Stationary logic

Barwise, Jon; Kaufmann, Matt; Makkai, Michael

doi:10.1016/0003-4843(78)90003-7

Cited by 66 publications

(45 citation statements)

References 12 publications

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“…The logic frame L(∃ ≥κ ) is (λ, ω)-compact for each λ < κ (see [11,22]). Example 2.5 Suppose κ is strong limit ω-Mahlo 3) cardinal. Let…”

Section: Example 23 Letmentioning

confidence: 99%

“…where M Q cub ℵ1 xy ϕ(x, y, z ) if and only if { x, y : M ϕ(x, y, z )} is an ℵ 1 -like linear order in which a cub of initial segments have a sup, and A(Q cub ℵ1 ) has as axioms the basic axioms of first order logic, and axioms from [3]. The logic frame L(Q cub ℵ1 ) is ℵ 0 -compact (see [24]), hence has finite character for a trivial reason.…”

Section: Example 23 Letmentioning

confidence: 99%

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Recursive logic frames

Shelah

Väänánen

2006

Math. Log. Quart.

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We define the concept of a logic frame, which extends the concept of an abstract logic by adding the concept of a syntax and an axiom system. In a recursive logic frame the syntax and the set of axioms are recursively coded. A recursive logic frame is called complete (recursively compact, ℵ0-compact), if every finite (respectively: recursive, countable) consistent theory has a model. We show that for logic frames built from the cardinality quantifiers "there exists at least λ" completeness always implies ℵ0-compactness. On the other hand we show that a recursively compact logic frame need not be ℵ0-compact.

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“…The logic frame L(∃ ≥κ ) is (λ, ω)-compact for each λ < κ (see [11,22]). Example 2.5 Suppose κ is strong limit ω-Mahlo 3) cardinal. Let…”

Section: Example 23 Letmentioning

confidence: 99%

Section: Example 23 Letmentioning

confidence: 99%

Recursive logic frames

Shelah

Väänánen

2006

Math. Log. Quart.

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“…. "; this logic was later investigated in [2]. However when talking about topological spaces we do not have at hand any structure in the sense of universal algebra.…”

Section: Proof It Is Clear That (B) Implies (A) and (D) Implies (C)mentioning

confidence: 99%

On families of Lindelöf and related subspaces of 2^ω₁

Junqueira

Koszmider

2001

Fund. Math.

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Abstract.We consider the families of all subspaces of size ω 1 of 2 ω 1 (or of a compact zero-dimensional space X of weight ω 1 in general) which are normal, have the Lindelöf property or are closed under limits of convergent ω 1 -sequences. Various relations among these families modulo the club filter in [X] ω 1 are shown to be consistently possible. One of the main tools is dealing with a subspace of the form X ∩M for an elementary submodel M of size ω 1 . Various results with this flavor are obtained. Another tool used is forcing and in this case various preservation or nonpreservation results of topological and combinatorial properties are proved. In particular we prove that there may be no c.c.c. forcing which destroys the Lindelöf property of compact spaces, answering a question of Juhász. Many related questions are formulated. Introduction.The results of this paper are related to several closely located topics. First, one can draw a connection with the general topic of reflection. A classical theorem in logic, called the lower Löwenheim-Skolem theorem, states that for any infinite mathematical structure and any infinite cardinal κ, smaller than the cardinality of the structure, there is a substructure of cardinality κ which satisfies the same first order formulas as the entire structure. For instance, every uncountable field has a subfield of cardinality ω 1 with the same first order properties as the entire field. Most topological and many other properties cannot be expressed by first order formulas, which makes the problem of reflection of these properties to smaller substructures more subtle. What we consider a small or a large substructure is a fundamental factor determining the flavor of the methods of research and the results. For example, here we consider small or large in the sense of infinite cardinality, and thus set-theoretic methods are central. Many deep results with this flavor have been obtained. Probably the most famous one is Shelah's compactness theorem for certain algebraic structures of singular

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“…What is the relationship between £neg and 7? Confer [2]. (c) Let Mod(wj) be the class of w,-standard models as in Keisler [5], i.e., (31, q) where q is the set of uncountable subsets of A.…”

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confidence: 99%

Ultraproduct invariant logics

Sgro¹

1980

Proc. Amer. Math. Soc.

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Abstract. In this paper we give a construction of logics via the property of being preserved from the models to their ultraproduct. Specific examples are given which include some cardinality quantifiers.We will assume that the reader is familiar with basic model theory including ultraproducts, Chang and Keisler Take Mod to be a class of generalized models of the same type closed under restriction and expansion of languages.Let SeqK(Mod) be the class of all sequences of length « of elements of Mod, Ult(«) be the set of ultrafilters on «, and Seq(Mod) = UK SeqK(Mod) and Ult= UKUlt(«).Definition.An ultraproduct on Mod is a partial function ULT from U, SeqK(Mod) X Ult(«) into Mod.We say that the pair is closed under «-ultraproducts for all cardinals « then it is said to be closed under ultraproducts. Let be the second-order logic with additional quantification over the generalized domains. That is: Let 7(Mod, ULT) be the sublogic of £2 consisting of the invariant sentences. Let Ult*(«) be the class of regular ultrafilters over «.

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Stationary logic

Cited by 66 publications

References 12 publications

Recursive logic frames

Recursive logic frames

On families of Lindelöf and related subspaces of 2^ω₁

Ultraproduct invariant logics

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Stationary logic

Cited by 66 publications

References 12 publications

Recursive logic frames

Recursive logic frames

On families of Lindelöf and related subspaces of 2ω1

Ultraproduct invariant logics

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On families of Lindelöf and related subspaces of 2^ω₁