 
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
