7. THIS STATEMENT IS FALSE
Gödel Gödel, Recursion and Attention
Bigand,
“the standard proof of the second incompleteness theorem assumes that
ProvA(P) satisfies that Hilbert–Bernays provability conditions. Letting #(P)
represent the Gödel number of a formula P, the derivability conditions say:
1. If T proves P, then T proves ProvA(#(P)).
2. T proves 1.; that is, T proves that if T proves P, then T proves
ProvA(#(P)). In other words, T proves that ProvA(#(P)) implies
ProvA(#(ProvA(#(P)))).
3. T proves that if T proves that (P → Q) and T proves P then T proves
Q. In other words, T proves that ProvA(#(P → Q)) and ProvA(#(P)) imply
ProvA(#(Q)).”
THIS STATEMENT IS FALSE