The Axiom of Choice, the Löwenheim-Skolem Theorem and Borel models

Jerome Malitz; Jan Mycielski; William Reinhardt

Displaying similar documents to “The Axiom of Choice, the Löwenheim-Skolem Theorem and Borel models”

The equivalence of definable quantifiers in second order arithmetic

Wojciech Guzicki (1981)

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Erratum to the paper "The Axion of Choice, the Löwenheim–Skolem Theorem and Borel models" (Fund. Math. 137 (1991), 53-58)

J. Malitz, Jan Mycielski, W. Reinhardt (1992)

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Borel Sets and Countable Models

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Matt Kaufmann (1984)

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Małgorzata Dubiel (1980)

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A contribution to Gödel's axiomatic set theory, I

Ladislav Rieger (1957)

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Non-Glimm–Effros equivalence relations at second projective level

Vladimir Kanovei (1997)

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A model is presented in which the $Σ_{2}^{1}$ equivalence relation xCy iff L[x]=L[y] of equiconstructibility of reals does not admit a reasonable form of the Glimm-Effros theorem. The model is a kind of iterated Sacks generic extension of the constructible model, but with an “ill“founded “length” of the iteration. In another model of this type, we get an example of a $Π_{2}^{1}$ non-Glimm-Effros equivalence relation on reals. As a more elementary application of the technique of “ill“founded Sacks iterations,...

Uncountable β-models with countable height

Wojciech Guzicki (1974)

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Borel sets with large squares

Saharon Shelah (1999)

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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...