Natural cybernetics and mathematical history: the principle of least choice in history

 The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices.

Key words: Gadamer, Hegel, Heidegger, Husserl, mathematical and historical dialectics, mathematical and historical hermeneutics, mathematical and historical phenomenology, information conservation, mathematical history, natural historical cybernetics, transcendental history, law (principle) of least choice

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The Completeness: from Henkin's Proposition to Quantum Computer

 A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user“ to infinity implied byHenkin's proposition as the only consistent ones as to infinity. 

An essential area of contemporary knowledge may be synthesized from a single viewpoint ...

Key words: EPR, Henkin's proposition, Lob's theorem, entanglement, quantum computer, quantum information, qubit

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Two deductions: (1) from the totality to quantum information conservation; (2) from the latter to dark matter and dark energy

 The paper discusses the origin of dark matter and dark energy from the concepts of time and the totality in the final analysis. Though both, and especially the latter, seem to be rather philosophical, nonetheless they are postulated axiomatically and interpreted physically, and the corresponding philosophical transcendentalism serves heuristically. The exposition of the article means to outline the “forest for the trees”, however, in an absolutely rigorous mathematical way, which to be explicated in detail in a future paper. The “two deductions” are two successive stage of a single conclusion mentioned above. The concept of “transcendental invariance” meaning ontologically and physically interpreting the mathematical equivalence of the axiom of choice and the well-ordering “theorem” is utilized again. Then, time arrow is a corollary from that transcendental invariance, and in turn, it implies quantum information conservation as the Noether correlate of the linear “increase of time” after time arrow. Quantum information conservation implies a few fundamental corollaries such as the “conservation of energy conservation” in quantum mechanics from reasons quite different from those in classical mechanics and physics as well as the “absence of hidden variables” (versus Einstein’s conjecture) in it. However, the paper is concentrated only into the inference of another corollary from quantum information conservation, namely, dark matter and dark energy being due to entanglement, and thus and in the final analysis, to the conservation of quantum information, however observed experimentally only on the “cognitive screen” of “Mach’s principle” in Einstein’s general relativity therefore excluding any other source of gravitational field than mass and gravity. Then, if quantum information by itself would generate a certain nonzero gravitational field, it will be depicted on the same screen as certain masses and energies distributed in space-time, and most presumably, observable as those dark energy and dark matter predominating in the universe as about 96% of its energy and matter quite unexpectedly for physics and the scientific worldview nowadays. Besides on the cognitive screen of general relativity, entanglement is available necessarily on still one “cognitive screen” (namely, that of quantum mechanics), being furthermore “flat”. Most probably, that projection is confinement, a mysterious and ad hoc added interaction along with the fundamental tree ones of the Standard model being even inconsistent to them conceptually, as far as it need differ the local space from the global space being definable only as a relation between them (similar to entanglement). So, entanglement is able to link the gravity of general relativity to the confinement of the Standard model as its projections of the “cognitive screens” of those two fundamental physical theories.

Key words: confinement, dark energy, dark matter, entanglement, general relativity, physical and mathematical transcendentalism, quantum information, the Standard model, transcendental invariance

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Mathematical history as counterfactual history. Prolegomena to natural cybernetics

The mathematization of history needs all the class of possible histories, which all without only one are counterfactual or alterntive. The mathematical formalism of the separable complex Hilbert space of quantum mechanics can be applied to that mathemazable class similarly to the alternative Feynman pathways in quantum mechanics. One can infer an operator (non-Hermitian, in general) transforming our real historical pathway into any other thus being counterfactual to ours as real.

Key words: Feynman interpretation of quantum mechanics, alternative history, counterfactual history, global and local space, historical space, mathematical history, transcendental history, transcendental space

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The case of quantum mechanics mathematizing reality: the “superposition” of mathematically modelled and mathematical reality. Is there any room for gravity?

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Gödel’s opinion) can be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined.

Key words: confinement, entanglement, general relativity, model, reality, quantum gravity, quantum information

 
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Quantum-information conservation. The problem about “hidden variables”, or the “conservation of energy conservation” in quantum mechanics: A historical lesson for future discoveries

 The explicit history of the “hidden variables” problem is well-known and established. The main events of its chronology are traced. An implicit context of that history is suggested. It links the problem with the “conservation of energy conservation” in quantum mechanics. Bohr, Kramers, and Slaters (1924) admitted its violation being due to the “fourth Heisenberg uncertainty”, that of energy in relation to time. Wolfgang Pauli rejected the conjecture and even forecast the existence of a new and unknown then elementary particle, neutrino, on the ground of energy conservation in quantum mechanics, afterwards confirmed experimentally. Bohr recognized his defeat and Pauli’s truth: the paradigm of elementary particles (furthermore underlying the Standard model) dominates nowadays. However, the reason of energy conservation in quantum mechanics is quite different from that in classical mechanics (the Lie group of all translations in time). Even more, if the reason was the latter, Bohr, Cramers, and Slatters’s argument would be valid. The link between the “conservation of energy conservation” and the problem of hidden variables is the following: the former is equivalent to their absence. The same can be verified historically by the unification of Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics in the contemporary quantum mechanics by means of the separable complex Hilbert space. The Heisenberg version relies on the vector interpretation of Hilbert space, and the Schrödinger one, on the wave-function interpretation. However the both are equivalent to each other only under the additional condition that a certain well-ordering is equivalent to the corresponding ordinal number (as in Neumann’s definition of “ordinal number”). The same condition interpreted in the proper terms of quantum mechanics means its “unitarity”, therefore the “conservation of energy conservation”. In other words, the “conservation of energy conservation” is postulated in the foundations of quantum mechanics by means of the concept of the separable complex Hilbert space, which furthermore is equivalent to postulating the absence of hidden variables in quantum mechanics (directly deducible from the properties of that Hilbert space). Further, the lesson of that unification (of Heisenberg’s approach and Schrödinger’s version) can be directly interpreted in terms of the unification of general relativity and quantum mechanics in the cherished “quantum gravity” as well as a “manual” of how one can do this considering them as isomorphic to each other in a new mathematical structure corresponding to quantum information. Even more, the condition of the unification is analogical to that in the historical precedent of the unifying mathematical structure (namely the separable complex Hilbert space of quantum mechanics) and consists in the class of equivalence of any smooth deformations of the pseudo-Riemannian space of general relativity: each element of that class is a wave function and vice versa as well. Thus, quantum mechanics can be considered as a “thermodynamic version” of general relativity, after which the universe is observed as if “outside” (similarly to a phenomenological thermodynamic system observable only “outside” as a whole). The statistical approach to that “phenomenological thermodynamics” of quantum mechanics implies Gibbs classes of equivalence of all states of the universe, furthermore representable in Boltzmann’s manner implying general relativity properly … The meta-lesson is that the historical lesson can serve for future discoveries.

Key words: BKS theory. class of equivalence, energy conservation in quantum mechanics, fourth uncertainty relation, general relativity and quantum gravity, Gibbs and Boltzmann thermodynamics, Heisenberg’s matrix mechanics, pseudo-Riemannian space, Schrödinger’s wave mechanics, separable complex Hilbert space, unitarity

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A CLASS OF EXAMPLES DEMONSTRATING THAT “P≠NP ” IN THE “P VS NP” PROBLEM

 The CMI Millennium “P vs NP Problem” can be resolved e.g. if one shows at least one counterexample to the “P=NP ” conjecture. A certain class of problems being such counterexamples will be formulated. This implies the rejection of the hypothesis “P=NP” for any conditions satisfying the formulation of the problem. Thus, the solution “P≠NP ” of the problem in general is proved. The class of counterexamples can be interpreted as any quantum superposition of any finite set of quantum states. The Kochen-Specker theorem is involved. Any fundamentally random choice among a finite set of alternatives belong to “NP’ but not to “P”. The conjecture that the set complement of “P” to “NP” can be described by that kind of choice exhaustively is formulated.

 

 

 

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A Book Written by (means of) “I Ching": "The Man in the High Castle"

 The structure and content of the presentation:

I A formal model of divination by means of “I Ching“
II The “I Ching” divination in terms of quantum measurement
III The scientific condition for any “I Ching“ divination to be relevant
IV How might one write a book by “I Ching“?
V How might “I Ching” write another book?
VI A few excerpts from “The Man in the High Castle” with or by “I Ching”
VII The precedent of Hermann Hesse’s “Das Glasperlenspiel” (“The Glass Beed Game”)

Philip K. Dick, “The Man in the High Castle,” Berkeley, 1982 (reprint) ISBN: 0-425-05051-3

 

 

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The relationship of arithmetic as two twin Peano arithmetic(s) and set theory: A new glance from the theory of information

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested.

Key words: Peano arithmetic, nonstandard interpretation of Peano arithmetic, two complimentary standard interpretations of Peano arithmetic, Hilbert arithmetic, consistent completeness of mathematics and physics, the unification of mathematics and physics, information, quantum information

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Fermi's paradox and a space travel from … a reality to another

“The Fermi paradox, named after Italian-American physicist Enrico Fermi, is the apparent contradiction between the lack of evidence for extraterrestrial civilizations and various high estimates for their probability” {“Fermi paradox,” in Wikipedia). The objection is to be represented as a not less paradoxical, “quantum-mechanical explanation” for it. The explanation relies on the non-invariance of physical laws after the space travel if quantum mechanics is valid.
A necessary condition for any space travel is the constancy of all physical laws during it. It is not satisfied in the “ideological universe” in general. Thus, “they are not here” because they cannot come here in the “ideological universe”.. The space travels in the “ideological universe” are possible only at a distance short enough for the physical laws to be constant approximately. The formulation of Fermi’s paradox does not admit for the physical laws not to be constant during the travel, which is the formal and logica reason for the contradiction.

Key words: Fermi's paradox, field of physical laws, the "ideological universe". Noether's consevation theorems, space tpavel




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The indeterminist objectivity of quantum mechanics versus the determinist subjectivity of classical physics

Indeterminism of quantum mechanics is considered as an immediate corollary from the theorems about absence of hidden variables in it, and first of all, the Kochen – Specker theorem. The base postulate of quantum mechanics formulated by Niels Bohr that it studies the system of an investigated microscopic quantum entity and the macroscopic apparatus described by the smooth equations of classical mechanics by the readings of the latter implies as a necessary condition of quantum mechanics the absence of hidden variables, and thus, quantum indeterminism. Consequently, the objectivity of quantum mechanics and even its possibility and ability to study its objects as they are by themselves imply quantum indeterminism. The so-called free-will theorems in quantum mechanics elucidate that the “valuable commodity” of free will is not a privilege of the experimenters and human beings, but it is shared by anything in the physical universe once the experimenter is granted to possess free will. The analogical idea, that e.g. an electron might possess free will to “decide” what to do, scandalized Einstein forced him to exclaim (in a letter to Max Born in 2016) that he would be а shoemaker or croupier rather than a physicist if this was true. Anyway, many experiments confirmed the absence of hidden variables and thus quantum indeterminism in virtue of the objectivity and completeness of quantum mechanics. Once quantum mechanics is complete and thus an objective science, one can ask what this would mean in relation to classical physics and its objectivity. In fact, it divides disjunctively what possesses free will from what does not. Properly, all physical objects belong to the latter area according to it, and their “behavior” is necessary and deterministic. All possible decisions, on the contrary, are concentrated in the experimenters (or human beings at all), i.e. in the former domain not intersecting the latter. One may say that the cost of the determinism and unambiguous laws of classical physics, is the indeterminism and free will of the experimenters and researchers (human beings) therefore necessarily being out of the scope and objectivity of classical physics. This is meant as the “deterministic subjectivity of classical physics” opposed to the “indeterminist objectivity of quantum mechanics”.

Keywords: choice, determinism and indeterminism free will, free will theorems, Kochen-Specker theorem

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The Frontier of Time: The Concept of Quantum Information

A concept of formal transcendentalism is utilized. The fundamental and definitive property of the totality suggests for “the totality to be all”, thus, its externality (unlike any other entity) is contained within it. This generates a fundamental (or philosophical) “doubling” of anything being referred to the totality, i.e. considered philosophically. Thus, that doubling as well as transcendentalism underlying it can be interpreted formally as an elementary choice such as a bit of information and a quantity corresponding to the number of elementary choices to be defined. This is the quantity of information defined both transcendentally and formally and thus, philosophically and mathematically. If one defines information specifically, as an elementary choice between finiteness (or mathematically, as any natural number of Peano arithmetic) and infinity (i.e. an actually infinite set in the meaning of set theory), the quantity of quantum information is defined. One can demonstrate that the so-defined quantum information and quantum information standardly defined by quantum mechanics are equivalent to each other. The equivalence of the axiom of choice and the well-ordering “theorem” is involved. It can be justified transcendentally as well, in virtue of transcendental equivalence implied by the totality. Thus, all can be considered as temporal as far anything possesses such a temporal counterpart necessarily. Formally defined, the frontier of time is the current choice now, a bit of information, furthermore interpretable as a qubit of quantum information.

Key words: axiom of choice, choice, formal transcendentalism, the totality, time, information, quantum information, well-ordering, well-ordering principle

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Indeterminism in quantum mechanics: beyond and/or within causation

The problem of indeterminism in quantum mechanics usually being considered as a generalization determinism of classical mechanics and physics for the case of discrete (quantum) changes is interpreted as an only mathematical problem referring to the relation of a set of independent choices to a well-ordered series therefore regulated by the equivalence of the axiom of choice and the well-ordering “theorem”. The former corresponds to quantum indeterminism, and the latter, to classical determinism. No other premises (besides the above only mathematical equivalence) are necessary to explain how the probabilistic causation of quantum mechanics refers to the unambiguous determinism of classical physics. The same equivalence underlies the mathematical formalism of quantum mechanics. It merged the well-ordered components of the vectors of Heisenberg’s matrix mechanics and the non-ordered members of the wave functions of Schrödinger’s undulatory mechanics. The mathematical condition of that merging is just the equivalence of the axiom of choice and the well-ordering theorem implying in turn Max Born’s probabilistic interpretation of quantum mechanics. Particularly, energy conservation is justified differently than classical physics. It is due to the equivalence at issue rather than to the principle of least action. One may involve two forms of energy conservation corresponding whether to the smooth changes of classical physics or to the discrete changes of quantum mechanics. Further both kinds of changes can be equated to each other under the unified energy conservation as well as the conditions for the violation of energy conservation to be investigated therefore directing to a certain generalization of energy conservation.

Key words: causation, choice and well ordering, determinism, Hilbert space of quantum mechanics, indeterminism, probabilistic causation



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A Model of Causal and Probabilistic Reasoning in Frame Semantics

Quantum mechanics admits a “linguistic interpretation” if one equates preliminary any quantum state of some whether quantum entity or word, i.e. a wave function interpretable as an element of the separable complex Hilbert space. All possible Feynman pathways can link to each other any two semantic units such as words or term in any theory. Then, the causal reasoning would correspond to the case of classical mechanics (a single trajectory, in which any next point is causally conditioned), and the probabilistic reasoning, to the case of quantum mechanics (many Feynman trajectories). Frame semantics turns out to be the natural counterpart of that linguistic interpretation of quantum mechanics.

Key words: frame, frame and reference frame, frame semantics, formal and mathematical semantics, entanglement, quantum information

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Universal Logic in terms of Quantum Information

Any logic is represented as a certain collection of well-orderings admitting or not some algebraic structure such as a generalized lattice. Then universal logic should refer to the class of all subclasses of all well-orderings. One can construct a mapping between Hilbert space and the class of all logics. Thus there exists a correspondence between universal logic and the world if the latter is considered a collection of wave functions, as which the points in Hilbert space can be interpreted. The correspondence can be further extended to the foundation of mathematics by set theory and arithmetic, and thus to all mathematics.

Key words: bit and choice, Hilbert space, information, quantum information, qubit

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Time and information in the foundations of physics

The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and
thus information in turn. Quantum information and its units, the quantum bits, can be
interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information to the substance of the world is constructed. The article is based only on the well-known facts and definitions and is with no premises in this sense. Nevertheless it is naturally situated among works and ideas of Husserl and Heidegger, linked to the foundation of mathematics by the axiom of choice, to the philosophy of quantum mechanics and information.

Key words: choice, order, quantum information, time, totality, well-ordering

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Natural Argument by a Quantum Computer

Natural argument is represented as the limit, to which an infinite Turing process converges. A Turing machine, in which the bits are substituted with qubits, is introduced.
That quantum Turing machine can recognize two complementary natural arguments in any data. That ability of natural argument is interpreted as an intellect featuring any quantum computer. The property is valid only within a quantum computer: To utilize it, the observer should be sit-ed inside it. Being outside it, the observer would obtain quite different result depending on the degree of the entanglement of the quantum computer and observer. All extraordinary properties of a quantum computer are due to involving a converging infinite computational process con-tenting necessarily both a continuous advancing calculation and a leap to the limit. Three types of quantum computation can be distinguished according to whether the series is a finite one, an infinite rational or irrational number.

Keywords: infinite computation, quantum computer, quantum Turing machine, quantum entanglement, qubit, natural argument



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The Gödel incompleteness theorems (1931) by the axiom of choice

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint the relation of set theory and arithmetic are demonstrated.

Key words: choice, arithmetic, set theory, well-ordering, information



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Main concepts in philosophy of quantum information

Quantum mechanics involves a generalized form of information, that of quantum information. It is the transfinite generalization of information and representable by transfinite ordinals. The physical world being in the current of time shares the quality of “choice”. Thus quantum information can be seen as the universal substance of the world serving to describe uniformly future, past, and thus the present as the frontier of time. Future is represented as a coherent whole, present as a choice among infinitely many alternatives, and past as a well-ordering obtained as a result of a series of choices. The concept of quantum information describes the frontier of time, that “now”, which transforms future into past. Quantum information generalizes information from finite to infinite series or collections. The concept of quantum information allows of any physical entity to be interpreted as some nonzero quantity of quantum information. The fundament of quantum information is the concept of ‘quantum bit’, “qubit”. A qubit is a choice among an infinite set of alternatives. It generalizes the unit of classical information, a bit, which refer to a finite set of alternatives. The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen.

Key words: quantum information, free will, choice, qubit, quantity of quantum information, transfinite generalization of information

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The Kochen - Specker theorem in quantum mechanics: a philosophical comment (part 1 & part 2)

PART 1:
Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem
Keywords: Kochen – Specker theorem, generalized relativity, Hilbert space, Minkowski space, world line by psi-function, psi-function by qubits

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PART 2:The text is a continuation of the article of the same name published in the previous issue of Philosophical Alternatives. The philosophical interpretations of the Kochen- Specker theorem (1967) are considered. Einstein's principle regarding the,consubstantiality of inertia and gravity" (1918) allows of a parallel between descriptions of a physical micro-entity in relation to the macro-apparatus on the one hand, and of physical macro-entities in relation to the astronomical mega-entities on the other. The Bohmian interpretation ( 1952) of quantum mechanics proposes that all quantum systems be interpreted as dissipative ones and that the theorem be thus derstood. The conclusion is that the continual representation, by force or (gravitational) field between parts interacting by means of it, of a system is equivalent to their mutual entanglement if representation is discrete. Gravity (force field) and entanglement are two different, correspondingly continual and discrete, images of a single common essence. General relativity can be interpreted as a superluminal generalization of special relativity. The postulate exists of an alleged obligatory difference between a model and reality in science and philosophy. It can also be deduced by interpreting a corollary of the heorem. On the other hand, quantum mechanics, on the basis of this theorem and of V on Neumann's (1932), introduces the option that a model be entirely identified as the modeled reality and, therefore, that absolutely reality be recognized: this is a non-standard hypothesis in the epistemology of science. Thus, the true reality begins to be understood mathematically, i.e. in a Pythagorean manner, for its identification with its mathematical model. A few linked problems are highlighted: the role of the axiom of choice forcorrectly interpreting the theorem; whether the theorem can be considered an axiom; whether the theorem can be considered equivalent to the negation of the axiom.

Keywords: Kochen- Specker theorem relativity entanglement model and reality Bohmian interpretation of quantum mechanics axiom of choice

The published paper (part 2: (2013) Philosophical Altertnatives 22 (3): 74-83) as a PDF or @ PhilPapers

Continuity and Continuum in Nonstandard Universum

One can consider two complementary Peano arithmetics both staandard, but well-ordered oppositely to each other. For examplle, the one starts from "1" by the function of suucessor interpreted as "+1", and the other one, from the odinal of the countable set, "omega" by the function of successor interpreted as "-1". The former needs only the Peano axiioms, and the latter (or both consistently) needs them and (ZFC) set theory. Thus, those two complimentary Peano arithmetics are a tool for studyimg problems and paradoxes about the foundations of mathematics (e.g. Gödel's incopletenes/ inconsitency of Peano arithmetic to ZFC set theory).

Furthermore, one can admit a "nonstandard interpretation" of Peano arithmetic to reconcile the two complimentary Peano arithmetics to each other in order to be valid simultaneously: its first element can be interpreted both as "1" or as "omega", and the function of successor as "= successor". This implies the cyclicicity of that nonstandard interpretation, as well as a "topological" or ("pre-topological") meaning of the axiom of choice as the "topological cut" of the cyclicity or coherence of the whole into a well-ordering. "Choice", "(quantum) information", "bit", and "qubit" can asquire a relevant topological (or pre-topological) meaning.

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Quantum Measure from a Philosophical Viewpoint

The paper discusses the philosophical conclusions, which the interrelation between quantum mechanics and general relativity implies by quantum measure. Quantum measure is three-dimensional, both universal as the Borel measure and complete as the
Lebesgue one. Its unit is a quantum bit (qubit) and can be considered as a generalization of the unit of classical information, a bit. It allows quantum mechanics to be interpreted in terms of quantum information, and all physical processes to be seen as informational in a generalized sense. This implies a fundamental connection between the physical and material, on the one hand, and the mathematical and ideal, on the other hand. Quantum measure unifies them by a common and joint informational unit.
Quantum mechanics and general relativity can be understood correspondingly as the holistic and temporal aspect of one and the same, the state of a quantum system, e.g. that of the universe as a whole.

Keywords: measurement, quantum mechanics, general relativity, quantum information, entanglement

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The sequrity of quantum cryptography

A prsentation 12 years ago:

Quantum cryptography (or QKD) offers an automated procedure for distributing secret keys utilizing generally used communication fibres. The revolutionary characteristic of Q KD is that it is intrinsically secure: the key cannot be acquired by an eavesdropper without the sender and addressee's knowledge. Moreover, QKD permits the key to be changed often.The philosophical meaning if quantum cryptography consists of existing an absolute secure communication which guarantees on private inviolability from Big Brother's infringement.

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Time: From the Totality to Quantum Information

The paper justifies the following theses: The totality can found time if the latter is axiomatically represented by its “arrow” as a well-ordering. Time can found choice and thus information in turn. Quantum information and its units, the quantum bits, can be interpreted as their generalization as to infinity and underlying the physical world as well as the ultimate substance of the world both subjective and objective. Thus a pathway of interpretation between the totality via time, order, choice, and information to the substance of the world is constructed. The article is based only on the well-known facts and definitions and is with no premises in this sense. Nevertheless it is naturally situated among works and ideas of Husserl and Heidegger, linked to the foundation of mathematics by the axiom of choice, to the philosophy of quantum mechanics and information.

Key words: choice, order, quantum information, time, totality, well-ordering




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Mass at rest after quantum information

The way, in which quantum information can unify quantum mechanics (and therefore the Standard model) and general relativity, is investigated. Quantum information is defined as the generalization of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit. The invariance to the axiom of choice shared by quantum mechanics is introduced: It constitutes quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical ensemble of the measurement of the quantum system at issue). This allows of equating the classical and quantum time correspondingly as the well-ordering of any physical quantity or quantities and their coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying their unification. Its deformation is representable correspondingly as gravitation in the deformed pseudo-Riemannian space of general relativity and the entanglement of two or more quantum systems. The Standard model studies a single quantum system and thus privileges a single reference frame turning out to be inertial for the generalized symmetry U(1)XSU(2)XSU(3) “gauging” the Standard model. As the Standard model refers to a single quantum system, it is necessarily linear and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism U(1) → U(1)XSU(2) confirmed enough already experimentally describes exactly the choice of the initial position of a privileged reference frame as the corresponding breaking of the symmetry. The Standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the “Big Bang”. It serves also in order to reconcile the linear Standard model in the singularity of the “Big Bang” with the observed nonlinearity of the further expansion of the universe described very well by general relativity. Quantum information links the Standard model and general relativity in another way by mediation of entanglement. The linearity and absoluteness of the former and the nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same whole divided into parts entangled in general.

Keywords: general relativity, the Standard model, quantum information, mass at rest, qubit, the Big Bang

The paper as a PDFor @ repositories: @ EasyChair, @ SocArxiv;
the published paper "(Is Mass at Rest One and the Same? A Philosophical Comment: on the Quantum Information Theory of Mass in General Relativity and the Standard Model," Журнал Сибирского федерального университета. Гуманитарные науки. Journal of Siberian Federal University. Humanities & Social Sciences (2014) 7 (4): 704-720.)

Being and Knowledge along any postmetaphysical context (new, 2020 edition)

1. Being&Probability. 2. Time&Fractal
What means “Being and Knowledge along any post-metaphysical context”?
I mean “Being&Knowledge along of any post-metaphysical contexts”:
“Being&Knowledge” means: Being and Knowledge are the same: Being, which is Knowledge&Knowledge, which is Being, i.e. Being=Knowledge
From Being&KnowledgetoBeing&Probability:
It is Informationthat is Substance of Being, which is Knowledge.
Informationis a relation between probabilities.
Granted is: two kinds of probabilties –subjective and objective. We reckon objective probabilities for Being and we reckon subjective probabilities for Knowledge.
1. Being&Probability.



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2. Time&Fractal



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CYCLIC MECHANICS: THE PRINCIPLE OF CYCLICITY

Cyclic mechanic is intended as a suitable generalization both of quantum mechanics and general relativity apt to unify them. It is founded on a few principles, which can be enumerated approximately as follows:
1. Actual infinity or the universe can be considered as a physical and experimentally verifiable entity. It allows of mechanical motion to exist.
2. A new law of conservation has to be involved to generalize and comprise the separate laws of conservation of classical and relativistic mechanics, and especially that of conservation of energy: This is the conservation of action or information.
3. Time is not a uniformly flowing time in general. It can have some speed, acceleration, more than one dimension, to be discrete.
4. The following principle of cyclicity: The universe returns in any point of it. The return can be only kinematic, i.e. per a unit of energy (or mass), and thermodynamic, i.e. considering the universe as a thermodynamic whole.
5. The kinematic return, which is per a unit of energy (or mass), is the counterpart of conservation of energy, which can be interpreted as the particular case of conservation of action “per a unit of time”. The kinematic return per a unit of energy (or mass) can be interpreted in turn as another particular law of conservation in the framework of conservation of action (or information), namely conservation of wave period (or time). These two counterpart laws of conservation correspond exactly to the particle “half” and to the wave “half” of wave-particle duality.
6. The principle of quantum invariance is introduced. It means that all physical laws have to be invariant to discrete and continuous (smooth) morphisms (motions) or mathematically, to the axiom of choice.
The list is not intended to be exhausted or disjunctive, but only to give an introductory idea about the text, which follows:

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#
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Free Will in Human Behavior and Physics

If the concept of “free will” is reduced to that of “choice” all physical world share the latter quality. Anyway the “free will” can be distinguished from the “choice”: The “free will” involves implicitly a certain goal, and the choice is only the mean, by which the aim can be achieved or not by the one who determines the target. Thus, for example, an electron has always a choice but not free will unlike a human possessing both. Consequently, and paradoxically, the determinism of classical physics is more subjective and more anthropomorphic than the indeterminism of quantum mechanics for the former presupposes certain deterministic goal implicitly following the model of human freewill behavior. Quantum mechanics introduces the choice in the fundament of physical world involving a generalized case of choice, which can be called “subjectless”: There is certain choice, which originates from the transition of the future into the past. Thus that kind of choice is shared of all existing and does not need any subject: It can be considered as a low of nature. There are a few theorems in quantum mechanics directly relevant to the topic: two of them are called “free will theorems” by their authors (Conway and Kochen 2006; 2009). Any quantum system either a human or an electron or whatever else has always a choice: Its behavior is not predetermined by its past. This is a physical law. It implies that a form of information, the quantum information underlies all existing for the unit of the quantity of information is an elementary choice: either a bit or a quantum bit (qubit).

Key words: axiom of choice, choice, free will, free will theorems, goal, hidden variables, quantum information, quantum mechanics

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The published paper (Vasil Penchev. Free will in human behavior and physics [Свободная воля в поведении человека и физике]. Labour and Social Relations. 2020.Vol. 30 Issue 6. P. 185-196. DOI 10.20410/2073-7815-2020-30-3-185-196) as a PDF, @ PhilPapers, @ SSRN, or @ JournalSite

“Схизмата във физиката” и ... неравенствата на Бел

Квантовата механика предлага множество "скандални" за здравия разум твърдения. Сред тях е и нарушаването на "неравенствата на Бел": достатъчно, но не необходимо усложие за "квантово сдвояване" - загадъчното действие на разстояние, "призрачно" по думите на Айншайн. Той, заедно с Подолски Розен го извежда още в 1935 г.като аргумент, доказващ математически "непълнотата на квантовата механика". "Схизмата във физиката" е прочута книга на Сър Карл Попър, посветена на философията и методологията на квантовата механика, в която той поддържа опозицията на Айншайн срешу квантовата механика. Обаче днес нарушаването на неравенствата на Бел е общоприет и многократно потвърден експериментален факт.

Презентацията и като PDF и като видео

Fleeting thoughts: A physical interpretation of Heidegger’s distinction of “existence” and ‘being’ by neutrinos

Neutrino exists only within the framework of energy conservation. (Wolfgang Pauli arguing with Niels Bohr forecast their necessary existence due to energy conservation.) If so, what would be representation of "neutrino" out of that framework? If one answers the latter, what would be the physical meaning of existence out of (energy) conservation? Does it correspond to Heidegger's distinction of "existence" ("conservation", "metaphysics") and "being" (generalizing "conservation", "ontology", eventually "fundamental")?

Well, neutrino is the difference between any "path of Feynman" and the privileged one of energy conservation. However, if the neutrino is observed experimentally, this implies that Feynman's paths

are not (only) virtual or a merely thought construction, but actual ones therefore generating all "immortal" neutrinos penetrating the universe.

So, one should search for neutrinos in the actual (rather than Born's probabilistic) viewpoint of Hugh Everett III to quantum mechanics postulating all "paths" as real "worlds": then neutrino would be a certain difference between a given "world" ("path") and the privileged "zero" one of energy conservation.

Furthermore, that privileged "zero" world of energy conservation can be identified as a privileged reference frame consistent to general rather than to special relativity and determined unambiguously by the "cosmological constant" even if its value is zero. That viewpoint of general relativity interprets

the neutrinos as direct physical equivalents of the certain curvatures of space-time in each point of it.

Thus, the mysterious neutrinos however interpreted both within and out of the framework of energy conservation are able to bridge over the gap between quantum mechanics and general relativity. Thus, the neutrinos offer a physical and very fruitful interpretation of Heidegger's distinction between "existence" and 'being’:

Existence corresponds to the “zero” reference frame of energy conservation, and ‘being’, to all possible variations about it (illustrable by Feynman’s paths) or, to all possible curvatures of pseudo-Riemannian space to the “zero” curvature of the cosmological constant.

Well, that being as a single one as the totality implies the equivalence of the set of all paths of Feynman (quantum mechanics) to all curvatures of space-time (general relativity) under the condition of that totality or “fundamentality” (in Heidegger’s word).

The physical meaning of the condition is the totality of the universe: its externality (as far as the universe is the physical totality) is described by quantum mechanics within it. Thus, it corresponds necessarily to its internality as it is described by general relativity.

An illustration: all worlds (in the interpretation of Hugh Everett III) or all possible discretely separated “universes” out of ours should be represented equivalently within it and thus describable by general relativity as all possible reference frames arbitrarily accelerated. The discrete transition between any two separated “universes” turns out to equivalent to smooth transition between any two reference frames!

Key words: Heidegger, fundamental ontology, being, existence, neutrino

Gravity as entanglement, entanglement as gravity

A generalized and unifying viewpoint to both general relativity and quantum mechanics and information is investigated. It may be described as a generaliztion of the concept of reference frame from mechanics to thermodynamics, or from a reference frame linked to an element of a system, and thus, within it, to another reference frame linked to the whole of the system or to any of other similar systems, and thus, out of it. Furthermore, the former is the viewpoint of general relativity, the latter is that of quantum mechanics and information.
Ciclicity in the manner of Nicolas Cusanus (Nicolas of Cusa) is complemented as a fundamental and definitive property of any totality, e.g. physically, that of the universe. It has to contain its externality within it somehow being namely the totality. This implies a seemingly paradoxical (in fact, only to common sense rather logically and mathematically) viewpoint for the universe to be repesented within it as each one quant of action according to the fundamental Planck constant.
That approach implies the unification of gravity and entanglement correspondiing to the former or latter class of reference frames. An invariance, more general than Einstein's general covariance is to be involved as to both classes of reference frames unifying them. Its essence is the unification of the discrete and cotnitinuous (smooth). That idea underlies implicitly quantum mechanics for Bohr's principle that it study the system of quantum microscopic entities and the macroscopic apparatus desribed uniformly by the smmoth equations of classical physics.

Key words: entanglement, gravity, invariance to the discrete and continuous, quantum gravity, reference frame




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Prolegomena: The quantum-information link between the Schrödinger equation (QM) and the Einstein filed equation (GR)

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Quantum Mechanics as a Measure Theory: the Theory of Quantum Measure

Quantum mechanics is well representable as a theory of a new kind of measure: quantum measure. It can be interpreted as a covariant counterpart of its usual consideration. The unit of quantum measure is a qubitand thus all physical quantities share a common fundament, that of quantum information, and all physical processes can be interpreted as computational ones. That interpretation of quantum mechanics is fruitful and heuristic: This can be visualized by an explanation of the Aharonov–Bohm effect.

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Prolegomena: Quantum measurement

The concept of measurement in quantum mechanics is seen from a few philosophical and mathematical viewpoints further: quantum invariance, the foundation of mathematics and the axiom of choice, and quantum measure. Then a few general and more particular philosophical problems can be guided in a new direction. Those are:

1. Quantum gravity: Can general relativity be considered as a theory of quantum gravity?
2. The laws of conservation: Can quantum mechanics offer a more general law than that of conservation of energy (energy-momentum) so that to be consistent with general relativity where energy-momentum is conserved only locally?
3. The principle of relativity: Can the Einstein relativity principle be generalized in a way to comprise quantum movements as well?
4. Invariance to discreteness or continuity: Can quantum mechanics offer a more general viewpoint to unify continuous (or smooth) and discrete (quantum) motions?
5. The alternatives of the “Big Bang”: Can the universe arise necessarily from nothing, in mathematical laws?
6. Nothing and anything: Can nothing and anything have a common measure?
7. Quantum information: Is quantum information that quantity both physical and mathematical one, which uses that measure?
8. Mathematics and physics: Is there a smooth transition between them, in which a mathematical structure or its element like a number or a probability distribution can transform into a physical entity like a particle?
9. The foundation of mathematics: Can set theory be “repaired” in a way to include directly quantum correlations or entanglement between the elements of a primary mathematical structure like ‘set’?
10. The interpretation of Hilbert space: Can one think Hilbert space as that structure in the frameworks of the set-theory based mathematics, which is both the simplest and may involve quantum correlations or entanglement?

The list can be continued rather. One should demonstrate that quantum measurement is relevant to it after all:

1. Quantum measurement and the axiom of choice:

The theorem of Kochen – Specker (1968) and yet John von Neumann’s (1932) one before it require the absence of any hidden variables in any quantum state before measurement and thus they exclude any well-ordering in it. In fact any well-ordering before measurement is excluded by any theory utilizing differential equations to describe the states in the studied area. It can happen only if the axiom of choice is added externally for the smooth continuum necessary for any differential equation cannot be well-ordered otherwise than by it. However the difference between a point of the continuum and a corresponding point in a well-ordered subset of it can converge to zero. That way out is forbidden to quantum mechanics because of the Plank constant for it imposes a finite difference between the states before and after measurement. Furthermore any set of measured results is well-ordered, e.g. by the exact time of recording in a central computer.
If one combines the Kochen – Specker theorem with the well-ordering after measurement, the well-ordering theorem equivalent to the axiom of choice is unavoidable: So quantum measurement involves necessarily the axiom of choice. One can object that the sets before and after measurement are different so that the well-ordering theorem is irrelevant. In fact even then the mapping between a set, which cannot be well-ordered in principle, and that, which is always well-ordered, requires the axiom of choice for the Cartesian product between them to exist, of which the mapping is a subset. So the measurement of any quantum system or state designated shortly as ‘quantum measurement’ cannot be free of the axiom of choice.
Since the Kochen – Specker theorem does not admit the axiom of choice in any quantum system or state by itself, i.e. before measurement, and quantum measurement requires it as above, quantum mechanics has to be invariant to the axiom of choice: Any statement or equation in it has to be equally valid both if the axiom of choice is accepted or not.

2. Quantum measurement and quantum invariance:

This unique relation of quantum mechanics to the axiom of choice is due to the true fundament of it, wave-particle duality and thus, to involving quantum leaps in mechanical motion. One can speak of quantum invariance underlying the above noticed invariance to the axiom of choice for quantum measurement. The wave-particle duality has another side, which can be designated as the discrete-smooth duality of mechanical motion, since there is an exact correspondence of a wave function to a quantum leap (a discrete motion) as well as of a particle to a smooth trajectory (a world line). Thus that discrete-smooth duality is a kind of generalization as to Einstein’s (1918) principle of relativity, which requires the invariance of all physical laws only to smooth relative motions (reference frames), so that it comprises already also quantum leaps.
That new and more general invariance is quantum invariance: It means that all physical laws have to be invariant to any, both discrete and continuous (smooth), transformation between two or more reference frames. The generalization pioneers the simplest pathway between general relativity and quantum mechanics, i.e. that of quantum gravity.
Any discrete motion does not allow defining a finite value of relative speed. If yet it is defined, this excludes to determine the distance of the leap. Both complementary restrictions constitute together the Heisenberg principle of uncertainty. This uncertainty is a new and unique physical variable, a free variable unboundable in principle, by a natural law such as uncertainty. Its physical dimension is action: The physical quantity of action is exceptional and singular since it is as the dimension of that unboundable free variable (in quantum mechanics) as the dimension of the corresponding bound variable (in general relativity) of the same name and dimension. Consequently what is conserved passing from each to the other theory is action in a rather extraordinary way of conservation: A dimensionless physical quantity like entropy is transformed in its reciprocal also dimensionless physical quantity like information and vice versa. Philosophically said the disorder of entropy is transformed in the order of information and vice versa conserving action as in each of both theories as between them: The theory of quantum gravity turns out to be the theory of quantum information.
Furthermore the mutual transformation between entropy and information or between disorder and order in the framework of conservation of action describes well quantum measurement and quantum invariance, and even the invariance to the axiom of choice:  Indeed quantum measurement transforms an initial fundamental disorder of coherent state into the order of a well-ordering of measured results, and quantum invariance means the equivalence of the disorder of probable values in a quantum leap and of the determined order of a smooth trajectory.

3. Quantum measurement and the foundation of mathematics:

The mutual transformation of order and disorder as the invariance to the axiom of choice deserves to be independently described for it can refer to the foundation of mathematics and thus to a new understanding how mathematics and physics are connected in reality and in quantum measurement. In fact the invariance to the axiom of choice is well known in the foundation of mathematics for a long time as the so-called paradox of Skolem (1922). He introduced the “relativity of the notion of set” meaning that any infinite set can be enumerated by the axiom of choice or even to be interpreted as a finite one. So a bridge exists from quantum measurement to a new, quantum foundation of mathematics. Its main idea is to borrow from nature the way how mathematics is founded in it generalizing the well-ordered set of natural numbers to that of qubits, which is equivalent to Hilbert space:
In other words, the idea is the founding set of natural numbers, which is always countable and thus it is noninvariant to the axiom of choice, to be replaced by the simplest one, which is invariant to the axiom of choice being uncountable and countable as a quantum coherent state before and after measurement. Such a one is Hilbert space, or the set of all well-ordered series of qubits. A point in it (or a sequence of qubits, or a wave function) can represent equally well both a coherent state before measurement and its corresponding statistical ensemble after measurement. That point is invariant to the transformation between entropy (E) and information (I) if and only if the definition of entropy and information is modified in a way to be invariant to the reciprocal transformation of their variables: . The Shannon type definition is not invariant so. However, the definition of physical quantity or observable in quantum mechanics by a selfadjoint operator is invariant just so. Consequently the latter is to be accepted as the relevant definition of information at least as to the quantum foundation of mathematics.
That mathematics founded in this way cannot involve undecidable statements since decidability can be generalized as the invariance of entropy and information to the reciprocal transformation of their variables as above. Furthermore any statement Gödel codable can be coded as a series of qubits and thus undecidable statements cannot exist in that mathematics. Furthermore it cannot be divided from physics in the bridge of quantum mechanics and thus from reality. The conception of quantum measurement serves as the base of total decidability. It leads to some kind of quantum Pythagoreanism.

4. Quantum measurement and quantum measure:

The unit of quantum measurement is specific, quite different from the classical one even as the mathematical notion of measure: It is not a Lebesgue or Borel measure. This is quantum measure, which is complete as the former and universal as the latter. It is neither arbitrarily dimensional nor one-dimensional. It is three-dimensional. Its universal unit is a qubit isomorphic to a unit ball. This is the unit of quantum information: So quantum measure serves as the quantity of quantum information of any measured. Indeed it is really universal. It can measure equally well both the disorder of entropy and the order of information, both anything, which exists being actual, and nothing, which does not exist, being virtual, only possible and probable. Consequently, quantum measure can describe the “genesis from nothing” as a process, of course not as a process in time, because that is the genesis of time: For example, the “Big Bang” is not more than a visualization of quantum measure onto the “screen” of the usual understanding of measure.