(i) Give an axiom system and rules of inference for the classical propositional calculus, and explain the notion of syntactic entailment. What does it mean to say that a set of propositions is consistent? Let $P$ be a set of primitive propositions and let $\Phi$ be a maximal consistent set of propositional formulae in the language based on $P$. Show that there is a valuation $v: P \rightarrow\{T, F\}$ with respect to which all members of $\Phi$ are true.

[You should state clearly but need not prove those properties of syntactic entailment which you use.]

(ii) Exhibit a theory $T$ which axiomatizes the collection of groups all of whose nonunit elements have infinite order. Is this theory finitely axiomatizable? Is the theory of groups all of whose elements are of finite order axiomatizable? Justify your answers.