1. Back to formal theories. I'm going to modify some notions involving formal theories. Suppose Summary: 1. Back to formal theories. I'm going to modify some notions involving formal theories. Suppose T = (L, A, R) is a formal theory where L is a language on the alphabet A and R is the set of rules of inference. Definition 1.1. Suppose L. We say the finite sequence A1, A2. . . . , An is a primary proof using if for each j {1, . . . , n} either Aj or there is a rule of inference (H, C) such that H {Ai : i < j} and C = Aj. We say the statement A is a theorem (of T ) using if there is a primary proof A1, . . . , An using such that An = A in which case we write A. We say the finite sequence A1, A2. . . . , An is a proof using if for each j {1, . . . , n} either Aj , or {Ai : i < j} Aj or there is a rule of inference (H, C) such that H {Ai : i < j} and C = Aj. Proposition 1.1. Suppose L and A L then A if and only if there is a proof A1, A2, . . . , An using such that An = A. Proof. Just replace each Aj such that {Ai : i < j} Aj be a primary proof Collections: Mathematics