> On 8 May 2018, at 02:20, Lawrence Crowell <goldenfieldquaterni...@gmail.com>
> wrote:
>
> On Monday, May 7, 2018 at 11:51:46 AM UTC-5, Bruno Marchal wrote:
>
>> On 7 May 2018, at 03:19, Brent Meeker <meek...@verizon.net <javascript:>>
>> wrote:
>>
>>
>>
>> On 5/6/2018 6:40 AM, Bruno Marchal wrote:
>>>
>>>> On 2 May 2018, at 02:28, Lawrence Crowell <goldenfield...@gmail.com
>>>> <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.
All theories of arithmetic are incomplete. More generally all theories in which
we can define a Universal machine is incomplete.
Then Peano arithmetic is just not talking about real numbers.
BTW, the first order theory of the real is complete, but that is why it is too
weak to define a universal machine.
Then real numbers + trigonometry is Turing complete, but that is because
trigonometry can bu used to define the natural numbers in the real numbers.
Logicians have understood that the real numbers is only a big simplification of
the natural numbers. It is a simple exercise to solve Fremat in the real, and a
very difficult subject research to solve the same problem for the natural
numbers (cf Wiles).
> 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.”
? (You can name each of them). If we give you enough time. It is different from
the thing we can really not name or define in arithmetic, or in any theory
having arithmetic as a sub theory.
> 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.
Assuming such domain exists, but with mechanism, it can’t.
> 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.
With mechanism, I prefer (and is probably forced) to not install sets or real
numbers in the ontology. They are only a way the natural numbers or digital
machine simplify their “lives”.
If you believe in a richer ontology than one finite universal machine, or one
enumerable universal machinery, your theory will violate the consequences of
mechanism. That is not obvious, see my papers, or the archive in this list.
Bruno
>
> LC
>
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