The concept of compactness is a necessary condition of any system
that is going to call itself a finitary method of proof. However, it
can also apply to predicates of sets of formulas in general and in
that manner it can be used in relation to level functions, a flavor
of measure functions. In what follows we will tie these concepts of
measure and compactness together and expand some concepts which
appear in d'Entremont's master's thesis, "Inference and
Level." We will also provide some applications of the concept of
level to the "preservationist" program of paraconsistent logic. We
apply the finite level compactness theorem in this paper to get a
Lindenbaum flavor extension lemma and a maximal "forcibility"
theorem. Each of these is based on an abstract deductive system X
which satisfies minimal conditions of inference and has
generalizations of 'and' and 'not' as logical words.