Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?

Question

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? If so, how?

I was reading the Wiki page on the , and it says "Moreover, $\mathbb{R}$ has the same number of elements as the power set of $\mathbb{N}$", but I don't see how this is true?

I feel like it has something to do with binary, but I'm not too sure how it works? Do I have to show a map of all reals can be done in binary? I'm just very confused, and any advice would be appreciated!

Answer 1

In short: A binary number $0.a_1a_2a_3\ldots$ can be identified with the set $\{n\in \mathbb N\mid a_n\ne 0\}$. A few details have to be checked, though

Answer 2

You can have a bijection from $\{0,1\}^{\mathbb N}$ to $\mathbb P(\mathbb N)$. Take the vector $a\in \{0,1\}^{\mathbb N} $ and map it to $A\subset \mathbb N$ with $i \in A \Leftrightarrow a_i = 1$. (It's easy to show that is a bijection)

Now as $|\mathbb P(\mathbb N)| = |\mathbb R|$, you can construct a bijection from $\{0,1\}^{\mathbb N}$ to $\mathbb R$. Note that I have used $\{0,1\}$ instead of your 2.