An inductive proof for characterizations of maximal consistent theories over the system L

By induction on the complexity of formulas,a structural characterization of maximal consistent theories over L* is given without using the strong completeness theorem of L*.It is proved that each maximal consistent theory must be the deductive closure of some set with the form S(α)={φ1,φ2,…} satisfying φi∈{pi,pi,(p2i)((pi)2)} for all i=1,2,…,where p1,p2,… are the propositional variables of L*.Several necessary and sufficient conditions for a consistent theory to be maximal are obtained.The satisfiability theorem and compactness theorem for L* are also obtained.The obtained results improve the theoretical system for L*.