User blog comment:LittlePeng9/Long hierarchies of functions (on my other blog)/@comment-11227630-20171031155258/@comment-1605058-20171113193613

First, a clarification: \(\omega_1\) does not correspond in any direct way to \([0,1[\), but rather it is the set of all countable ordinals. Either way, the second problem is more serious: what your argument establishes is that \(|B|=|A|+1\). However, with cardinal numbers, it is a, perhaps surprising, fact that we don't necessarily have \(|A|<|A|+1\), and this has to do with what \(|A|<|B|\) actually means for cardinal numbers, and it means that there is no injection from \(B\) to \(A\). But in this case, there is such an injection.

Actually, as I show above, we in fact have \(|A|=|B|\), which by definition means that there is a bijection from \(A\) to \(B\).