On 21 Dec 2013, at 23:28, meekerdb wrote:
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,
In set theory, OK. But not in arithmetic.
And in a set theory (like ZF) you cannot define a set theoretical
predicate for set theoretical truth.
In ZF+kappa, you can define truth for ZF, but not for ZF+kappa. (ZF
+kappa can prove the consistency of ZF).
Shortly put, no correct machine can *define* a notion of truth
sufficiently large to encompass all its possible assertions.
Self-consistency is not provable by the consistent self (Gödel)
Self-correctness is not even definable by the consistent self (Tarski,
and also Gödel, note).
but not all models or arithmetic are the same as the standard model.
Computationalism uses only the standard model of arithmetic, except
for indirect metamathematical use like proof of independence of
axioms, or for modeling the weird sentences of G*, like <>[]f (the
consistency of inconsistency).
Bruno
Brent
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