 
Summary: 1. GĻodel's Theorem.
1.1. consistency.
Definition 1.1. We say formal number theory (FNT) is consistent if whenever
A is a statement, x free(A) and
Axk for all k N
it is not the case that
x A.
Mendelson remarks after this definition on page 142 of Mendelson that we accept
that the standard interpretation as a model then FNT is consistent.
Proposition 1.1. If FNT is consistent then it is consistent.
Proof. Suppose FNT is consistent. Let x X and let A be the statement (x = x).
Then, as we have seen,
Axk for all k N.
Since FNT is consistent it is not the case that
x A.
Were FNT inconsistent we would have B for all statements B. Thus FNT is
consistent.
1.2. The relation W1. Let g be the GĻodel numbering function.
Let W1 be the logical function of two arguments defined by requiring that
W1(u, v) = 1 if and only if there are a statement U and a finite sequence of state
