State and prove the Upward Löwenheim-Skolem Theorem.

[You may assume the Compactness Theorem, provided that you state it clearly.]

A total ordering $(X,<)$ is called dense if for any $x<y$ there exists $z$ with $x<z<y$. Show that a dense total ordering (on more than one point) cannot be a well-ordering.

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 in the language of posets.

(i) The theory of dense total orderings.

(ii) The theory of countable dense total orderings.

(iii) The theory of uncountable dense total orderings.

(iv) The theory of well-orderings.