4 ms·
Well my math is quite rusty, but if I remember correctly, you can "define" reals using Dedekind cuts[1] made of rational numbers - in this way you can trivially
by yongjik 6y ago
Well my math is quite rusty, but if I remember correctly, you can "define" reals using Dedekind cuts[1] made of rational numbers - in this way you can trivially prove that a bounded increasing sequence converges to a real number (i.e., the reals are complete), without assuming anything about uncountability of reals.
[1] https://en.wikipedia.org/wiki/Dedekind_cut https://en.wikipedia.org/wiki/Dedekind_cut
- eru 6y agoYou can also not define anything up front; then take the proof that Bob can enforce that there's no convergence to any rational number. Next step, look at the process of playing the game, and show that you can define operations like adding any two games or comparing them for equality etc. (In some sense, that would be similar to showing that Dedekind cuts define a sensible structure. With the added complication that the game doesn't have to converge to arbitrary small intervals, because the players could agree to leave a finite interval like [1/4, 3/4] untouched, eg Alice plays 1/4-1/n and Bob plays 3/4+1/n. You'd just be defining interval arithmetic on real numbers.)