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> It is more than formal consistency because to show a whole number solution to a^n + b^n = c^n and n > 2, one doesn't need to use a high powered formal framework like ZFC. Thus Wiles' proof of Fermat's last theorem, which uses all kinds of infinite sets and other features of ZFC, can be falsified by a simple manipulation of finite numbers.

This follows from consistency (and is one reason why consistency is important). If there were some whole number solution to a^n+b^n=c^n for n>2, then this would be provable in Peano Arithmetic (a much simpler system than ZFC, that axiomatizes natural number arithmetic). ZFC extends PA (every theorem of PA is a theorem of ZFC, interpreted correctly; this is relatively easy to show); thus, anything ZFC proves is not falsifiable in PA, otherwise we'd have a contradiction provable in ZFC (& it wouldn't be consistent).

It doesn't follow from consistency in itself. It is possible to imagine a formal system that is consistent but nevertheless doesn't predict the behaviour of calculators. If you presume that ZFC correctly models the behaviour of calculators, or that PA models calculators and inconsistency of PA implies inconsistency of ZFC, then sure it follows from consistency of ZFC, but that was the point, namely that there is a difference between a purely formal system that only exposes itself to inconsistency tests, and a formal system that exposes itself to external tests. Numbers aren't the only way that math does that. Geometry makes predictions about the behaviour of rulers and compasses, for instance.

Contrast this with a discipline in which there is jargon and arguments about that jargon. It might be the case that the jargon and the basic ways of arguing about the jargon forms a consistent system in some sense, even though the whole enterprise is meaningless.

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