$Π^1_2$ singletons and $O^#$

Leo Harrington; Alexander Kechris

Displaying similar documents to “ $Π_{2}^{1}$ singletons and $O$ ”

Provident sets and rudimentary set forcing

A. R. D. Mathias (2015)

Fundamenta Mathematicae

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Using the theory of rudimentary recursion and provident sets expounded in [MB], we give a treatment of set forcing appropriate for working over models of a theory PROVI which may plausibly claim to be the weakest set theory supporting a smooth theory of set forcing, and of which the minimal model is Jensen’s $J_{ω}$ . Much of the development is rudimentary or at worst given by rudimentary recursions with parameter the notion of forcing under consideration. Our development eschews the power...

Pressing Down Lemma for $λ$ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

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For any ordinal $λ$ of uncountable cofinality, a $λ$ -tree is a tree $T$ of height $λ$ such that $| T_{α} | < cf (λ)$ for each $α < λ$ , where $T_{α} = {x \in T : ht (x) = α}$ . In this note we get a Pressing Down Lemma for $λ$ -trees and discuss some of its applications. We show that if $η$ is an uncountable ordinal and $T$ is a Hausdorff tree of height $η$ such that $| T_{α} | \leq ω$ for each $α < η$ , then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C \subset T$ and for each limit ordinal $α \leq η$ with $cf (α) > ω$ , ${ht (c) : c \in C} \cap α$ is not stationary in $α$ . In the last part of this note, we investigate...

Borel sets with large squares

Saharon Shelah (1999)

Fundamenta Mathematicae

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For a cardinal μ we give a sufficient condition $\oplus_{μ}$ (involving ranks measuring existence of independent sets) for: $\otimes_{μ}$ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a $2^{ℵ_{0}}$ -square and even a perfect square, and also for $\otimes_{μ}^{'}$ if $ψ \in L_{ω_{1}, ω}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $M A + 2^{ℵ_{0}} > μ$ for transparency, those three conditions ( $\oplus_{μ}$ , $\otimes_{μ}$ and $\otimes_{μ}^{'}$ ) are equivalent, and from this we...

Chain conditions in maximal models

Paul Larson, Stevo Todorčević (2001)

Fundamenta Mathematicae

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We present two $ℙ_{m a x}$ varations which create maximal models relative to certain counterexamples to Martin’s Axiom, in hope of separating certain classical statements which fall between MA and Suslin’s Hypothesis. One of these models is taken from [19], in which we maximize relative to the existence of a certain type of Suslin tree, and then force with that tree. In the resulting model, all Aronszajn trees are special and Knaster’s forcing axiom ₃ fails. Of particular interest is the still...

Ordinal indices and Ramsey dichotomies measuring c₀-content and semibounded completeness

Vassiliki Farmaki (2002)

Fundamenta Mathematicae

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We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index $ξ^{(χ ₙ)} ₀$ and the semibounded completeness index $ξ_{b}^{(χ ₙ)}$ , and we examine their relationship. The countable ordinal values that these indices can take are always of the form...

Proof of a conjecture of McKay

Krister Segerberg (1974)

Fundamenta Mathematicae

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OCA and towers in $𝒫 (ℕ) / f i n$

Ilijas Farah (1996)

Commentationes Mathematicae Universitatis Carolinae

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We shall show that Open Coloring Axiom has different influence on the algebra $𝒫 (ℕ) / f i n$ than on $ℕ^{ℕ} / f i n$ . The tool used to accomplish this is forcing with a Suslin tree.

A new construction of non-constructible ${Δ^{1}}_{3}$ subset of ω

Ronald Jensen, Håvard Johnsbråten (1974)

Fundamenta Mathematicae

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The strength of the projective Martin conjecture

C. T. Chong, Wei Wang, Liang Yu (2010)

Fundamenta Mathematicae

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We show that Martin’s conjecture on Π¹₁ functions uniformly $\leq_{T}$ -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant $Π ¹_{2 n + 1}$ functions is equivalent over ZFC to $Σ ¹_{2 n + 2}$ -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.

On countable cofinality and decomposition of definable thin orderings

Vladimir Kanovei, Vassily Lyubetsky (2016)

Fundamenta Mathematicae

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We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and Σ¹₂ thin sets under the assumption that $ω ₁^{L [x]} < ω ₁$ for all reals x. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.

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...

Squares with diamonds and Souslin trees with special squares

Uri Abraham, Saharon Shelah, R. Solovay (1987)

Fundamenta Mathematicae

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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 $2^{κ} = θ$ .

Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

Saharon Shelah, R. Jin (1992)

Fundamenta Mathematicae

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By an $ω_{1}$ - tree we mean a tree of power $ω_{1}$ and height $ω_{1}$ . Under CH and $2^{ω_{1}} > ω_{2}$ we call an $ω_{1}$ -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between $ω_{1}$ and $2^{ω_{1}}$ . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{ω_{1}} > ω_{2}$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{ω_{1}} = ω_{4}$ that there only exist Kurepa trees with $ω_{3}$ -many branches, which answers...

A note on the Mac Dowell-Specker theorem

Peter Clote (1987)

Fundamenta Mathematicae

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Displaying similar documents to “ Π 2 1 singletons and O ”

Displaying similar documents to “ $Π_{2}^{1}$ singletons and $O$ ”