Exercise 2101Logic, sets and proofs › Countability and cardinalityM1Added 3 Nov 2026

Infinite subsets of $\mathbb{N}$

#infinite subsets#uncountable#power set

Statement

Show that the set of infinite subsets of $\mathbb{N}$ is uncountable.

Hint

$\mathcal{P}(\mathbb{N})$ = finite subsets $\cup$ infinite subsets.

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Partition. $\mathcal{P}(\mathbb{N})$ = (finite subsets) $\cup$ (infinite subsets), disjoint.

Counting. $|\mathcal{P}(\mathbb{N})|=2^{\aleph_0}$ (uncountable). Finite subsets are countable.

Conclusion. Infinite subsets are uncountable: $|\text{infinite subsets}|=2^{\aleph_0}$ .