Exercise 817Logic, sets and proofs › Countability and cardinalityL1Added 3 Nov 2026

The cardinal $\aleph_0$

#aleph zero#notation#cardinality

Statement

Define $\aleph_0$ and compute $|\mathbb{N}\cup\{\star\}|$ where $\star$ is an extra element.

Hint

$\aleph_0=|\mathbb{N}|$ .

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Definition. $\aleph_0=|\mathbb{N}|$ : the first infinite cardinal.

Compute. $|{\mathbb{N}}\cup\{\star\}|=|{\mathbb{N}}|$ : shift $n\mapsto n+1$ and $\star\mapsto0$ is a bijection.

Conclusion. $\aleph_0+1=\aleph_0$ ; adding finitely many elements to a countably infinite set leaves it countably infinite.