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.




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