A set of formulae is maximally consistent iff:

- It is consistent.
- No additional formula can be added to the set and result in a consistent set.

I formalise^{1} this as:

Given a set of formulae S

S is consistent iff ¬(S |- ¬(a→a))

S is maximally consistent iff S is consistent and (∀p)((p ∉ S)→((S ∪ {p}) |- ¬(a→a)))

Any consistent set of formulae can generate a maximally consistent set by adding all other consistent formulae. This can be used to prove the Completeness Theorem.

Aside: some philosophers define a possible world as a maximally consistent set of true propositions/statements.

1: Partial source: A Problem Course in Mathematical Logic by Stefan Bilaniuk <http://www.trentu.ca/mathematics/sb/pcml/>