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Intermediate predicate logics determined by ordinals

Published online by Cambridge University Press:  12 March 2014

Pierluigi Minari ,
Mitio Takano  and
Hiroakira Ono
Pierluigi Minari
Affiliation:
Department of Philosophy, University of Florence, 50139 Firenze, Italy
Mitio Takano
Affiliation:
Department of Mathematics, Niigata University, Niigata 950-21, Japan
Hiroakira Ono
Affiliation:
Department of Mathematics, Hiroshima University, Hiroshima 730, Japan
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Abstract

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

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 3 , September 1990 , pp. 1099 - 1124
Copyright
Copyright © Association for Symbolic Logic 1990

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References

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