Minimality

If $S$ is an infinite definable set of some structure (mathematical logic) and $A:=\{a_1,a_2,\ldots,a_n\}$ is any finite subset, then $A$ is definable via the formula (mathematical logic) $(x=a_1) \lor (x=a_2) \lor \ldots \lor (x=a_n)$ .

Similarly, any subset of $S$ which is cofinite, that is whose compliment is finite, is definable.

$S$ is said to be minimal if and only if these are the only definable subsets of $S$ .

A structure is said to be minimal if and only if its domain is a minimal set.

See also

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