Journal of Symbolic Logic

Intermediate Predicate Logics Determined by Ordinals

Abstract

For each ordinal $\alpha > 0, L(\alpha)$ is the intermediate predicate logic characterized by the class of all Kripke frames with the poset $\alpha$ and with constant domain. This paper will be devoted to a study of logics of the form $L(\alpha)$. It will be shown that for each uncountable ordinal of the form $\alpha + \eta$ with a finite or a countable $\eta (> 0)$, there exists a countable ordinal of the form $\beta + \eta$ such that $L(\alpha + \eta) = L(\beta + \eta)$. On the other hand, such a reduction of ordinals to countable ones is impossible for a logic $L(\alpha)$ if $\alpha$ is an uncountable regular ordinal. Moreover, it will be proved that the mapping $L$ is injective if it is restricted to ordinals less than $\omega^\omega$, i.e. $\alpha \neq \beta$ implies $L(\alpha) \neq L(\beta)$ for each ordinal $\alpha,\beta < \omega^\omega$.

Article information

Source
J. Symbolic Logic, Volume 55, Issue 3 (1990), 1099-1124.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183743408

Mathematical Reviews number (MathSciNet)
MR1071317

Zentralblatt MATH identifier
0745.03022

JSTOR