Löwenheim-Skolem/downward/Field: says that there must be no uncountable models for 1st order consistent theories.
Compactness theorem/Löwenheim-Skolem/upward: 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._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
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
"Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67
Theories of Truth, Paul Horwich, Aldershot 1994