Vol. 43, No. 3, 1972

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A complete countable Lω1Q theory with maximal models of many cardinalities

Jerome Irving Malitz and William Nelson Reinhardt

Vol. 43 (1972), No. 3, 691–700

Because of the compactness of first order logic, every structure has a proper elementarily equivalent extension. However, in the countably compact language La!IQ obtained from first order logic by adding a new quantifier Q and interpreting Qx as “there are at least ω1x’s such that…,” the situation is radically different. Indeed there are structures of countable type which are maximal in the sense of having no proper Lω1Q-extensions, and the class S of cardinals admitting such maximal structures is known to be large. Here it is shown that there is a countable complete Lω1Q theory T having maximal models of cardinality κ for each κ 21 which is in S. The problem of giving a complete characterization of the maximal model spectra of Lω1Q theories T remains open: what classes of cardinals have the form Sp (T) = {κ: there is a maximal model of T of cardinality κ} for T a (complete, countable) Lω1Q theory.

Mathematical Subject Classification
Primary: 02H10
Secondary: 02B20
Received: 4 May 1971
Published: 1 December 1972
Jerome Irving Malitz
William Nelson Reinhardt