Vol. 40, No. 1, 1972

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Maximal models in the language with quantifier “there exist uncountably many”

Jerome Irving Malitz and William Nelson Reinhardt

Vol. 40 (1972), No. 1, 139–155

Fuhrken has shown that the language Lω1Q, obtained from first order logic by adding a new quantifier Q and interpreting QX as “there are at least ω1x’s such that …” is countably compact but not fully compact. The countable compactness is not enough to yield strong analogs of the upward Löwenheim-Skolem theorem, and the amalgamation property. In fact, it is shown that for “most” cardinals κ, there are structures of power κ with countable type that are maximal in the sense of having no proper extensions with the same Lω1Q theory. From this the failure of the amalgamation property is obtained. There is still the possibility that the model theory of LκQ (with QX interpreted as “there are at least κx’s such that …”) for κ > ω1, is more analogous to the model theory of first order logic.

Mathematical Subject Classification
Primary: 02B20
Secondary: 02H10
Received: 12 October 1970
Published: 1 January 1972
Jerome Irving Malitz
William Nelson Reinhardt