Journal of Symbolic Logic

Intermediate Predicate Logics Determined by Ordinals

Pierluigi Minari, Mitio Takano, and Hiroakira Ono

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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 $\alpha,\beta < \omega^\omega$.

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

First available in Project Euclid: 6 July 2007

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Minari, Pierluigi; Takano, Mitio; Ono, Hiroakira. Intermediate Predicate Logics Determined by Ordinals. J. Symbolic Logic 55 (1990), no. 3, 1099--1124.

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