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I am teaching myself real analysis, and in this particular set of lecture notes, the introductory chapter on set theory when explaining that not all sets are countable, states as follows:

If $S$ is a set, $\operatorname{card}(S) < \operatorname{card}(\mathcal{P}(S))$.

Can anyone tell me what this means? It is theorem 1.5.2 found on page 13 of the article.

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4  
Power set, the set of subsets. Also $\mathfrak{P}(S)$ –  Daniel Fischer Jul 15 '13 at 20:32
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It’s the set of all subsets of $S$, often called the power set of $S$. –  Brian M. Scott Jul 15 '13 at 20:33
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Why do you say that your set of lecture notes "never explains exactly what $\mathcal P$ means" ? The explanation is given on page 2 in Definition 1.1.1. It is extremely clear and followed by excellent examples. Asking a question answered in detail at the very beginning of your reference, in a section named 1.1.1 (!) makes me think you should try to improve your way of self-teaching. –  Georges Elencwajg Jul 15 '13 at 21:27
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Maybe because your method of self-teaching doesn't seem to be working well. –  MJD Jul 16 '13 at 15:53
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@MJD Well, the question was about the symbol, not about the quality of my self teaching, so why are you concerned with whether or not my method of self teaching is "working well"? What relevance does it have to the question at hand? –  Ataraxia Jul 16 '13 at 16:37

2 Answers 2

up vote 7 down vote accepted

The set $\mathcal P(X)$ also denoted by $\frak P,\wp,\it P$ or as $2^X$ sometimes, is the set of all subsets of the set $X$. Formally: $$\{A\mid A\subseteq X\}$$

Note that this is never an empty set, because $X\in\mathcal P(X)$. Cantor's theorem, which you quote in your question, states that no matter what set $X$ is, the cardinality of its power set is strictly larger.

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This is the set of all subsets of $S$.

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