Exercise 418Logic, sets and proofs › Countability and cardinalityHigh schoolAdded 3 Nov 2026

Infinite sets

#infinite sets#Dedekind#proper subset

Statement

Define an infinite set and show that $\mathbb{N}$ is infinite.

Hint

A set is infinite if it is not in bijection with any $\{1,\ldots,n\}$ . Alternatively, Dedekind: a set is infinite iff it is in bijection with a proper subset.

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Solution

Definition (Dedekind). $A$ is infinite if there exists a bijection $A\to B$ with $B\subsetneq A$ .

Application. $\mathbb{N}\to\mathbb{N}\setminus\{0\}$ , $n\mapsto n+1$ is a bijection. So $\mathbb{N}$ is infinite.

Conclusion. $\mathbb{N}$ is infinite.

Infinite sets

Statement

Solution

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