On 12/21/2013 1:26 AM, Jason Resch wrote:
If there exists a mathematical theorem that requires a countable infinity of
integers to represent, no finite version can exist of it, in other words,
can its
proof be found?
If its shortest proof is infinitely long, or if the required axioms needed to develop a
finite proof are infinite, (or instead of infinite, so large we could not represent them
in this universe), then its proof can't be found (by us), but there is a definite answer
to the question.
The other possibility is that there are mutually inconsistent axioms that can be added. As
I understand it, that was the point of
http://intelligence.org/wp-content/uploads/2013/03/Christiano-et-al-Naturalistic-reflection-early-draft.pdf
A truth predicate can be defined for arithmetic, but not all models or arithmetic are the
same as the standard model.
Brent
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