I'm not asking how to do these exercises. But I'm very confused about a thing.
Question: By hypothesis $T$ is an axiomatization of the class of $L$-structure with an even cardinality. So if $\mathcal{M} $ is a model of $T$ then it has an even cardinality. Also $T_{\infty} $ is an axiomatization of the class of $L$-structure that are infinite. So how it is possible that $T \cup T_{\infty} $ is consistent? Said in an ugly way: how can infinity be even?
Let $L$ a language consisting only with the symbol equal. And consider any $L$-theory $T$ for which its models are exactly the structure with an even cardinality.
- Find a theory $T_{\infty}$ whose models are exactly the $L$-structure whose domain is infinite.
- Prove that $T \cup T_{\infty} $ is consistent.
- What are the cardinaities of the domains of models of the theory $T \cup T_{\infty} $
- Prove that every infinite $L$-structure is a model of $T \cup T_{\infty}$ and deduce that every infinite $L$-structure is a model of $T$.
My reasoning: I proved that $T \cup T_{\infty} $ is finitely satisfiable, then by compactness theorem it is satisfiable. Okay, true! But I'm struggling to understand how it is possible that a model of $T \cup T_{\infty} $ could even exist. A model $ \mathcal{M} $ of $T \cup T_{\infty} $ has to be also a model of $T_{\infty} $ and a model of $T$ since $ T, T_{\infty} \subseteq T \cup T_{\infty} $. Since $ \mathcal{M} $ is a model of $T_{\infty}$ has to be infinite, since $ \mathcal{M} $ is a model of $T$ it has to be even. It seems to me more a proof of the fact that $T \cup T_{\infty}$ is not consistent. Clearly, if an infinite model exists then by downward Löwenheim–Skolem Theorem and by the fact the language is finite, then we have a countable model, then by upward Löwenheim–Skolem Theorem we have that every infinite model is model of $T \cup T_{\infty} $.
Theorem (overspill principle). If a set of sentences has arbitrarily large finite models, then it has a denumerable model.
Proof sketch. For each $n$, add a sentence which is true in and only in models of size $\geq n$. Then apply the compactness theorem.
Corollary. There exists no set of statements $T$ s.t. all of its models have even (and thus finite) cardinality and for every even number $n \in \mathbb{N}$, it has a model of cardinality $n$, since then $T$ would also have a denumerable model.