On 22 Dec 2013, at 20:55, meekerdb wrote:
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.
OK. It is not a *truth* predicate, only an approximation. In fact, for
each sigma_i or pi_i sentence, Löbian theories or machine can define a
corresponding sigma_i or pi_i truth, and even limiting approximations,
which are far enough for practical purposes, but our concern here is
not a practical one, but a conceptual one.
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.
Exactly, and that is why I "don't allow them". This leads to a
technical difficulty in AUDA here (alluded to in Torkel's book on the
misuse of Gödel's incompleteness), which is solved by the use of the
intensional nuances, but that would be lengthy and technical to
describe here right now.
I am not sure we are disagreeing on something here, but if it is the
case, let me know, thanks.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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