On Tuesday, June 9, 2020 at 5:57:38 AM UTC-5, Bruno Marchal wrote:
>
>
> On 9 Jun 2020, at 02:27, Lawrence Crowell <[email protected]
> <javascript:>> 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]>
>> 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
>>>
>>>
>>> "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.
>
>
>
Yes it is that sort of construction.
>
> 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).
>
>
>
Matiyasevich showed the Hilbert's 10 problem can't be solved. This was the
existence of a global single solution system for Diophantine equations.
DIophantine equation are equivalent to p-adic sets by Robinson, Davis and
others.
>
>
> 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.
>
>
>
>
As a physicist I tend to have my heaviest foot on the side of Babylonian
math, which is more applied and something of a tool. I have some weight on
a foot on the Greek math side as well, which is the axiomatic, theorem and
proof mathematics.
As for further down, Godel's theorem works with reals. In effect Cantor
diagonization shows the set of reals are not enumerable, and the people who
get into set theory deeply use the concept of forcing. This takes one from
integers or natural numbers to the reals, and the same concept can lead to
complex numbers and so forth. Bernays and Cohen used Godel's theorem in
this form to show the continuum hypothesis is consistent with ZF set
theory. However, it is not provable. Don't ask me details on this because I
am no expert.
LC
>
>
>
>>
>> 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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