Download PDFOpen PDF in browserThe Relationship of Arithmetic as Two Twin Peano Arithmetic(s) and Set Theory: a New Glance from the Theory of InformationEasyChair Preprint no. 391133 pages•Date: July 20, 2020AbstractThe 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, fourcolor theorem as well as its newformulated generalization as “fourletter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the newfounding 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 mathematicsphysics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. Keyphrases: consistent completeness of mathematics and physics unification of mathematics and physics, Hilbert arithmetic, nonstandard Peano arithmetic, Peano arithmetic, quantum information, two complimentary arithmetics
