[sharpener]

> Theorem: There is no bijection between ℕ and {0, 1}*.

So no bijection between \bb{N} in an unspecified number base and some binary strings that could represent integers? That's a nonstarter for me.

Please read my article. Thanks.

[myWindoonn]

You misread their type. There is no bijection N -> (N -> 2); {0,1}* = 2* = N -> 2.

Note that the given proof is a version of Lawvere's fixed-point theorem. The trick is noticing that for some x : N, f : N -> (N -> 2) can be applied onto it twice, giving f(x) : N -> 2 and f(x)(x) : 2.

Note further that infinite binary strings don't just represent natural numbers. Indeed, every natural number has a finite binary string, and that is the bijection that you're imagining. The question is, which natural number is represented by 111...? This leads to the difference between the natural numbers, their one-point compactification, and Cantor space in terms of searchability. Escardó has an article on this: http://math.andrej.com/2007/09/28/seemingly-impossible-funct...

Please read Lawvere's article. Thanks.