> On 9 Jun 2020, at 02:27, Lawrence Crowell <[email protected]>
> wrote:
>
> On Monday, June 8, 2020 at 9:39:30 AM UTC-5, Bruno Marchal wrote:
>
>> On 7 Jun 2020, at 13:25, Lawrence Crowell <[email protected]
>> <javascript:>> wrote:
>>
>> On Saturday, June 6, 2020 at 5:25:23 PM UTC-5, Philip Thrift wrote:
>>
>> As for Hossenfelder's fav quantum mechanics semantics, she has stated many
>> times on her blog, it's superdeterminism.
>>
>> https://arxiv.org/abs/1912.06462 <https://arxiv.org/abs/1912.06462>
>>
>> "A superdeterministic theory is one which violates the assumption of
>> Statistical Independence (that distributions of hidden variables are
>> independent of measurement settings). Intuition suggests that Statistical
>> Independence is an essential ingredient of any theory of science (never mind
>> physics), and for this reason Superdeterminism is typically discarded
>> swiftly in any discussion of quantum foundations. The purpose of this paper
>> is to explain why the existing objections to Superdeterminism are based on
>> experience with classical physics and linear systems, but that this
>> experience misleads us. Superdeterminism is a promising approach not only to
>> solve the measurement problem, but also to understand the apparent
>> nonlocality of quantum physics. Most importantly, we will discuss how it may
>> be possible to test this hypothesis in an (almost) model independent way."
>>
>> @philipthrift
>>
>>
>> Superdeterminism is just a form of hidden variable theory. This invariant
>> set theory of Palmer and Hossenfelder as a means of connecting nonlinearity
>> with QM is interesting. The approach with Cantor sets connects with
>> incomputability.
>
>
> The Mandelbrot Set (its complement) has been shown indecidable, but in a vary
> peculiar theory of computability, which has not so many relation with Turing.
> It is “computability in a ring”.
>
> In the Turing theory, it is an open problem if he complement of the
> rational-complex Mandelbrot set is undecidable. That is a conjecture in my
> long thesis. Penrose has come up with a similar (less precise) hypothesis.
>
>
>
> The p-adic ring is what determines the trajectory of a point. where in the
> case of a Cantor set the divisor of the quotient ring is the map from one
> point to another. The Cantor set then has a set of orbits given by a set of
> p-adic rings.
Do you mean the triadic Cantor set? It is a set of reals. This play some role
for the isolation of the measure, thanks to relation between Baire Space,
Cantor triadic set. This requires ZF + Projective Determinacy (the existence of
some winning strategy for some infinite game). It is not directly related to
computability theory, except that we get them from the union of all sigma_set
relative to all oracles. That leads to complex set theory. Here, I use only N,
never R, nor bare space.
> The result of Matiyaesivich is there is no global method for solving these or
> the Diophantine equations they correspond to. This is the approach that I
> take.
I would like to see a pair on this. Matiyasevic’s paper and books do not refer
to p-adic structure, nor to real numbers. There is no Church’s thesis for the
notion of computability with real number. Constructive reals can be represented
by total computable functions (with a computable modulus so that + and * remain
computable).
> With the Mandelbrot set the "black bit" has periodic orbits or maps which
> correspond to periods of Julius sets. The points outside are chaotic and are
> in a sense "beyond chaos" and are not computable.
That is proved with the notion of computability on a ring, but like you, I
prefer to not use such notion. I see some application in theoretical numerical
analysis, but not much for computability theory in general. Then the measure
problem is enforce to use all set of (usual) real numbers, except that we can
make the closed set “perfect”, which helps to neglect the infinite countable
set of “isolated points”, but I am not there already.
>
>
>> I prefer a more standard definition of incomputability than what P&H appeal
>> to. This works invariant set theory does imply a violation of statistical
>> independence, but it does so as a hidden variable.
>>
>> The complement of a fractal set is undecidable.
>
> The complement of some fractal set have been shown undecidable in a theory of
> computability on a ring. This has been shown by Blum, Smale and Shub, if I
> remember well.
>
>
>
> The incomputability if with the fractal set itself. The incomputability
> occurs because with a finite cut off you have uncertainty whether points or
> regions are in or outside the Mandelbrot set. In this somewhat different
> meaning the Mandelbrot set is considered incomputable by Blum, Smale and Shub,
But only that meaning makes sense to me, and is of no use with respect to the
problem I am working on. I have only numbers (natural numbers!), and a real
number is (coddle by) any subset of N.
>
>
>> A fractal set is recursively enumerable, which means we can compute it in a
>> finite automata up to some point, and “in principle” a Turing machine that
>> runs eternally could compute the whole thing.
>
> Yes, but it uses only the potential infinite. We get all element in the
> enumeration after a finite time (except that here we use computability on a
> ring, which is not so easy to compare with Turing computability).
>
>
>
>
>
>> The complement of this is not computable. The complement of a recursive set
>> is recursive, but the complement of a recursively enumerable set is not
>> recursively enumerable and is incomputable.
>
> You mean “ … is not necessarily recursively enumerable”. Of course a
> complement of a recursively enumerable set can be recursively enumerable.
> That is always the case with recursive set.
>
>
> Yes, if the RE set is recursive.
OK.
>
>
>
>> The invariant set in this superdeterminism is a form of Cantor set or
>> related to a fractal. The results of Matiyasevich showed that p-adic sets
>> have no global solution method, where p-adic sets are equivalent to
>> Diophantine equations.
>
>
> I would be interested in a precise statement of this, and some link to a
> proof. What has a p-adic set? Set of what?
>
> Cantor sets are related to self-reference in many ways. For example through
> the topological semantic of G and S4Grz, but also through the “fuzzification”
> of Gödel or Löb theorem, like in a paper by Grim.
>
>
>
> A p-adic set is a quotient ring with the Z_p for p a prime. The Chinese
> remainder theorem guarantees that all quotient rings are equivalent to the
> product of quotient rings with primes that are the prime decomposition of the
> quotient ring. In other words for the quotient group ℤ_n = ℤ_{p1}×ℤ_{p2}× …
> ×ℤ_{p} for n = {p1}×{p2}× … ×{p} the prime factorization. There is a lot
> there and quotient rings define an elementary aspect of cohomology that leads
> to p-adic topology and with complex rings algebraic geometry.
OK. I like very much the Chinese lemma, if only through its use by Gödel to
code “digital machine” into numbers, without using exponentiation. But it is
basically only a representation trick. What you say here seems to have some
interest, but cohomology is a complex matter. Keep in mind that everything I
say can be translated faithfully in the elementary arithmetic of the natural
numbers, or in combinator theory. I avoid algebra, category theory, set theory,
even if those comes back at some point in the phenomenology. But I did not have
to use this to get the quantum phenomenology from arithmetic.
>
>
>
>
>> This means that dynamical maps from one point to another on the Cantor set
>> are not given by the same quotient group and in general there is no single
>> decidable system for such maps. In effect this means it is not observable.
>
> What is the relation between observable and decidable? If you study my
> papers, this is the most difficult thing to do. It is possible, and
> necessary, though, by the fact intensional variant of G and G*, which makes
> the logic of the observable/predictibvle obeying a quite different logic than
> G (indeed, a quantum logic).
>
>
>
> That is a part of the issue, and as I have worked things, hidden variables
> are unobservable and incomputable.
Here you are too much quick. Keep in mind that I have only natural numbers, and
that the observable is defined by what digital machine (number) can predict
about their accessible computational states. The point is that we cannot invoke
an ontological universe, given that we have arithmetic (just to define what is
a digital machine), and then we are confronted to the fist person indeterminacy
on all computations (in arithmetic) going through our actual states.
> The superdeterminism 'tHooft advanced and that others have taken up is really
> a form of hidden variable, and is not computable.
>
>
>>
>> So, while superdeterminism violates statistical independence this is all a
>> nonlocal hidden variable and thus unobservable. In ways this is where I
>> depart from Hossenfelder and Palmer, where Palmer uses a different concept
>> of incomputability, based on the idea of Smale et al on the need to compute
>> a fractal an infinite amount.
>
>
> I thought you did this.
>
>
> I worked this in a way similar to Hossenfelder and Palmer, but with out the
> appeal to Blum, Smale and Shub.
You will need to define “computable” in the context of the real number, for
which there is no CT thesis. I refer to avoid this. The real numbers exists in
arithmetic only as a kind of whole.
>
>
>
>> I appeal to the complement of a fractal, a fractal being a recursively
>> enumerable set
>
> Set of what?
>
>
>
> A fractal is a region of space that has a boundary with a Hausdorff dimension
> that is not integral. So it is a set of points or orbits under maps.
To solve the mind-body problem, such a notion of space can only be
phenomenological. It belongs to the imagination of the natural numbers, and
this has to be taken into account (and indeed that plays the main role in
making the logic of the observable into a quantum logic. My approach has to be
bottom up, where the bottom is elementary arithmetic.
>
>
>> and computable in a standard sense, but where the complement is not
>> computable.
>
> The complement of a creative set (the set-definition of a universal machine,
> due to Emil Post, and done before Church and Turing) is always non
> recursively enumerable.
> The complement of any set of all theorems of an axiomatic rich enough to
> prove the axiom of RA is automatically non recursively enumerable.
>
> Thanks to the work of Myhill, we know that a theory is Turing complete
> (Turing universal) iff and only the complement in N of the set of (Gödel
> number of) its theorem is constructively not recursively enumerable.
> A set S of numbers is constructively not recursively enumerable, or called
> also productive, means that for any W_i subset of S, you can find some x in
> S, but not in W_i.. That x serves as counter-example of the recursive
> enumerability of S. You can extend the extension of S in the constructive
> transfinite, by reiterating this trasnfinitely (on the recursive ordinals, or
> beyond).
>
> A Recursively enumerable set with a productive complement is called creative,
> by Emil Post, and is the set theoretical definition of Turing universality,
> by a result of Myhill.
>
>
>
>
> This I am not that familiar with. I tend to prefer to stay as much as
> possible within more standard mathematics instead of set theory. Set theory I
> will appeal to somewhat, but I prefer to stay more within algebra and
> geometry.
I have no other choice (given my goal and methodology) to never go outside
arithmetic. (Algebra and geometry use already to much of (naive) set theory.
>
>
>> The fractal emerges from QM in a singular perturbation series and the
>> complement comes with the dual of a convex set with measure L^p is L^q with
>> 1/p + 1/q = 1.
>
>
> I fail to see what are the elements of the sets you are talking. The standard
> notion of computability concerns set of natural numbers (or of things
> encodable into finite numbers, like strings (the computer science one!),
> words, formula, etc.
>
> Bruno
>
>
> The set of maps for any point is something computable or not. The Cantor set
> is not because there is not a single algorithm for solving all orbits that
> hop from one point to the other. This is because there is no global solution
> method for all p-adic quotient rings. The elements are really maps, maps that
> take a point here to there and then to elsewhere in an iterative manner.
I can understand that the complex-rational Mandelbrot set is or not computable,
but once real numbers are considered, I am lost. Just lost. You need a
definition of computability for real. I can imagine why you avoid Blum, Shub
and Small, but I have no real clue which notion you are using. It might be
interesting, we will see, or not.
Bruno
>
> LC
>
>
>
>>
>> LC
>>
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