This is the definition of a maximally consistent set from Logic and Structure by Van Dalen
A set Γ is maximally consistent iff
(a) Γ is consistent,
(b) Γ ⊆ Γ ′ and Γ ′ consistent ⇒ Γ = Γ ′.
Remark. One could replace (b) by (b’): if Γ is a proper subset of Γ ′, then Γ ′ is inconsistent. I.e., by just throwing in one extra proposition, the set becomes inconsistent.
This, at first, made me assume that Γ, in this case, must be finite, since a new formula can't be added. But a colleague told me that, for example, the set for every theorem is maximally consistent and infinite. Is that true?