 
Summary: 1. The tough one.
We introduce the following notation. Suppose A is a statement and
free(A) = {xi1 , . . . , xin }
where
i1 < i2 < · · · < in
and suppose t1, . . . , tn are terms. We let
A(t1, . . . , tn) = Axi1
t1,...,xin tn .
We will frequently write
x for (x1, . . . , xn).
Let
V = {xi : i {1, . . . , n}.
Proposition 1.1. Suppose N N and N > 4. Let u be the term
(1 + (x3 + 1) · x2)
and let Bt be the statement
xN (((x1 = (u · xN ) + x4) (x4 < u))
Then Bt strongly represents the GĻodel function.
Proof. See p. 131 in Mendelson for the straightforward proof.
Let us suppose that
(i) g Nn
