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Löwenheim-Skolem / down / Field: says that there must be no uncountable models for 1st order consistent theories. - Compactness theorem / Löwenheim-Skolem / up: says that each 1st order space-time theory, according to which there are infinitely many space-time points, will have models, in which the set of the space-time points is mightier than the set of real numbers - problem: then the Representation theorem does not apply.
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980