Is the theory of finite sets in the language of pure equality the minimal theory?

Question

Consider the first-order language of pure equality. Consider $Th(K)$, where $K$ is the class of finite sets. I conjecture that it is the minimal theory, that is to say, the theory axiomatized by the empty set of axioms. Is this true?

Answer 1

Yes, this is correct.

By definition, the empty theory in the empty language is the theory of all sets of all cardinalities, so it suffices to show that every satisfiable pure-equality sentence has a finite model.

So let $\varphi$ be a sentence in the empty language with no finite models. By compactness, $\neg\varphi$ has an infinite model; by Lowenheim-Skolem (and the fact that a pure set is determined up to isomorphism by its cardinality) all infinite pure sets are elementarily equivalent, so every infinite set satisfies $\neg\varphi$. Consequently $\varphi$ is unsatisfiable.

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In fact, a more general result holds: every satisfiable sentence in a language consisting only of $1$-ary relation symbols and $0$-ary function symbols (= constant symbols) has a finite model. This is largely the same argument, the point being that a structure in such a language is really just a list of pure sets (we need higher arity relations/functions to be able to have any real "interactivity").