On 22 Dec 2013, at 20:55, meekerdb wrote:

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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 countableinfinity of integers to represent, no finite version can exist ofit, in other words, can its proof be found?If its shortest proof is infinitely long, or if the requiredaxioms needed to develop a finite proof are infinite, (or insteadof infinite, so large we could not represent them in thisuniverse), then its proof can't be found (by us), but there is adefinite answer to the question.The other possibility is that there are mutually inconsistentaxioms that can be added. As I understand it, that was the pointof http://intelligence.org/wp-content/uploads/2013/03/Christiano-et-al-Naturalistic-reflection-early-draft.pdfA 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 bedefined for arithmetic. I put "truth" in scare quotes because thepredicate 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 theoreticalpredicate 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 truthsufficiently 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 independenceof 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 theunprovable truths of arithmetic. If you allowed other models thisset 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/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.