On 5 Mar 2020, at 12:42, ronaldheld <ronal...@gmail.com> wrote:Any comments, especially from Bruno, and the Physicalists?
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<2003.01807.pdf>
Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½
measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.
These are exciting developments.
LC
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On 3/6/2020 3:40 AM, Lawrence Crowell wrote:
Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½
Which would give 1/p + 1/q = 4 ??
Brent
--measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.
These are exciting developments.
LC
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While programming/computing in (hypothetical) infinite domains is interesting ...Computing in Cantor’s Paradise With λ_ZFChow any of this relates in any way to physical reality (the stuff of nature that is actually around us in the universe, vs. just some theoretical, mathematical concoction someone may come up with) is dubious.(Things like consciousness is another thing, or subject: It may be "beyond" Turing, bit in a way that has nothing to do with "super" or "hyper" Turing or Cantor or Godel.)@philipthrift
This is about the λ_ZFC calculus, not the λ calculus.λ_ZFC contains infinite terms. Infinitary languages are usefuland definable: the infinitary lambda calculus [10] is an example, and Aczel’sbroadly used work [2] on inductive sets treats infinite inference rules explicitly.@philipthrift
On 5 Mar 2020, at 12:42, ronaldheld <ronal...@gmail.com> wrote:
Any comments, especially from Bruno, and the Physicalists?
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<2003.01807.pdf>
On 5 Mar 2020, at 12:42, ronaldheld <ronal...@gmail.com> wrote:
Any comments, especially from Bruno, and the Physicalists?--
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A simple way to read the meaning of a computation is to consider it as a mechanical procedure consisting of a finite number of steps that, when completed, yields a result. This interpretation is inadequate, however, for procedures that do not terminate after a finite number of steps, but nonetheless have an intuitive meaning. Consider, for example, a procedure for computing the decimal expansion of π; if implemented appropriately, it can provide partial output as it "runs", and this ongoing output is a natural way to assign meaning to the computation. This is in contrast to, say, a program that loops infinitely without ever providing output. These two procedures have very different intuitive meanings.
Since a computation described using lambda calculus is the process of reducing a lambda term to its normal form, this normal form itself is the result of the computation, and the entire process may be considered as "evaluating" the original term. For this reason Church's original suggestion was that the meaning of the computation (described by) a lambda term should be the normal form it reduces to, and that terms which do not have a normal form are meaningless.[1] This suffers exactly the inadequacy described above. Extending the π analogy, however, if "trying" to reduce a term to its normal form would give "in the limit" an infinitely long lambda term (if such a thing existed), that object could be considered this result.
No such term exists in the lambda calculus, of course, and so Böhm trees are the objects used in this place.
On 6 Mar 2020, at 12:40, Lawrence Crowell <goldenfield...@gmail.com> wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.
LC
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On 6 Mar 2020, at 12:57, Philip Thrift <cloud...@gmail.com> wrote:While programming/computing in (hypothetical) infinite domains is interesting ...Computing in Cantor’s Paradise With λ_ZFChow any of this relates in any way to physical reality (the stuff of nature that is actually around us in the universe, vs. just some theoretical, mathematical concoction someone may come up with) is dubious.
(Things like consciousness is another thing, or subject: It may be "beyond" Turing, bit in a way that has nothing to do with "super" or "hyper" Turing or Cantor or Godel.)
@philipthrift
On Friday, March 6, 2020 at 5:40:08 AM UTC-6, Lawrence Crowell wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.LC
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On 6 Mar 2020, at 21:07, ronaldheld <ronal...@gmail.com> wrote:interesting responses I did expect. From the physical universe POV, CT is relevant?
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On 7 Mar 2020, at 00:25, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Friday, March 6, 2020 at 5:57:34 AM UTC-6, Philip Thrift wrote:While programming/computing in (hypothetical) infinite domains is interesting ...Computing in Cantor’s Paradise With λ_ZFChow any of this relates in any way to physical reality (the stuff of nature that is actually around us in the universe, vs. just some theoretical, mathematical concoction someone may come up with) is dubious.(Things like consciousness is another thing, or subject: It may be "beyond" Turing, bit in a way that has nothing to do with "super" or "hyper" Turing or Cantor or Godel.)@philipthriftλ-calculus is equivalent to Turing computation.
In fact it is similar to Assembly language. It might be that some of these problems could be looked at according to λ-calculus.
LC
On Friday, March 6, 2020 at 5:40:08 AM UTC-6, Lawrence Crowell wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.LC
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On 7 Mar 2020, at 18:33, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Saturday, March 7, 2020 at 6:07:26 AM UTC-6, Philip Thrift wrote:This is about the λ_ZFC calculus, not the λ calculus.λ_ZFC contains infinite terms. Infinitary languages are usefuland definable: the infinitary lambda calculus [10] is an example, and Aczel’sbroadly used work [2] on inductive sets treats infinite inference rules explicitly.@philipthriftI am aware of this, It is a bit like considering Peano arithmetic in a domain where the axioms of infinity and choice hold.
LC
On Friday, March 6, 2020 at 5:25:13 PM UTC-6, Lawrence Crowell wrote:On Friday, March 6, 2020 at 5:57:34 AM UTC-6, Philip Thrift wrote:While programming/computing in (hypothetical) infinite domains is interesting ...Computing in Cantor’s Paradise With λ_ZFChow any of this relates in any way to physical reality (the stuff of nature that is actually around us in the universe, vs. just some theoretical, mathematical concoction someone may come up with) is dubious.(Things like consciousness is another thing, or subject: It may be "beyond" Turing, bit in a way that has nothing to do with "super" or "hyper" Turing or Cantor or Godel.)@philipthriftλ-calculus is equivalent to Turing computation. In fact it is similar to Assembly language. It might be that some of these problems could be looked at according to λ-calculus.LC
On Friday, March 6, 2020 at 5:40:08 AM UTC-6, Lawrence Crowell wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.LC
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On 6 Mar 2020, at 12:40, Lawrence Crowell <goldenfield...@gmail.com> wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.The Digital Mechanist thesis enforces that physics is derivable from Gödel’s and Löbs’ theorem, and indeed we find quantum logic exactly were expected.All computations are executed/emulated, in the mathematical sense of the Logicians, in arithmetic. The physical appearances have to be justified by the calculus of the 1p (plural) indeterminacy in arithmetic.There is a natural, canonical “many-wold” interpretation of arithmetic, developed by the “majority of universal numbers in arithmetic.
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.Sure.Bruno
LC--
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I will have to write more if possible. I am not sure that all of physics is derived from Gödel’s theorem. I see is as more that from classical to quantum mechanics there is a sort of forcing, to borrow from set theory, to extend a model with undecidable propositions. Where this undecidable matter enters in is with the problem of measurement and decoherence.
On 5 Mar 2020, at 12:42, ronaldheld <ronal...@gmail.com> wrote:
Any comments, especially from Bruno, and the Physicalists?--
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@philipthrift
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On 10 Mar 2020, at 01:52, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, March 9, 2020 at 6:42:41 AM UTC-5, Bruno Marchal wrote:On 6 Mar 2020, at 12:40, Lawrence Crowell <goldenfield...@gmail.com> wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.The Digital Mechanist thesis enforces that physics is derivable from Gödel’s and Löbs’ theorem, and indeed we find quantum logic exactly were expected.All computations are executed/emulated, in the mathematical sense of the Logicians, in arithmetic. The physical appearances have to be justified by the calculus of the 1p (plural) indeterminacy in arithmetic.There is a natural, canonical “many-wold” interpretation of arithmetic, developed by the “majority of universal numbers in arithmetic.I will have to write more if possible. I am not sure that all of physics is derived from Gödel’s theorem.
I see is as more that from classical to quantum mechanics there is a sort of forcing, to borrow from set theory, to extend a model with undecidable propositions. Where this undecidable matter enters in is with the problem of measurement and decoherence.
As for an earlier comment, Turing's model is in a grey zone between mathematics often seen as pure and with physics. The tape and reader, appearing a bit like a sort of cart on a track, is a model of a physical system. That system is a computer.
LC
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.Sure.BrunoLC--
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LC--
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Terms such as physicalists are not used, and materialism sometimes comes up and it is not clear to me how this deviates from the term physicalist.
The first of Gödel’s theorem comes in with looking at a list of observations of a quantum system, such as a list of probabilities on the abscissa and actual measurements on the ordinate.
This can then be used to perform the Cantor diagonal trick, which flips the outcome, and this is then not predicable. The inability to predict the outcome of a particular measurement of a quantum system can then be expressed according to a Cantor diagonal argument. This then leads to a form of the incomputable nature of QM.
This then leads to the observation that measurements in QM require that if one is to measure a spin in the z direction this means any prior knowledge of spin in the x direction is to be lost. In general relativity there are also event horizons that restrict knowledge one can have of the quantum state of a black hole. Jacobson showed how spacetime can be viewed from statistical mechanics as composed of a distribution of states. An event horizon is also a surface of reduced dimension that has quantum information. Raamsdonk also illustrated how spacetime can be looked at as due to large N-entanglement. So the loss of knowledge of a quantum spin in one direction in a measurement along another is in a general setting much the same as the red shifting of information from n event horizon that restricts access to information.
This then suggests with the Cantor diagonalization that the relationship between stochasticity and its dual in determinism has an incomputable relationship between quantum and spacetime physics. For stochasticity a p = 1 in a convex set with a dual q = ∞ and 1/p + 1/q = 1 there is are associated L^2 systems for p = q = 2, or 1/p = 1/q = ½, which are relativity as a metric space and QM as a system of probabilities determined by the square of amplitudes.
LC
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On 10 Mar 2020, at 01:52, Lawrence Crowell <goldenfield...@gmail.com> wrote:On Monday, March 9, 2020 at 6:42:41 AM UTC-5, Bruno Marchal wrote:On 6 Mar 2020, at 12:40, Lawrence Crowell <goldenfield...@gmail.com> wrote:Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form of the Cantor diagonalization for quantum measurements. As yet a full up form of the CHSH or Bell inequality violation result is waiting. There are exciting possibilities for connections between quantum mechanics, in particular the subject of quantum decoherence and measurement, and Gödel’s theorem.The Digital Mechanist thesis enforces that physics is derivable from Gödel’s and Löbs’ theorem, and indeed we find quantum logic exactly were expected.All computations are executed/emulated, in the mathematical sense of the Logicians, in arithmetic. The physical appearances have to be justified by the calculus of the 1p (plural) indeterminacy in arithmetic.There is a natural, canonical “many-wold” interpretation of arithmetic, developed by the “majority of universal numbers in arithmetic.I will have to write more if possible. I am not sure that all of physics is derived from Gödel’s theorem.That might not be the case, but it has to be so when we assume the digital mechanist hypothesis? Of course it is not a direct derivation from Gödel, but from all nuances that incompleteness impose to the provability notion. Incompleteness gives sense to the Theatetus-like variant of the rational opinion/belief, namely:TRUTH (p, God, the One, …)BELIEF ([]p, provability, Incompleteness forbids to see this as a knowledge)KNOWLEDGE ([]p & p, true belief, the soul, the first person, the owner of consciousness)And the “two matters”:INTELLIGIBLE MATTER ([]p & <>t) (with p restricted to sigma_1 propositions, the partial computable one)SENSIBLE MATTER ([]p & <>t & p) (idem)The soul provides an intuitionist logic, the two matters and the soul provides quantum logics when p is restricted on the partial computable (sigma_1) propositions (the true and the false one, which makes things more complex).See may papers for more on this. This needs some understanding of the existence of all computations in the models of arithmetic).I see is as more that from classical to quantum mechanics there is a sort of forcing, to borrow from set theory, to extend a model with undecidable propositions. Where this undecidable matter enters in is with the problem of measurement and decoherence.With mechanism, physics must become a study of the relative computational histories statistics.As for an earlier comment, Turing's model is in a grey zone between mathematics often seen as pure and with physics. The tape and reader, appearing a bit like a sort of cart on a track, is a model of a physical system. That system is a computer.A computer is universal implemented in the physical reality, but with mechanism, the physical reality is what emerge statistically from the first person points of view of the computer executed in the arithmetical reality (or in any reality related to any universal machine). I use numbers only because most people are familiar with them.I sum up 40 years of work, in a taboo domain, and all this is build on not so well known, or understood, theorem in mathematical logic, so ask any question, and maybe read also the papers. The first thing to understand is the incompatibility between (very weak form of) Digital Mechanism and (very weak form of ) Materialism or Physicalism.Bruno
LC
If we think of all physics as a form of convex sets of states, then there are dualisms of measures p and q that obey 1/p + 1/q = 1. For quantum mechanics this is p = ½ as an L^2 measure theory. It then has a corresponding q = ½ measure system that I think is spacetime physics. A straight probability system has p = 1, sum of probabilities as unity, and the corresponding q → ∞ has no measure or distribution system. This is any deterministic system, think completely localized, that can be a Turing machine, Conway's <i>Game of life</i> or classical mechanics. A quantum measurement is a transition between p = ½ for QM and ∞ for classicality or 1 for classical probability on a fundamental level.
What separates these different convex sets are these topological obstructions, such as the indices given by the Kirwan polytope. The distinction between entanglements is also given by these topological indices or obstructions. How these determine a measurement outcome, or the ontology of an element of a decoherent sets is not decidable. This is where Gödel’s theorem enters in. A quantum measurement is a way that quantum information or qubits encode other qubits as Gödel numbers.
The prospect spacetime, or the entropy of spacetime via event horizon areas, is a condensate or large N-entanglement of quantum states then implies there is a connection between quantum computation and information accessible in spacetime configurations. These configurations may either be the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum Church-Turing thesis is I think equivalent to the information processing of spacetime as black holes and maybe entire cosmologies.These are exciting developments.Sure.BrunoLC--
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My intent is more limited in showing that Gödel's theorem results in epistemic horizons or limitation on knowledge any observer can have.
This then in general is a basis for uncertainty principle and limits with observing qubits with black holes.
I am not sure this encompasses the entirety of physical principles.
It does not for instance tell us why dynamics involves the time derivative of momentum or second order differentiation of position.