Is there a reason why we only consider the 'standard" models to apply
when we are considering foundation theory (or whatever you might denote
what we are studying)? Have you ever looked at the Tennenbaum
wondered if it could be weakened to allow for computations that are
"outside" of the countable recursive functions?
I suspect that the "standard" model (of arithmetic) is a type of
invariant under a strong restricted group of transformations, I do not have
the proper language to explain this further at this time. :_( There is more
Just because we can prove that N x N ->N mapping can represent all
possible computations we forget that the proof assumes that the quantity of
resources and the number of computational steps is irrelevant. As a
researcher of computer science and physics, why is the tractability of a
computation given a quantity of resources not relevant in considerations of
what, say, the UD* can accomplish?
I believe that the quest of a universal rule or "measure" that determines
"Everything" is already excluded as a possibility; have we not learned that
a measure zero set is? Why not instead look at how computations can
"interact" with each other, how they might "evolve", how entropy may be
involved, what does it mean for truths to be finitely accessible and
infinite truths to be inaccessible, etc.
We look like monkeys chasing the weasel 'round the mulberry tree...
On Sun, Dec 22, 2013 at 2:55 PM, meekerdb <meeke...@verizon.net> 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
> 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
> 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
> 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.
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Stephen Paul King
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