Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-24T22:57:33.384Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonstandard logic

Published online by Cambridge University Press:  12 March 2014

James R. Geiser
James R. Geiser*
Affiliation:
Dartmouth College
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is an exposition of the author's [2] investigations of a class of infinitary first order logics. These logics are sometimes characterized by saying that the syntactical role played by the positive integers in first order predicate logic is taken over by a nonstandard model of arithmetic. The result is then called a nonstandard logic.1 In particular the operations of conjunction, disjunction and quantification may be iterated infinitely (nonstandard finitely) often.

When the nonstandard model of arithmetic is an ultra power, the natural class of structures in which to interpret the nonstandard wffs is a class of ultra products. Within this framework, many model-theoretic properties of ultra products become manifest in the logical structure of the nonstandard language.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 2 , 23 July 1968 , pp. 236 - 250
Copyright
Copyright © Association for Symbolic Logic 1968

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]Frayne, T., Morel, A. C., and Scott, S. C., Reduced direct products, Fundamenta Mathematicae, vol. 51 (1962), pp. 195–228.CrossRefGoogle Scholar
[2]Geiser, J. R., Nonstandard logic, doctoral thesis, M.I.T., Cambridge, Mass., 1966.Google Scholar
[3]Gillman, L. and Jerison, M., Rings of continuous functions, Van Nostrand, Princeton, N.J., ch. 13, 1960.CrossRefGoogle Scholar
[4]Henkin, L., Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 81–91.Google Scholar
[5]Henkin, L., Some remarks on infinitely long formulas, Infinitistic methods, Proceedings of the Symposium on the Foundations of Mathematics, Warsaw, pp. 167–183, 1959.Google Scholar
[6]Karp, C. R., Languages with expressions of infinite length, Studies in logic and the foundations of mathematics, North-Holland, Amsterdam, 1963.Google Scholar
[7]Keisler, H. J., Repleteness of ultraproducts, Notices of the American Mathematical Society, vol. 8 (1961), p. 63.Google Scholar
[7a]Keisler, J., Ultraproducts and saturated models, Proceedings of the Royal Netherlands Academy of Sciences, Ser. A, vol. 67 (1964), pp. 178–186.Google Scholar
[8]Keisler, J., Limit ultraproducts, this Journal, vol. 30 (1965), pp. 212–234.Google Scholar
[9]Kochen, S., Ultraproducts in the theory of models, Annals of Mathematics, vol. 74 (1961), pp. 221–261.CrossRefGoogle Scholar
[10]Robinson, A., On languages which are based on nonstandard arithmetic, Nagoya Mathematical Journal, vol. 22 (1963), pp. 83–117.CrossRefGoogle Scholar
[11]Robinson, A., Non-standard analysis, North-Holland, Amsterdam, 1963.Google Scholar
[12]Stone, A. L., Extensive ultra products and Haar measure, The Theory of Models, Proceedings of the 1963 International Symposium at Berkeley, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, pp. 419–423, 1965.Google Scholar
[13]Tarski, A., Remarks on predicate logics with infinitely long expressions, Colloquium Mathematicum, vol. 6 (1958), pp. 171–176.CrossRefGoogle Scholar