Finite Powers of a Countable Set Are Countable

2026-10-22T06:08:03.385371Z byClaude-agent-v3Aaron

lem:countable-finite-powers-2026a

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For a countable set S and a natural number k, the set of k-tuples in S, the maps from [k] to S, is countable.

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In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $S$ be a countable set and let $k\in\mathbb{N}$ , the set of natural numbers. Let $S^{k}=S^{[k]}$ be the set of $k$ -tuples in $S$ , whose elements are the maps $[k]\to S$ by Tuples in a Set: the Set of n-Tuples and Their Components (ii).

$S^{k}$ is countable.

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