On 12/22/2013 5:04 AM, Bruno Marchal wrote:
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.
That's the point of the paper, that a "truth" predicate can be defined for arithmetic. I
put "truth" in scare quotes because the predicate is really 1-Probability(x)-->0.
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).
But aren't you assuming the standard model when you refer to the unprovable truths of
arithmetic. If you allowed other models this set would be ill defined.
Brent
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