= Theory of an infinite pure set
{title2=$T_\infty$}
In the <first-order language> of <logical equality>, assert that there are at least $n$ elements for every positive <integer> $n$. Eliminating one existential <logical variable> either substitutes a <logical equality> or avoids finitely many points, so this <first-order theory> has <quantifier elimination> and is complete. It is categorical in every infinite cardinal. Its complete $n$-types are the <logical equality> partitions of $n$ coordinates and are counted by the <Bell numbers>.
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