The Relationship of Arithmetic As Two Twin Peano Arithmetic(s) and Set Theory: A New Glance From the Theory of Information

Metaphilosophy eJournal (Elseviers: SSRN) 12 (10):1-33 (2020)
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Abstract

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.

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Vasil Penchev
Bulgarian Academy of Sciences

Citations of this work

Both Classical & Quantum Information; Both Bit & Qubit: Both Physical & Transcendental Time.Vasil Penchev - 2021 - Philosophy of Science eJournal (Elsevier: SSRN) 14 (22):1-24.

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References found in this work

A Mathematical Theory of Communication.Claude Elwood Shannon - 1948 - Bell System Technical Journal 27 (April 1924):379–423.
Was Sind und was Sollen Die Zahlen?Richard Dedekind - 1888 - Cambridge University Press.
Neuer Beweis für die Möglichkeit einer Wohlordnung.Ernst Zermelo - 1907 - Mathematische Annalen 65 (1):107-128.

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