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Infinitary logic and admissible sets1

  • Jon Barwise (a1)
  • DOI:
  • Published online: 01 March 2014

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is

(I) strong enough to express interesting properties not expressible by the classical language, but

(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

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This paper contains the principal results of the first half of the author's Ph.D. thesis [1], submitted to Stanford University in August, 1967. We wish to thank our thesis advisor, Professor Solomon Feferman, for the considerable time, advice, direction and encouragement which we received. We also thank Professors Georg Kreisel and Dana Scott, as well as Kenneth Kunen, for many interesting discussions and helpful suggestions.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2]J. Barwise , Implicit definability and compactness in infinitary languages, The syntax and semantics of infinitary languages, Lecture Kotes in Mathematics, vol. 72, Springer-Verlag, 1968 pp. 135.

[3]S. Feferman and G. Kreisel , Persistent and invariant formulas relative to theories of higher type, Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 480485.

[4]K. Gödel , The consistency of the axiom of choice and of the generalized continuum hypothesis, Proceedings of the National Academy of Sciences, vol. 24 (1938), pp. 556557.

[20]R. Solovay , A Δ31 non-constructible set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 5075.

[21]G. Takeuti and A. Kino , On predicates with infinitely long expressions, Journal of the Mathematical Society of Japan, vol. 15 (1963), pp. 176190.

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