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Semantic ascent / Field: Not a set of axioms (not logically true), but weaker: with modal operator (diamond) "It is possible that" in front - then logically true that it is logically possible.
Semantic ascent: "not every axiom of this theory is true" solution for infinitely axiomatized theories.
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