Is the class of infinite sets along with some certain other finite sets an axiomatizable class?

Let $L$ be the language of pure equality, and let $F$ be a finite set of natural numbers. I define the class $K(F)$ of $L$-structures $K$ associated with $F$ to be the class of infinite sets united with the class of sets whose cardinality is one of the members of $F$. For example, if $F$ is empty, then $K(F)$ is simply the class of infinite sets, and if $F=\{2,4\}$, then $K(F)$ is the class of infinite sets united with the class of two-element sets united with the class of four-element sets. Now, my question is, for any finite set $F$ of natural numbers, is the associated class $K(F)$ axiomatizable but not finitely axiomatizable?

Solution 1:

Yes.

Axiomatizability is easy: since $F$ is finite, for each $n\in\mathbb{N}$ a we can express "Either there are at least $n$ elements in the universe, or the cardinality of the universe is $\in F$" as a single first-order sentence $\varphi_n$. The theory $\{\varphi_n:n\in\mathbb{N}\}$ axiomatizes the class $K(F)$ as desired.

• In fact, however, with a bit of care we can show that $K(F)$ is axiomatizable even if $F$ is infinite! This involves a cute trick: instead of the $\varphi_n$s defined above, consider the sentences $\psi_n$ saying "Either there are at least $n$ elements in the universe, or the cardinality of the universe is $\in F\cap\{1,...,n\}$."

Meanwhile, non-finite-axiomatizability is a quick consequence of compactness: the complement of a finitely axiomatizable class is axiomatizable (indeed finitely axiomatizable), but if $F$ is finite then the complement of $K(F)$ contains arbitrarily large finite structures but no infinite structures. In fact, all that's needed here is that $F$ be co-infinite (if $F$ is cofinite then $K(F)$ is clearly finitely axiomatizable).