Field on Consistency

2 02 2005

In “Realism, Mathematics, and Modality”, pp. 31-32, Hartry Field makes what seems to me to be a muddled argument about the consistency of set theory on platonistic grounds. He starts by suggesting that it is unproblematic that if a theory T is true, then it must be consistent. However, he then suggests that the Gödel and Henkin proofs of the completeness theorem, guaranteeing the existence of a model of T, are a sort of “accident of first order logic”, as if this were an argument for the platonist against identifying logical consistency with the existence of a model.

If S is a sentence that is false in set theory, then set theory shouldn’t imply S. But if no set – no proper part of the set-theoretical reality – could serve as a model for the entire set-theoretical reality, as seems prima facie possible, then set theory would Tarski-imply S. Again, if we are platonists we are saved from this possible extensional divergence between implication and Tarski-implication, by virtue of [the completeness theorem].

I suppose that here by “set theory” he means to refer to the collection of all true first-order statements in the language of set theory (working from a platonistic perspective). What makes this make sense is the fact that “set theory” is not recursively axiomatizable (the way ZFC or any of its standard extentions are), so Gödel’s Incompleteness Theorem doesn’t apply here. Thus, “set theory” can be a theory that implies its own consistency, and thus has models of itself inside any model. I don’t see why the Completeness Theorem is seen as somehow “accidental” and thus not reason enough for the platonist to accept the Tarski definition of implication.

Rather, what seems to me a better argument against the platonist set theorist using the Tarski definition of logical consequence (which relies on the existence of models, which are set-theoretic constructs) is that logical consequence should be prior even to set theory even for the platonist. This is certainly the case for the logicist, who says that the way we know things about the set-theoretic universe is because they are logical truths. Thus, to know that truths about sets are logical truths, we must have some way of recognizing logical truth independent of the existence of set-theoretic entities whose existence has not been justified.

I think even other non-logicist platonists would have to accept the primacy of logical consequence over set theory, because the ideas of model theory are taken to apply to models like “the actual universe of sets” rather than just set-theoretic models. This is an interesting recursivity in foundational mathematics that I have noted, that model theory talks about set-theoretic entities, while set theory assumes that we are working within a model. I think if both are taken literally, then it really is a vicious circle, but we can get out of it by making a move like the one Field argues for in a confused way, by saying that logical consequence is basic, and Tarski merely gave us an extensionally adequate analysis of it.

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