Is the class of finite sets a pseudo-elementary class?

Question

Consider the class $K$ of finite sets, just bare sets without any structure. Certainly, it is not an elementary class. Is it pseudo-elementary?

Answer 1

To move this one off the unanswered queue: as bof observes, the answer is negative due to the compactness theorem. Specifically, by compactness every pseudoelementary class either has a finite bound on the size of its members or has infinite members (and indeed members of arbitrarily large infinite cardinality). So no "class of all finite $X$s" is ever pseudoelementary (unless of course there is a finite bound on the size of the finite $X$s).

Since pseudoelementary classes may look weird at first, let me point out in some detail how compactness can be applied to them too. The general result (which is overkill here, but important to internalize) is the following:

Suppose $\mathfrak{K}$ is a pseudoelementary class and $T$ is a set of sentences such that for every finite $T_0\subseteq T$ there is some $\mathcal{A}\in\mathfrak{K}$ such that $\mathcal{A}\models T_0$. Then there is some $\mathcal{B}\in\mathfrak{K}$ such that $\mathcal{B}\models T$.

This becomes more intuitive once we get used to interpretability, and realize that being a pseudoelementary class can be reexpressed as "is the class of '$\Phi$-interpretations' of members of some elementary class $\mathfrak{E}$ for some tuple of formulas $\Phi$." The point is that we can relativize the sentences in $T$ to the interpretation $\Phi$; let $T^\Phi=\{\varphi^\Phi: \varphi\in T\}$ be the so-relativized version of $T$. Since each finite subtheory of $T$ is satisfied by some member of $\mathfrak{K}$, we know that each finite subtheory of $T^\Phi$ is satisfied by some member of $\mathfrak{E}$. We now apply compactness in $\mathfrak{E}$ to get an $\mathcal{M}\in\mathfrak{E}$ with $\mathcal{M}\models T^\Phi$. But then $\Phi^\mathcal{M}\models T$ and $\Phi^\mathcal{M}\in\mathfrak{K}$.

In general, your first idea for testing whether something is "first-order describable" in any sense should always be the compactness theorem. (And your second idea should be the downward Lowenheim-Skolem theorem, which also applies to pseudoelementary classes by the same line of reasoning - "push through the interpretation.")

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As a coda, note that the class $\mathfrak{Inf}$ of all infinite sets is much better behaved. $\mathfrak{Inf}$ is both an elementary class (via $\{\exists x_1,...,x_n(\bigwedge_{1\le i<j\le n}x_i\not=x_j): n\in\mathbb{N}\}$) and a "finitely axiomatizable" pseudoelementary class (I don't actually know a term for this, which is to say that it's the class of reducts of models of a single sentence. $\mathfrak{Inf}$ is not however itself a finitely axiomatizable class (since the complement of a finitely axiomatizable class is itself an elementary class).