Is $|P(\mathbb{N})| = |\mathbb{R}|$?
If so, what is the argument? I know that the cardinality of the power set greater than the cardinality of the natural numbers, and I assume that the cardinality of the power set of the natural numbers is smaller than that of the power set of the real numbers, though I have not been able to prove this to myself definitively (it simply seems intuitive that the power set of a countable set is smaller than the power set of an uncountable set).
The two approaches that I have considered have been:
(1) Create a bijection between $P(\mathbb{N})$ and $\mathbb{R}.$
(2) Show that there exists no uncountable set smaller than the power set of the real numbers.
One argument used to count $\mathbb{R}$ is to express real numbers between $0$ and $1$ using binary decimals. Those correspond to subsets of $\mathbb{N}$ - think about the decimal places with value $1$. (You have to be a little careful with the few reals = countably many - that have two binary decimal representations).