Kleshev D.S. Pythagorean axioms of arithmetic: the historical roots of the second problem of Hilbert

Abstract: the article deals with an ancient theorem of the incommensurability of side and diagonal of a square, particular attention is drawn to the axiom of the indivisibility of units (μονάς), that plays a key role in the ancient Pythagorean theory of evidence of disparate segments. Subsequent development of this theory led to the formation of the theory of irrational numbers and the theory of infinite sets of Cantor. However, in modern mathematics the continuous decimal fractions are used, which were not used in Pythagorean arithmetic. The operation of the infinite division of a unit, through which the continuous use of decimal fractions was introduced, is contrary to the axiom of indivisible units. Consequently, there is an axiomatic contradiction in the grounds of the standard of mathematics, which recognizes the validity of the Pythagorean theory of incommensurability, which led to three crises in the foundations of mathematics: ancient, associated with the discovery of incommensurable line segments, the new European associated with infinitesimal greatness, modern, getting out of which, as proved by Godel, is impossible within the framework of the standard mathematics.

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