A Scrabble on Cantor, Hilbert, and Zermelo (1)
なんとなく、突然、英語。ってかルーズに書くときは英語のほうが書きやすいような気がする。これは書こうとしている論文の内容ではないです。
What kind of view did Cantor take concerning the existence of mathematical object? Here's a sketch:
A mathematical object exists if the forming process of it is consistent.
First, for Cantor mathematics is the displine of `law of thought'.*1 And objects of thought*2 form a region/world -- inner world. Thus, if the forming process of an object is consistent, it has existence in the inner world. But this means only that this obejct is thought to exist.
Cantor's claim ought to be taken to say that if a mathematical object is thought to exist (consistently), it exists. Why? Roughly, he says:
Concerning mathematics, inner world and outer world has harmony. They are in the unity of all (Einheit des Alles).
Because of this harmony, the mathematical object which is consistently thought to exist in inner world really exists in the outer world, in other words, it simply exists. But one may ask again, what gurantees this harmony? Searching the reason, one finally arrives at the core of Cantor's belief:
Because God makes it held.
Alas, we cannot go further.
