[FOM] Strong forms of Gödel's incompletenewss theorem
Martin Davis
martin at eipye.com
Wed Mar 22 00:22:49 EST 2006
In a recent FOM post, Harvey Friedman noted the theorem:
>THEOREM 3. Let T be a consistent extension of Q, whose language contains
>L(Q). Assume that the set of axioms of T is recursive. Then there is an
>A...A sentence of L(Q) that is neither provable nor refutable in T. (No
>bounded existential quantifiers allowed).
Using Yuri Matiyasevich's reduction of the number of unknowns in a
universal Diophantine equation to 9, we can easily see that the prefix
A...A in the above can be assumed to be of length 9 provided we replace
"consistent" by "omega-consistent" in the statement. Can the same be said
if we stick to simple consistency?
Martin
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