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Infinitary logic and admissible sets1

  • Jon Barwise (a1)

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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[1]Barwise J., Infinitary logic and admissible sets, Doctoral Dissertation, Stanford Univ., Stanford, Calif., 1967.
[2]Barwise J., 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]Feferman S. and Kreisel G., Persistent and invariant formulas relative to theories of higher type, Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 480485.
[4]Gödel K., 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.
[5]Hanf W., Incompactness in languages with infinitely long expressions, Fundamenta mathematicae, vol. 53 (1964), pp. 309324.
[6]Jensen R. and Karp C., Primitive recursive set functions (to appear).
[7]Karp C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.
[8]Karp C., Non-axiomatizability results for infinitary systems, this Journal, vol. 32 (1967), pp. 367384.
[9]Kreisel G., Set theoretic problems suggested by the notion of potential totality, Infinitistic methods, Warsaw, 1961, pp. 103140.
[10]Kreisel G., Model theoretic invariants: Applications to recursive and hyperarithmetic operations, The theory of models, edited by Addison L., Henkin L., and Tarski A., North-Holland, Amsterdam, 1965, pp. 190205.
[11]Kreisel G. and Sacks G. E., Metarecursive sets, this Journal, vol. 30 (1965), pp. 318337.
[12]Kripke S., Transfinite recursion on admissible ordinals. I, II (abstracts), this Journal vol. 29 (1964), pp. 161162.
[13]Kripke S., Admissible ordinals and the analytic hierarchy (abstract), this Journal, vol. 29 (1964), p. 162.
[14]Kunen K., Implicit definability and infinitary languages, this Journal, vol. 33 (1968), pp. 446451.
[15] E. Lopez-Escobar G. K., An interpolation theorem for denumerably long formulas, Fundamenta mathematicae, vol. 58 (1965), pp. 254272.
[16] E. Lopez-Escobar G. K., Remarks on an infinitary language with constructive formulas, this Journal, vol. 32 (1967), pp. 305319.
[17]Malitz J., Problems in the model theory of infinite languages, Doctoral Dissertation, Univ. of Calif., Berkeley, Calif., 1965.
[18]Platek R., Foundations of recursion theory, Doctoral Dissertation and Supplement, Stanford Univ., Stanford, Calif., 1966.
[19]Scott D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models, edited by Addison J., Henkin L., and Tarski A., North-Holland, Amsterdam, 1965, pp. 329341.
[20]Solovay R., A Δ31 non-constructible set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 5075.
[21]Takeuti G. and Kino A., On predicates with infinitely long expressions, Journal of the Mathematical Society of Japan, vol. 15 (1963), pp. 176190.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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