(i) State the Completeness Theorem and the Compactness Theorem for the predicate calculus.

(ii) Show that if a theory has arbitrarily large finite models then it has an infinite model. Deduce that there is no first order theory whose models are just the finite fields of characteristic 2 . Show that the theory of infinite fields of characteristic 2 does not have a finite axiomatisation.

(iii) Let $\mathcal{T}$ be the collection of closed terms in some first order language $\mathcal{L}$. Suppose that $\exists x . \phi(x)$ is a provable sentence of $\mathcal{L}$ with $\phi$ quantifier-free. Show that the set of sentences $\{\neg \phi(t): t \in \mathcal{T}\}$ is inconsistent.

[Hint: consider the minimal substructure of a model.]

Deduce that there are $t_{1}, t_{2}, \ldots, t_{n}$ in $\mathcal{T}$ such that $\phi\left(t_{1}\right) \vee \phi\left(t_{2}\right) \vee \cdots \vee \phi\left(t_{n}\right)$ is provable.