The Arithmetic of the Even and the Odd

Abstract

We present several formal theories for the arithmetic of the even and the odd, show that the irrationality of $\sqrt 2$ can be proved in one of them, that the proof must involve contradiction, and prove that the irrationality of $\sqrt {17}$ cannot be proved inside any formal theory of the even and the odd.

A Problem in Pythagorean Arithmetic

“…If d were odd, then, by (9), bearing in mind that P T (a), we would have d|b, but that would contradict the fact that P T (b). This proves (2). Suppose now a < b, P T (a) and P T (b).…”

Another Arithmetic of the Even and the Odd

“…Axiom A17 is missing in Pambuccian (2016), although it cannot be deduced from the axiom system presented there for OE + < . The statement in Pambuccian (2016), that A18 follows from {A1-A16} is incorrect.…”

“…The oldest evidence of number-theoretical thinking appears to go back to the Pythagoreans. The evidence itself, which has left traces in the studies of Plato and Aristotle, as well as in Euclid's Elements, Book IX, Propositions 21-34, has been summarized in Pambuccian (2016). The number-theoretical story focuses upon the parity of natural numbers, in which the even and the odd are the main characters.…”

“…The number-theoretical story focuses upon the parity of natural numbers, in which the even and the odd are the main characters. Of the various formal systems for the arithmetic of the even and the odd put forward in Pambuccian (2016) and Menn & Pambuccian (2016), the richest axiom system consists of the following axioms, in a language with 0, 1, +, •, <, − -in which 0 and 1 are individual constants, +, •, and − are binary operations, and < is a binary relation-as well as the unary operation • 2 , and the two binary operations κ and μ:…”

“…In a slight departure from Pambuccian (2016), let OE + < stand for the theory axiomatized by {A1-A17}, where (here, as in Pambuccian (2016), 2x stands for x + x)…”