Infinitary logic and admissible sets

Barwise, Jon

doi:10.2307/2271099

Cited by 105 publications

(260 citation statements)

References 8 publications

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“…It works also in the countable admissible fragments of L ω 1 ω , as demonstrated by Barwise (1969). The best result in this direction in the classical higher infinitary logics is the following result of Malitz (1971):…”

Section: Positive Resultsmentioning

confidence: 90%

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The Craig Interpolation Theorem in abstract model theory

Väänánen

2008

Synthese

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The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying "natural," robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

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Section: Positive Resultsmentioning

confidence: 90%

“…The admissible fragment Barwise (1969Barwise ( , 1975 Second order logic √ * * * Separation, * * Single-sorted…”

Section: Amentioning

confidence: 99%

The Craig Interpolation Theorem in abstract model theory

Väänánen

2008

Synthese

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“…It was shown by López-Escobar (1965) that L HC has the interpolation property and more generally by Barwise (1969) that the same holds for every L A with A a countable admissible subset of HC. Barwise also generalized the r.e.…”

Section: Lemmamentioning

confidence: 92%

Harmonious logic: Craig’s interpolation theorem and its descendants

Feferman

2008

Synthese

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Though deceptively simple and plausible on the face of it, Craig's interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory.

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“…Hanf numbers. In Barwise [1] it is shown that the Hanf number of LA is 3^n0n, for all countable admissible sets A. Is this theorem true for all admissible A with countable A n Onl…”

Section: Harvey Friedmanmentioning

confidence: 99%

Large Models of Countable Height

Friedman¹

1975

Transactions of the American Mathematical Society

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ABSTRACT. Every countable transitive model M of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power MnO/i" An aPPucation t0 infinitary logic is given.Any transitive model M of ZFC with countably many ordinals must be countable. The situation is quite different when the axiom of choice is dropped.The first examples of transitive models of ZF of power Wj with countably many ordinals were constructed by Cohen. Later Easton, Solovay, and Sacks showed that every countable transitive model of ZF has an ordinal-preserving extension satisfying ZF, of power 2". We prove here that every countable transitive model M of ZF has an ordinal preserving extension satisfying ZF, of power 3Mno".Theorem 1 is probably in the folklore. However, the proof of its first part is apparently not standard. The method used in that proof and the combinatorial construction of §2 form the crux of the proof of the main theorem.

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Infinitary logic and admissible sets

Cited by 105 publications

References 8 publications

The Craig Interpolation Theorem in abstract model theory

The Craig Interpolation Theorem in abstract model theory

Harmonious logic: Craig’s interpolation theorem and its descendants

Large Models of Countable Height

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