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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