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Definition ω model/Putnam: for a set theory is a model in which the natural numbers are ordered, "as it should be", that means, the sequence of "the natural numbers" of the model a ω-sequence.
Countable/over-countable/uncountable/infinity/Loeweheim/Putnam: E.g. an instrument that will detect the presence of a particle within a volume, will at most give countable measurements - but is the instrument shifted by r centimeters and r can be any real number, then there are over-countable many measurements - N.B.: then operational conditions cannot be identified with the totality of facts that can be observed, but only with the actual observed. - If the shifted intervals are then rational, there are only a countable number of facts. - Loewenheim: Then, a model can be constructed that matches with all facts. - Counterfactual conditional: with a predicate "makes subjunctive necessary" for not occurred cases a model can be constructed that induces an interpretation of counterfactual speech that makes precisely those counterfactual conditionals true that are according to some completion to our theory true - that means, the appeal to counterfactual observations cannot exclude models. -> Wittgenstein: the question of what God could calculate, is a matter within the math and cannot determine the interpretation of mathematics. (PU §§ 193,352,426).
There are possible set theory with and without the axiom of choice. - Skolem: we should assign a truth value only as a part of a previously accepted theory.
Von einem Realistischen Standpunkt Frankfurt 1993
Repräsentation und Realität Frankfurt 1999
Für eine Erneuerung der Philosophie Stuttgart 1997
Pragmatismus Eine offene Frage Frankfurt 1995
Vernunft, Wahrheit und Geschichte Frankfurt 1990