prove that the class of cyclic groups is not axiomatizable?

Question

1)prove that the class of finite groups is not axiomatizable? Suppose there is a set $\Sigma$ of first-order sentences such that $\mathrm{Mod}(\Sigma)$ is the class of all cyclic groups. and how to cntinue?

Answer 1

We know from the compactness theorem that any set of formulas which has arbitrary large finite models also has an infinite model. Suppose $T$ was the theory of finite groups. We know that $\mathbb{Z}/n\mathbb{Z} \vDash T$ for any $n \in \mathbb{N}$ with $n > 1$ and $|\mathbb{Z}/n\mathbb{Z}|=n$ for all such n. Hence $T$ has arbitrary large finite models and we know by compactness that $T$ has an infinite model $M$. But then $M$ is an infinite finite group. A contradiction.

Using the theorem of Löwenheim-Skolem, we know that a theory which has a countable model has models of uncountable cardinality. So, if $\Sigma$ was the theory of all cyclic groups, then $\mathbb{Z} \vDash \Sigma$ and hence there is a model $M \vDash \Sigma$ such that $M$ is uncountable. But we know that every cyclic group is at most countable. A contradiction.