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What is cardinality of $P(\mathbb{R})$? And $P(P(\mathbb{R}))$?

P is a Power Set, $\mathbb{R}$ is set of real numbers.

In general - how can find cardinality of different sets? Is/are there a good method(s)? And is there some kind of handout, that provides list of cardinality of popular sets ($\mathbb{R}$, $\mathbb{Q}$, $P(\mathbb{Q})$) etc.?

And, how to find cardinality of sets of functions? For example, what is the cardinality of set that contains all functions? What about set of functions $f: \mathbb{N} \to \mathbb{R}$?

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    cusier: The cardinality of ${\mathcal P}(X)$ is the same as the size of the set of functions from $X$ into $\{0,1\}$, usually denoted $2^{|X|}$. Beyond this, for infinite $X$ not much can be said. We know that $2^{|X|}>|X|$ but the usual axioms of set theory cannot tell you how much larger $2^{|X|}$ must be.2011-01-02
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    Should this be tagged [elementary-set-theory]?2011-01-02

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