ...

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...