| > I'll try to physicalize countability. > Start counting the naturals: 1, 2, 3, ... > At future timelike infinity you'll reach infinity. > Now for the reals. Your goal is to step from 0 to 1, by way of 0.1, 0.01, and so forth. > 0 0.0........ > At future timelike infinity you still haven't stopped adding in zeroes to the right of the decimal point. > In the first case, at any finite time before \breve{i}^+ you will have counted out some finite natural number. In the second case, you will not yet have counted out your first nonzero real. The same reasoning applies for rationals, yet they can still be counted. The definition of «can be counted» means there is a bijection between your set and the naturals. Such kind of bijection can easily be created for the rationals[1] and Cantor's diagonal argument[2] shows that you can't create such bijection for reals. There is nothing really intuitive about this concept, but fortunately the proofs are pretty straigtforward which give a kind of «intuition» around this. [1] https://en.wikipedia.org/wiki/Pairing_function#/media/File:D... [2] https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument |
It is only straightforward once you accepted infinity and all other definitions/description based on infinity.
If you, like me, cannot accept infinity, then all the proofs/descriptions that contain infinity become apparent non-sensical.
>> Start counting the naturals: 1, 2, 3, ... >> At future timelike infinity you'll reach infinity.
This highlights the flaw. You reach infinity with infinity. Nothing is really being said about infinity. But somehow if you accepted the understanding infinity here, the rest of thesis such as cantor's diagonal argument may seem to be natural, except you forget that you really didn't know what infinity is.
All property of infinity cannot be finitely described. So anything about infinity is built on top of infinity. Turtle all the way down (or up), and we don't really know what it is.