Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

So is G a theorem of FA? We assume first that FA is consistent. If it were not consistent, it
would obviously be useless for mathematics, and the game would be over. Assume next
that G is provable in FA. That means that some sequence of formulas S of FA is a proof of
G. Using the abbreviation we introduced earlier, that means that Bew(S, G). But we saw
earlier that Godel had shown that the function, Bew(x, y), when coded into natural
numbers, yields a recursive function and is thus rep-resentable in FA. The representation
occurs via the arithmetization of the syntax of FA, so corresponding to a given syntactical
truth Bew(x, y) of MFA, there is an arithmetical truth Bew(x, y) of IA that corresponds to a
formula Bew(x, y) in FA that can be interpreted as saying that the sequence of formulas
with Godel number x is a proof of the formula with Godel number y, and this formula,
Bew(x, y), is a theorem of FA. Thus if G is provable in FA for some s, then Bew(s, g) is a
theorem of FA (where s is the numeral in FA for s, the Godel number of the sequence of
formulas S that constitutes the proof of G, and g is the numeral in FA for g, the Godel
number of G). But as we saw at the beginning, what G states is that formula number g
(i.e., G) is unprovable, that is, for any sequence x, it is not the case that Bew(x, g). Using
a little logic, however, it follows from Bew(s, g) that there exists a sequence of formulas x
such that Bew(x, g), hence, using a bit more logic, that it is not the case that for any
sequence x it is not the case that Bew(x, g). That is, if something is a proof of formula
number g, it can't be that there is no proof of g. But since G can be interpreted as saying
that there is no proof of g, this last
conclusion is evidently equivalent to not-G, the negation of G, and since it was derived,
logically, from Bew(s, g), which, by hypothesis, is a theorem, it too is a theorem. That is,
not-G is also a theorem. The assumption that G is provable, then, leads to the conclusion
that both G and not-G are theorems, i.e., that both are provable in FA. Hence FA is
inconsistent. But the assumption was that FA is consistent. If FA is consistent, then, G is
not provable. That is the first part of GodePs theorem.
How about not-G? Godel assumed for this half of his theorem that FA is not just consistent
but "omega-consistent." (Omega is the traditional symbol for the natural numbers.) If a
system is omega-consistent, the following cannot happen: some statement F( ) is provable
individually of each natural number, i.e., F(0), F(l), F(2), etc., but it cannot be proved
that for all x, the statement F(x) is true. Clearly, if FA is to represent all the natural
numbers via its coordination with IA, omega-consistency if nonnegotiable. If not-G is a
theorem, then, of FA, given what G says, it follows that the following is provable: it is not
the case that every sequence of formulas number x does not constitute a proof of formula
number g. But we have already learned that formula number g, i.e., G, is not provable in
FA. Thus, for every individual sequence of formulas x, not-Bew(x, g) holds. Given the
representability of the function Bew(x, y) in FA, it follows that for every natural number x,
it is provable in FA that not-Bew(x, g). Since we are assuming that FA is omega-consistent,
it follows that it cannot also be provable that it is not the case that for all sequences of
formulas x, not-Bew(x, g) holds. But the latter is precisely what we assumed when we
agreed for the sake of argument that not-G is a theorem. If FA is omega-consistent, then it
follows that not-G is also not provable. This is the second half of Godel's theorem. Barkley
Rosser later showed that one can construct an unprovable statement assuming only
consistency, not omega-consistency.
But Godel wasn't done yet. He was able to prove, quickly, his second incompleteness
theorem, also on the assumption that FA is consistent. Using techniques similar to his proof
of the first theorem, Godel proved easily that if FA is consistent, it cannot prove that it is