Some variants of Vaught's conjecture from the perspective of algebraic logic

Sági, Gábor; Sziráki, Dorottya

doi:10.1093/jigpal/jzr049

Cited by 8 publications

(5 citation statements)

References 17 publications

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“…If there are at least κ + many pairwise non (F, H)-elementarily embeddable models in Mod ψ κ , then there are perfectly many such models. Theorem 3.8 can be seen as uncountable version of [21,Theorems 5.8 and 5.9]. We note that these two cited theorems of [21] also follow from [12,Corollary 2.13] or from [23,Remark 1.14].…”

Section: Introductionmentioning

confidence: 78%

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A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

Sziráki

Väänánen

2017

Fund. Math.

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We consider the following dichotomy for Σ 0 2 finitary relations R on analytic subsets of the generalized Baire space for κ: either all R-independent sets are of size at most κ, or there is a κ-perfect Rindependent set. This dichotomy is the uncountable version of a result found in (W. Kubiś, Proc. Amer. Math. Soc. 131 (2003), no 2.:619-623) and in (S. Shelah, Fund. Math. 159 (1999), no. 1:1-50). We prove that the above statement holds assuming ♦ κ and the set theoretical hypothesis I − (κ), which is the modification of the hypothesis I(κ) suitable for limit cardinals. When κ is inaccessible, or when R is a closed binary relation, the assumption ♦ κ is not needed.We obtain as a corollary the uncountable version of a result by G. Sági and the first author (Log. J. IGPL 20 (2012), no. 6:1064-1082) about the κ-sized models of a Σ 1 1 (L κ + κ )-sentence when considered up to isomorphism, or elementary embeddability, by elements of a K κ subset of κ κ. The role of elementary embeddings can be replaced by a more general notion that also includes embeddings, as well as the maps preserving L λµ for ω ≤ µ ≤ λ ≤ κ and the finite variable fragments of these logics.

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Section: Introductionmentioning

confidence: 78%

“…Theorem 3.8 can be seen as uncountable version of [21, Theorems 5.8 and 5.9]. We note that these two cited theorems of [21] also follow from [12,Corollary 2.13] or from [23,Remark 1.14].…”

Section: Union) Of < κ Many Open (Closed) Sets Is Open (Closed)mentioning

confidence: 90%

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

Sziráki

Väänánen

2017

Fund. Math.

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“…As an immediate consequence we will see that Vaught's conjecture holds for any sentence of L ω 1 ,ω (L) which doesn't contain equations. This will answer a question in [6]. Our proof will also show that Martin's conjecture holds for sentences of this form.…”

Section: Introductionmentioning

confidence: 55%

Vaught’s Conjecture Without Equality

Ackerman

2015

Notre Dame J. Formal Logic

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Suppose σ ∈ L ω1,ω (L) is such that all equations occurring in σ are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that σ satisfies Vaught's conjecture. In particular this proves Vaught's conjecture for sentences of L ω1,ω (L) without equality.2010 Mathematics Subject Classification. 03C15, 03C30, 03C75.

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“…The idea of model points is central in this paper, and is identical in spirit to ultrafilters defined by Sági and Sziráki ([15], 2012). What we call the set of model points is denoted in [15] by the symbol H(A).…”

Section: Recall That ∃ *mentioning

confidence: 99%

Stone Space of Cylindric Algebras and Topological Model Spaces

Pinter¹

2016

J. symb. log.

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The Stone representation theorem was a milestone for the understanding of Boolean algebras. From Stone’s theorem, every Boolean algebra is representable as a field of sets with a topological structure. By means of this, the structural elements of any Boolean algebra, as well as the relations between them, are represented geometrically and can be clearly visualized. It is no different for cylindric algebras: Suppose that ${\frak A}$ is a cylindric algebra and ${\cal S}$ is the Stone space of its Boolean part. (Among the elements of the Boolean part are the diagonal elements.) It is known that with nothing more than a family of equivalence relations on ${\cal S}$ to represent quantifiers, ${\cal S}$ represents the full cylindric structure just as the Stone space alone represents the Boolean structure. ${\cal S}$ with this structure is called a cylindric space.Many assertions about cylindric algebras can be stated in terms of elementary topological properties of ${\cal S}$. Moreover, points of ${\cal S}$ may be construed as models, and on that construal ${\cal S}$ is called a model space. Certain relations between points on this space turn out to be morphisms between models, and the space of models with these relations hints at the possibility of an “abstract” model theory. With these ideas, a point-set version of model theory is proposed, in the spirit of pointless topology or category theory, in which the central insight is to treat the semantic objects (models) homologously with the corresponding syntactic objects so they reside together in the same space.It is shown that there is a new, purely algebraic way of introducing constants in cylindric algebras, leading to a simplified proof of the representation theorem for locally finite cylindric algebras. Simple rich algebras emerge as homomorphic images of cylindric algebras. The topological version of this theorem is especially interesting: The Stone space of every locally finite cylindric algebra ${\frak A}$ can be partitioned into subspaces which are the Stone spaces of all the simple rich homomorphic images of ${\frak A}$. Each of these images completely determines a model of ${\frak A}$, and all denumerable models of ${\frak A}$ appear in this representation.The Stone space ${\cal S}$ of every cylindric algebra can likewise be partitioned into closed sets which are duals of all the types in ${\frak A}$. This fact yields new insights into miscellaneous results in the model theory of saturated models.

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Some variants of Vaught's conjecture from the perspective of algebraic logic

Cited by 8 publications

References 17 publications

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

A dichotomy theorem for the generalized Baire space and elementary embeddability at uncountable cardinals

Vaught’s Conjecture Without Equality

Stone Space of Cylindric Algebras and Topological Model Spaces

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