= Prime-divisibility coding of a finite set
{title2=$e\in F\iff p_e\mid d$}
An internally finite subset of $\{0,\ldots,c-1\}$ can be represented in <Peano arithmetic> by a product of distinct <prime numbers>: multiply the $e$th <prime number> exactly when $e$ belongs to the subset. Divisibility by that prime recovers membership. Arithmetic induction proves existence of this code even for a definable subset with a nonstandard cutoff in a <Nonstandard model of Peano arithmetic>.
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