On Monday, May 7, 2018 at 11:51:46 AM UTC-5, Bruno Marchal wrote:
>
>
> On 7 May 2018, at 03:19, Brent Meeker <[email protected] <javascript:>>
> wrote:
>
>
>
> On 5/6/2018 6:40 AM, Bruno Marchal wrote:
>
>
> On 2 May 2018, at 02:28, Lawrence Crowell <[email protected]
> <javascript:>> 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?
>
>
> Yes. Where is the problem? I could do without, and use the Gaussian
> integers, where numbers can have an up and right successors, if you prefer.
>
> I could do this point on all inductive system having some,operations
> making them Turing universal. But elementary arithmetic, and its primary
> school interpretation (assuming students and teachers are not zombie!) is
> enough.
>
> Like I just said to John, I assume less, far less, than most scientists,
> despite feeling close to Moderatus, the advaita veda, Lao-Ze, etc. The
> universal machine which knows that she is universal is quite close, when
> you look at its G/G* theology.
>
> Bruno
>
Peano number theory is incomplete, for there is no way the system can prove
it will define every possible real number. A form of this could be seen as
a form of Berry paradox where most numbers between 10^{10^{10^{10}}} and
10^{10^{10^{10^{10}}}} have no possible way of being "named." From a
practical perspective there are not enough quantum bits or particles within
our causal domain one might use to name most of these numbers. What ever
limits you place on the size of the name that bound is inevitably violated,
which leads in the infinite sense to a Cantor-like diagonalization because
it is non-enumerable. It does mean we can't prove there are not oddball
situation way out there on the number line. This does not though mean
things are really that quirky.
LC
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