= Exactly one infinite equivalence class is not first-order axiomatizable
Suppose a <first-order theory> characterized <equivalence relations> with exactly one infinite class. Name two inequivalent points and require each named class to have at least $n$ elements for every positive <integer> $n$. Every finite <subset> is satisfied with one infinite class and a sufficiently large finite class. <Compactness theorem> would produce a <first-order model> with two infinite classes, a <contradiction>.
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