Exercise 818Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

The cardinality of the continuum

#continuum#cardinality#2^aleph0

Statement

Define $2^{\aleph_0}$ and give two sets with this cardinality.

Hint

$2^{\aleph_0}=|\mathcal{P}(\mathbb{N})|$ .

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Definition. $2^{\aleph_0}=|\mathcal{P}(\mathbb{N})|$ .

Examples. $\mathbb{R}$ and $\mathcal{P}(\mathbb{N})$ both have cardinality $2^{\aleph_0}$ .

Conclusion. $|\mathbb{R}|=2^{\aleph_0}$ .