A Scrabble on Cantor, Hilbert, and Zermelo (3)

This is not much fictitious story. Hilbert's idea of metamathematics is surely influenced by Cantor's philosophy of mathematical objects. But an important difference is this: Hilbert changed the question of existence of mathematical objects into the mathematical problems. Namely, if one can prove the consistency of the theory, then a model that the theory describes exists (through completeness). And moreover: a proof of consistency shall be treated as finite combinatorics. Finally, the notion of `consistency' becomes well-defined for the first time -- there is no proof-figure (or, no procedure to construct a proof-figure) of one ending with `bottom' by using only pre-determined set of deduction rules.

Take a stock. Cantor said that the consistency is key to justify the existence of mathematical objects, but at the same time he tried to justify it by `philosophical' argument.*1 On the contrary, Hilbert turned this problem into mathematical (or metamathematical) one, and made clear what it is to show the consistency.

*1:I mean `philosophical' here a kind of arguments that comes from outside mathematics and gives overall/general justification, containing more than mathematics. But I think: philosophy also lives just inside mathematical arguments.