Three red herrings around Vaught’s conjecture

Baldwin, John T.; Friedman, Sy D.; Koerwien, Martin; Laskowski, M.

doi:10.1090/tran/6572

Cited by 22 publications

(41 citation statements)

References 22 publications

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“…The rich history of the investigation related to that (still unresolved) conjecture includes a long list of results confirming the conjecture in particular classes of theories (see, for example, the introduction and references of [8]) and, on the other hand, intriguing results concerning the consequences of the existence of counterexamples and the properties of (potential) counterexamples (see, e.g., [1]).…”

Section: Introductionmentioning

confidence: 99%

Vaught's conjecture for monomorphic theories

Kurilić

2019

Annals of Pure and Applied Logic

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A structure Y of a relational language L is called almost chainable iff there are a finite set F ⊂ Y and a linear order < on the set Y \F such that for each partial automorphism ϕ (i.e., local automorphism, in Fraïssé's terminology) of the linear order Y \ F, < the mapping id F ∪ϕ is a partial automorphism of Y. By a theorem of Fraïssé, if |L| < ω, then Y is almost chainable iff the profile of Y is bounded; namely, iff there is a positive integer m such that Y has ≤ m non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory T having infinite models is called almost chainable iff all models of T are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of T . In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

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Section: Introductionmentioning

confidence: 99%

Vaught's conjecture for monomorphic theories

Kurilić

2019

Annals of Pure and Applied Logic

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“…This procedure was nondeterministic in the sense that he showed one of (countably many if α is infinite) sentences worked at each ℵ α ; it is conjectured [Sou13] that it may be impossible to decide in ZFC which sentence works. In [BKL15], we show a modification of the Laskowski-Shelah example (see [LS93,BFKL16]) gives a family of L ω1,ω -sentences φ r , such that φ r homogeneously characterizes ℵ r for r < ω. Thus for the first time [BKL15] establishes in ZFC, the existence of specific sentences φ r characterizing ℵ r .…”

Section: The Big Gapmentioning

confidence: 99%

Hanf numbers and presentation theorems in AECs

Boney¹,

Baldwin²

2017

Beyond First Order Model Theory

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This paper addresses a number of fundamental problems in logic and the philosophy of mathematics by considering some more technical problems in model theory and set theory. The interplay between syntax and semantics is usually considered the hallmark of model theory. At first sight, Shelah's notion of abstract elementary class shatters that icon. As in the beginnings of the modern theory of structures ([Cor92]) Shelah studies certain classes of models and relations among them, providing an axiomatization in the Bourbaki ([Bou50]) as opposed to the Gödel or Tarski sense: mathematical requirements, not sentences in a formal language. This formalism-free approach ([Ken13]) was designed to circumvent confusion arising from the syntactical schemes of infinitary logic; if a logic is closed under infinite conjunctions, what is the sense of studying types? However, Shelah's presentation theorem and more strongly Boney's use [Bon] of aec's as theories of L κ,ω (for κ strongly compact) reintroduce syntactical arguments. The issues addressed in this paper trace to the failure of infinitary logics to satisfy the upward Löwenheim-Skolem theorem or more specifically the compactness theorem. The compactness theorem allows such basic algebraic notions as amalgamation and joint embedding to be easily encoded in first order logic. Thus, all complete first order theories have amalgamation and joint embedding in all cardinalities. In contrast *

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“…There is a (class) function Sc whose domain is the class of all countable L-structures, and whose range is the set of all (codes for) L ω 1 ,ω sentences such that for all countable…”

Section: Proposition 25 Suppose L Is a Countable Languagementioning

confidence: 99%

“…Their proofs also use Scott sentences, but our approach is different (they use Fraïssé limits and we use atomic models), and the focuses of the two papers are completely different. Two other recent papers, one by Baldwin, Friedman, Koerwien and Laskowski [1], and another by Larson [10] have some overlapping with [6] (they all give a new proof of a result of Harrington regarding Vaught's conjecture using different methods).…”

Section: Problem 516mentioning

confidence: 99%

Forcing a countable structure to belong to the ground model

Kaplan

Shelah

2016

Math. Log. Quart.

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Suppose that P is a forcing notion, L is a language (in double-struckV), trueτ̇ a P‐name such that P⊩ “trueτ̇ is a countable L‐structure”. In the product P×P, there are names τ1̇,τ2̇ such that for any generic filter G=G1×G2 over P×P, trueτ̇1false[Gfalse]=τ̇false[G1false] and trueτ̇2false[Gfalse]=τ̇false[G2false]. Zapletal asked whether or not P×P⊩trueτ̇1≅trueτ̇2 implies that there is some M∈V such that P⊩τ̇≅M̌. We answer this question negatively and discuss related issues.

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Three red herrings around Vaught’s conjecture

Cited by 22 publications

References 22 publications

Vaught's conjecture for monomorphic theories

Vaught's conjecture for monomorphic theories

Hanf numbers and presentation theorems in AECs

Forcing a countable structure to belong to the ground model

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