A Scrabble on Cantor, Hilbert, and Zermelo (6)

やっぱ飽きてきた。支離滅裂になってきた。そろそろやめよう。

Zermelo's strong belief in inexhaustiblity of mathematics with finite means (which in some repect Gödels shared) came from reactions to Weyl and Skolem's critics to his axiomatization of set theory. Namely, so-called Skolem's paradox. Zermelo saw (finally) that its problem lies in the 'finiteness' (in what sense?) of language. To Zermelo, Skolem's paradox shows only that we cannot express the full content of universe of sets with our language, and not that there exists a certain relativity of set theoretic concept (as Skolem argued).

But it is not the end. We saw that the full content of mathematics, especially infinities cannot be captured with finite means. This attempt to capture mathematics with finite means was also one way to capture exact contents of mathematics. If we can express these contents with finite means, in some sense we can grasp what we are doing in mathematics (Gödel's representation theorem is relevant here). Then, in one sense, failure of Hilbert's attempt means that we cannot capture the content of mathematics fully with mathematical means. Actually, Gödel tried to connect the `mathematical intuition' in this respect. We cannot grasp with (meta-)mathematical means, but can do with philosophical means. Therefore, there remains a room for philosophy to come in here...