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For example

Finite models are not axiomatizable

I have been searching through the book after an explanation of what this is saying, but the word does not exist in the textbook. All I find online related to the word is "Elementary classes", does not seem to be related to the proposition that "Finite models are not axiomatizable".

Salviati
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    It means that there is no collection of formulas in predicate logic such that a model satisfies this collection if and only if the model is finite. In other words, we can not describe, using predicate logic, what it means for a model to be finite. – mrp Oct 15 '19 at 15:49
  • Aha, ok. Thank you! – Salviati Oct 15 '19 at 15:55
  • Related (but not duplicate) question and answers: https://math.stackexchange.com/q/1569449/97045 – DanielV Oct 15 '19 at 23:03

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More precisely, the property of being a finite model is not first-order axiomatizable. This means that there is no set $\Gamma$ of first order sentences such that a structure is finite if and only if it satisfies each sentence in $\Gamma.$

Note that it is the “only if” direction that is nontrivial. In fact it is easy to find a single first order sentence that implies a structure is finite, by, say, implying that it has exactly one element.

In fact, for any $n$, it is possible to write down a first order sentence involving only equality that says the structure has more than $n$ elements. And this is essentially how we prove the result: if such a $\Gamma$ existed, it would be consistent with any finite number of these statements, since it is possible for a structure to be finite and have more than $n$ elements for any given $n$. However by the compactness theorem, this means it has to be consistent with all of them, which is impossible since the sentences collectively say that the structure is infinite.

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