On 5/6/2018 6:40 AM, Bruno Marchal wrote:
On 2 May 2018, at 02:28, Lawrence Crowell
<goldenfieldquaterni...@gmail.com
<mailto:goldenfieldquaterni...@gmail.com>> wrote:
On Tuesday, May 1, 2018 at 3:37:15 PM UTC-5, Brent wrote:
An interesting proof by Hamkins and a lot of discussion of its
significance on John Baez's blog. It agrees with my intuition
that the
mathematical idea of "finite" is not so obvious.
Brent
This gets into the rarefied atmosphere of degrees of unprovability. I
have a book by Lerman on the subject, which I can read maybe 25 pages
into before I am largely confused and lost. I would really need to be
far better grounded in this. The idea is that one may ask if things
are diagonal up to ω ordinarlity, which is standard Gödel/Turing
machine stuff. Then we might however have Halting or provability out
to ω + n, or 2ω to nω and then how about ω^n and then n^ω and now
make is bigger with ω^ω and so forth. Then this in principle may
continue onwards beyond the alephs into least accessible cardinals
and so forth. One has this vast and maybe endless tower of greater
transfinite models.
Finite systems that are well defined are cyclic groups and related
structures. A mathematical system that has some artificial bound on
it is not going to satisfy any universal requirements. The most one
can have is finite but unbounded. So long as one does not have some
series or progression that grows endlessly this can work.
With mechanism, we don’t really have to take care of the non-standard
model of arithmetic, because it can be proved that even addition and
multiplication are not (Church-Turing) computable.
The church-Turing thesis makes computability absolute in the sense
that what can be proved to be computable or non computable will be
true in *all* models of any arithmetical theories. If a machine (or
number, to emphasise their finiteness) is universal, it is universal
in all models or interpretation of the ontological theories.
But don't you take all arithmetic theories to include the axioms that
say every number has a successor?
Brent
But this remark would become obsolete in case we add an infinity
axioms in the ontology, like in set theories. To take no risk, I
“forbid” even the induction axioms at the ontological level. So,
induction, infinities, second-order logic, real numbers, analysis and
physics will belong to the phenomenology of numbers, and this makes
mechanism into a finitism. 0, 1, 2, 3 …. exist, but N or ω do not,
except as number/machine’s mind’s tools.
Bruno
LC
-------- Forwarded Message --------
On Tue, May 1, 2018 at 1:13 PM, James wrote:
> On Tue, May 1, 2018 at 7:19 AM, Cris wrote:
>>
>> ... For any set of axioms, there is a Turing machine which 1)
never halts and 2) that set of axioms cannot prove that it never
halts. ...
>
>> But don’t you agree that the Halting Problem has a definite
truth value? In other words, that a given Turing machine (with a
given input) either runs forever or doesn’t, regardless of our
ability to prove it? ...
>
> To answer the question posed, shouldn't we ask if, given any
> *particular* TM, there exists *some* consistent system/set of
axioms
> that can prove whether it halts or not? I was under the
impression
> that the answer here was "yes", regardless of any individual
> consistent system being unable to tackle the general problem.
The problem is when you have nonstandard natural numbers. It's
perfectly valid, for instance, to have a Turing machine halt
after ω +
3 steps. You can say, "oh, but we use the unique standard model
defined by the second-order theory", but then the second order
theory
has to live in some universe, and there are universes in which
what's
uncomputable in your universe can be computable in mine:
http://jdh.hamkins.org/every-function-can-be-computable/
<http://jdh.hamkins.org/every-function-can-be-computable/>
https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/
<https://johncarlosbaez.wordpress.com/2016/04/02/computing-the-uncomputable/>
So as soon as you move away from "only physically implementable math
is real", then you have do deal with all these other models.
--
Mike
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