Addenda Et Corrigenda to “The Arithmetic of the Even and the Odd”

Abstract

We correct two typos and an error in The Arithmetic of the Even and the Odd, and provide an axiom system for a weak arithmetic in which the fact that a square root of an integer is either irrational or an integer can be proved.

Another Arithmetic of the Even and the Odd

“…This fact can be seen from the model C(Q Z [X ]) of OE + < (that this is a model of OE + < has been shown in Menn & Pambuccian (2016)…”

“…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:…”

“…, and κ and μ as defined in Menn & Pambuccian (2016). As in Pambuccian (2016), we have denoted here by K D [X ] the ring of polynomials in X with free term in D and with all other coefficients in K , ordered by n i=0 c i X i > 0 if and only if c n > 0 (c 0 ∈ D, c i ∈ K for all 1 ≤ i ≤ n, with c n = 0).…”