Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-23T02:24:21.378Z Has data issue: false hasContentIssue false
  • English
  • Français

Independence proofs in predicate logic with infinitely long expressions

Published online by Cambridge University Press:  12 March 2014

Carol R. Karp
Carol R. Karp*
Affiliation:
University of Maryland
Get access Rights & Permissions [Opens in a new window]

Extract

The systems of predicate logic considered here are extensions of ordinary first-order predicate logic. They were obtained by first admitting infinite conjunctions, disjunctions, and quantifications into formulas, and then adjusting the axiom schemata and rules of inference to accommodate them. In this paper, only systems with formulas of denumerable length and “formal proofs” with denumerably many steps will be discussed, though such restrictions are hardly essential. The systems with denumerable conjunctions and disjunctions, but finite quantifications, are semantically complete; i.e., valid formulas are provable. However, systems in which denumerably infinite quantifications are also allowed, are semantically incomplete.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 2 , June 1962 , pp. 171 - 188
Copyright
Copyright © Association for Symbolic Logic 1962

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Erd, P.ös and Tarski, A., On families of mutually exclusive sets, Annals of Mathematics, vol. 44 (1943), pp. 315–329.Google Scholar
[2]Ginsburg, S., On the existence of complete Boolean algebras whose principal ideals are isomorphic to each other, Proceedings of the American Mathematical Society, vol. 9 (1958), pp. 130–132.CrossRefGoogle Scholar
[3]Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159–166.Google Scholar
[4]Henkin, L., Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 81–91.Google Scholar
[5]Henkin, L., The representation theorem for cylindrical algebras, Mathematical interpretation of formal systems, Amsterdam (North-Holland) 1955, pp. 85–97.CrossRefGoogle Scholar
[6]Jordan, P., Zur Axiomatik der Verknüpfungsbereiche, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 16 (1949), pp. 54–70.CrossRefGoogle Scholar
[7]Karp, C., Formalizations of prepositional languages with wffs of infinite length, Abstract 542–56, Notices of the American Mathematical Society, vol. 5 (1958), p. 172.Google Scholar
[8]Karp, C., Formalizations of functional languages with wffs of infinite length, Abstract 542–57, Notices of the American Mathematical Society, vol. 5 (1958), pp. 172–173.Google Scholar
[9]Karp, C., Split semantic models, Abstract 550–20, Notices of the American Mathematical Society, vol. 5 (1958), p. 679.Google Scholar
[10]Keisler, H., The completeness of a certain class of first-order logics with infinitary relations, Abstract 569–46, Notices of the American Mathematical Society, vol. 7 (1960), p. 363.Google Scholar
[11]Kleene, S. C., Introduction to metamathematics, New York (Van Nostrana) 1952, 550 pp.Google Scholar
[12]Krasner, M., Une généralisation de la notion de corps, Journal de mathématiques pures et appliquées, ser. 9, 17 (1938), p. 165–170.Google Scholar
[13]Mostowski, A., Proofs of non-deducibility in intuitionistic functional calculus, this Journal, vol. 13 (1948), pp. 204–207.Google Scholar
[14]Rasiowa, H. and Sikorski, R., A proof of the completeness theorem of Gödel, Fundamenta mathematicae, vol. 37 (1950), pp. 193–200.CrossRefGoogle Scholar
[15]Rasiowa, H. and Sikorski, R., An algebraic treatment of the notion of satisfiability, Fundamenta mathematicae, vol. 40 (1943), pp. 62–95.CrossRefGoogle Scholar
[16]Rieger, L., On free ℵξ-complete Boolean algebras, Fundamenta mathematicae, vol. 38 (1951), pp. 35–52.CrossRefGoogle Scholar
[17]Scott, D. and Tarski, A., The sentential calculus with infinitely long expressions, Colloquium mathematicum, vol. 6 (1958), pp. 166–170.CrossRefGoogle Scholar
[18]Smith, E. C. and Tarski, A., Higher degrees of distributivity and completeness in Boolean algebras, Transactions of the American Mathematical Society, vol. 84 (1957), pp. 230–257.CrossRefGoogle Scholar
[19]Tarski, A., Remarks on predicate logic with infinitely long expressions, Colloquium mathematicum, vol. 6 (1958), pp. 171–176.CrossRefGoogle Scholar