Wyniki wyszukiwania

1

A López-Escobar theorem for metric structures, and the topological Vaught conjecture

100%

Coskey S., Lupini M.

Fundamenta Mathematicae

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2016

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tom 234

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nr 1

55-72

EN

We show that a version of López-Escobar's theorem holds in the setting of model theory for metric structures. More precisely, let 𝕌 denote the Urysohn sphere and let Mod(𝓛,𝕌) be the space of metric 𝓛-structures supported on 𝕌. Then for any Iso(𝕌)-invariant Borel function f: Mod(𝓛,𝕌) → [0,1], there exists a sentence ϕ of $𝓛_{ω₁ω}$ such that for all M ∈ Mod(𝓛,𝕌) we have $f(M) = ϕ^{M}$. This answers a question of Ivanov and Majcher-Iwanow. We prove several consequences, for example every orbit equivalence relation of a Polish group action is Borel isomorphic to the isomorphism relation on the set of models of a given $𝓛_{ω₁ω}$-sentence that are supported on the Urysohn sphere. This in turn provides a model-theoretic reformulation of the topological Vaught conjecture.

2

Generalized Choquet spaces

100%

Coskey S., Schlicht P.

Fundamenta Mathematicae

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2016

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tom 232

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nr 3

227-248

EN

We introduce an analog to the notion of Polish space for spaces of weight ≤ κ, where κ is an uncountable regular cardinal such that $κ^{<κ} = κ$. Specifically, we consider spaces in which player II has a winning strategy in a variant of the strong Choquet game which runs for κ many rounds. After discussing the basic theory of these games and spaces, we prove that there is a surjectively universal such space and that there are exactly $2^{κ}$ many such spaces up to homeomorphism. We also establish a Kuratowski-like theorem that under mild hypotheses, any two such spaces of size > κ are isomorphic by a κ-Borel function. We then consider a dynamic version of the Choquet game, and show that in this case the existence of a winning strategy for player II implies the existence of a winning tactic, that is, a strategy that depends only on the most recent move. We also study a generalization of Polish ultrametric spaces where the ultrametric is allowed to take values in a set of size κ. We show that in this context, there is a family of universal Urysohn-type spaces, and we give a characterization of such spaces which are hereditarily κ-Baire.

3

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

81%

Coskey S., Mátrai T., Steprāns J.

Fundamenta Mathematicae

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2013

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tom 223

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nr 1

29-48

EN

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality 𝔭 ≤ 𝔟 does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we use our results to give an embedding from the inclusion ordering on 𝒫(ω) into the Borel Tukey ordering on cardinal invariants.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Coskey S.

1 Lupini M.

1 Mátrai T.

1 Schlicht P.

1 Steprāns J.

2 2016

1 2013

A López-Escobar theorem for metric structures, and the topological Vaught conjecture

Generalized Choquet spaces

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum