(i) State and prove the Compactness Theorem for first-order predicate logic.

State and prove the Upward Löwenheim-Skolem Theorem.

[You may use the Completeness Theorem for first-order predicate logic.]

(ii) For each of the following theories, either give axioms (in the language of posets) for the theory or prove carefully that the theory is not axiomatisable.

(a) The theory of posets having no maximal element.

(b) The theory of posets having a unique maximal element.

(c) The theory of posets having infinitely many maximal elements.

(d) The theory of posets having finitely many maximal elements.

(e) The theory of countable posets having a unique maximal element.