Example of an abstract elementary class that is not even a pseudo-elementary class

Question

I have read the definition of an abstract elementary class, and I am told that all elementary class and even all pseudo-elementary classes are abstract elementary classes. That raises the question, is there an abstract elementary class which is not even a pseudo-elementary class? And if so, can someone exhibit an example?

Answer 1

First, note that there's a bit of a mismatch of terminology here. A pseudoelementary class is just a class of structures (satisfying some properties). An AEC, by contrast, is a class of structures together with an embedding relation (satisfying some properties). So strictly speaking, it is not the case that each pseudoelementary class is an AEC; rather, if $P$ is a pseudoelementary class then there is an AEC whose underlying class of structures is exactly $P$.

To make your question more precise, let me rephrase it as follows:

What are examples of AECs whose underlying classes of structures are not pseudoelementary?

For a silly example, take the class of all structures of cardinality $\le\aleph_{17}$ (say) and the identity embedding relation. This trivially satisfies the AEC axioms; in particular, its Lowenheim-Skolem number is $\aleph_{17}$.

(An earlier version of this answer made a very silly mistake here: if we take as the embedding relation the usual elementary embedding relation in the above, we do not get an AEC since the chain axiom isn't satisfied - consider a proper elementary chain of length $\omega_{17}$ consisting of structures of cardinality $\aleph_{16}$.)

Of course there are far more interesting ones. Even ignoring the silliness re: the embedding relation, this example is "eventually elementary" in the sense that there is a elementary class which coincides with it once we restrict attention to large enough structures (namely, $Mod(\perp)$). For a better example, pick some $\mathcal{L}_{\omega_1,\omega}$-sentence $\varphi$ and look at the models of $\varphi$ ordered by $\mathcal{L}_{\omega_1,\omega}$-elementary embeddings. This is an AEC, and if you pick $\varphi$ correctly the class of structures in it will not be "eventually pseudoelementary."

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Now as a cautionary tale, note that in general regular logics in the sense of non-AEC-based abstract model theory do not generally translate nicely into AECs. Suppose $\mathcal{L}$ is a regular logic and $T$ is a $\mathcal{L}$-theory. The pair $(Mod_\mathcal{L}(T),\preccurlyeq_\mathcal{L})$ is a natural thing to consider and trivially satisfies the AEC axioms except Tarski-Vaught and Lowenheim-Skolem. However, this still leaveslots of room for nastiness. For example, both $\mathcal{L}_{\omega_1,\omega_1}$ and $\mathsf{SOL}$ can detect when an ordering has countable cofinality, and this leads to the Tarski-Vaught axiom failing in their respective "candidate AECs."

This should actually be a promising sign, though: it indicates that the AEC axioms are strict enough to rule out a lot of the nastier logics that abstract model theory runs into, and raises the possibility of an actually nontrivial structure theory (which of course is what emerged).