Let P be the class of all predicates of a mathematical theory T. Say a subclass M of P to be compatible, provided that there is an object O in T satisfying the conjunction of all members of M. Say a compatible subclass M to be saturated provided that for every predicate p in P - M, either MU{p} is a non compatible class, or there is a subclass M(p) of M the conjunction of which implies p.

Can you define some saturated predicate class in number theory?

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P.S.

The concept of saturated predicate class is introduced by myself in order to state this question.