Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T08:16:15.637Z Has data issue: false hasContentIssue false
  • English
  • Français

A heterogeneous interpolant

Published online by Cambridge University Press:  22 January 2016

Walter Taylor
Walter Taylor*
Affiliation:
University of Colorado
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we exhibit an interpolant for a certain valid implication ╞ φ → ψ, where φ and ψ come from the infinitary language Lω1ω1. The existence of this interpolant follows from Takeuti’s heterogeneous interpolation theorem [5], but unfortunately the proof in [5] is not explicit enough to allow one to find the interpolant explicitly. Takeuti’s theorem asserts the existence of an interpolant in the class Lω1ω1 of heterogeneous formulas, which admits the rules of formation of Lω1ω1 plus the following additional rule:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 52 , December 1973 , pp. 31 - 33
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Barwise, J., ed., The syntax and semantics of infinitary languages, Lecture notes in mathematics #72, Springer-Verlag, Berlin, 1968.Google Scholar
[2] Henkin, L., Some remarks on infinitely long formulas, pp. 167–183 in: Infinitistic methods (Proceedings of the symposium on foundations of mathematics, Warsaw, September, 1959), Pergamon, Paōstwowe Wydawnictwo Naukowe, Warsaw, 1961.Google Scholar
[3] Malitz, J., Infinitary analogs of theorems from first order model theory, J. Symbolic Logic 36 (1971), 216–228.Google Scholar
[4] Mycielski, J., On the axiom of determinateness, Fund. Math., 53 (1964), 205–224.CrossRefGoogle Scholar
[5] Takeuti, G., A determinate logic, Nagoya Math. J. 38 (1970), 113–138. (Essentially the same article appears as pp. 237–264 in [1].) Added July 18, 1975 The following articles give further information on Takeuti’s (and other) interpolation theorems:Google Scholar
[6] Kueker, D. W., Löwenheim-Skolem and interpolation theorems in infinitary languages, Bull. Amer. Math. Soc. 78 (1972), 211–215.Google Scholar
[7] Nebres, B. F., Herbrand uniformity theorems for infinitary languages, J. Math. Soc. Japan 24 (1972), 1–19.Google Scholar
[8] Swett, A. K., Interpolation theorems for languages with game quantifiers, ms., Toronto, 1973.Google Scholar