I am reading Enderton's book, A mathematical introduction to logic, beggining of the section $2.6$. I have two following question (page 147).
First question
He says that some sentences have only infinite models and the negation of such a sentence is true in every finite structure.
- Is it there any proof of this statement? Or, what is the intuition behind this statement?
Second question
(Theorem 26A) If a set $\Sigma$ of sentences has arbitrarily large finite models, then it has an infinite model.
For a set $\Sigma$ of sentences, let $\text{Mod } \Sigma$ be the class of all models of $\Sigma$.
A class $K$ of structures for our language is an elementary class (EC) iff $K = \text{Mod } \tau$ for some sentence $\tau$.
A class $K$ of structures for our language is an elementary class in the wider sence ($EC_\Delta$) iff $K = \text{Mod } \Sigma$ for some set $\Sigma$ of sentences.
(Corollary 26B)
- The class of all finite structures (for a fixed language) is not $EC_\Delta$.
- The class of all infinite structures is not $EC$.
Proof: 1) follows immediately from theorem 26A. 2) [...]
We consider some fixed language and the class $K$ of all finite structure of the language (i.e., all structure of the language such that the domain of the structure is finite). By the definition we need to show that there is no set $\Sigma$ of sentences such that $K = \text{Mod }\Sigma$.
How can apply Theorem 26A to prove this part of the corollary?