Prove the set of all infinite binary sequences is uncountable by using Cantor's theorem.

Question

This is different then the other versions of the questions here. I'm not supposed to use the diagonal argument. I'm looking to write a proof based on Cantor's theorem, and power sets.

Answer 1

dfsfjn already provided you with a hint, but in case it isn't obvious to you what she meant, let me leave you the following spoiler:

For each $X \subseteq \mathbb N$ define $s_X \colon \mathbb N \to \{0,1\}$ by $s_N(n) = 1$ iff $n \in X$. Let $^{\mathbb N}\{0,1\}$ be the set of all functions $s \colon \mathbb N \to \{0,1\}$, i.e. the set of all infinite $0-1$ sequences. Define $\pi \colon \mathcal P(\mathbb N) \to ^{\mathbb N}\{0,1\}$ by $\pi(X) = s_X$ for all $X \subseteq \mathbb N$. Check that $\pi$ is a bijection and use Cantor's Theorem to conclude that $^{\mathbb N}\{0,1\}$ is uncountable.