Axiomatizability of class of the language $L=\{ = \} $ I was solving a mock exam and there is a "tricky" question, for me at least. It seem to me impossible that question 1) and 4) are both true. I would love to understand
Let $L$ a language with the symbol $ \{=\} $ and $C$ a class of $L$-structure. We say that $C$ is axiomatizable if there exists a theory $T$ of $L$ such that for all $L$-structure $\mathcal{M} $ we have that $ \mathcal{M} \in C $ if and only if $ \mathcal{M} \models T$.
We say that $ C$ is finitely axiomatizable if there exists a finite theory $T$.

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*Let $ L $ be the language consisting only of the symbol $ = $ and $ C$  the class of the finite $L$-structures. Is $C$ axiomatizable? Is $C$ finitely axiomatizable?
(Both answers: no)

*Let $ L$ be a language of the first order. Prove that if $C$ is finitely axiomatizable then the complementary class is also finitely axiomatizable.

*Let $L$ be the language consisting only of the symbol $ = $ and $C$ the class of the infinite $L$-structures. Is $C$ axiomatizable? Is $C$ finitely axiomatizable?
(First answer yes, the second answer no by point 2)

In the following let $ L $ be the language consisting only of the symbol $ = $


*Write a formula $F$ that axiomatizes the class of the $L$-structure for which the cardinality of its domains is $n$, with $n=1,2$ or $7$.

*Write a formula $G$ whose finite models are exactly the $L$-structure for which the cardinality of its domains is $n$, with $n\neq 1,2,7$.

*Give a theory whose finite models are exactly the $L$-structure with even cardinality.

 A: You're asking why the following two true statements are not in tension with each other:

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*The class $\mathbb{K}_{\mathit{fin}}$ of finite $\{=\}$-structures is not axiomatizable.


*The class $\mathbb{K}_{1,2,7}$ of $\{=\}$-structures of size either $1$, $2$, or $7$ is axiomatizable - indeed, can be axiomatized by a single sentence.
The silly-but-totally-correct response is simply: these are statements about different classes of structures! Since $\mathbb{K}_{\mathit{fin}}\not=\mathbb{K}_{1,2,7}$ - there are finite sets of size other than $1$, $2$, or $7$ (exercise) - there is no reason to expect them to have the same properties.
The more serious answer is that you should look at the proof that $\mathbb{K}_{\mathit{fin}}$ is not axiomatizable. It relies on a particular property of the set of finite cardinals (in order to let us apply compactness appropriately), which the set $\{1,2,7\}$ does not have. So there is in fact a "dividing line" here: there is a very simple characterization of those classes of cardinals $C$ whose associated classes of structures $$\mathbb{Card}_C:=\{\mathcal{A}: \vert\mathcal{A}\vert\in C\}$$ are axiomatizable. This is the real theorem which you should tease out here. I'm not going to spoil it entirely, but note that in this notation we have $\mathbb{K}_{\mathit{fin}}=\mathbb{Card}_{\mathbb{N}}$; now, as a set of cardinals, what nice property does $\mathbb{N}$ not have?
