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Or, very simply stated, given the finite set $S$ = {$a_1, ... , a_k$}, consider the diophantine equation: $$(n-a_1)...(n-a_n)=0$$. EDIT: Then we can write S as {$ \ n \ | \ \exists x : (n-a_1)...(n-a_k)=0$ }. (Thanks David)