Originally Posted by

**MoeBlee** Giraffro, to answer your question briefly: It is NOT the case that maximally consistent is equivalent to complete (since an inconsistent set of formulas is complete). But there are other equivalences. Which equivalence in particular are you having a hard time proving?

Definitions:

A set of formulas G is consistent iff there is not a formula P such that G |- P and G |- ~P.

A set of formulas G is complete iff for every sentence S we have either G |- S or G |- ~S.

A set of formulas G is maximally consistent iff (G is consistent & for all sentences S not in G we have Gu{S} is inconsistent).