Zeroth-rank operation and non transitive numbers

Download the pdf from the website Arxiv: http://arxiv.org/abs/1205.1703

Article by G.F. Romerio in the forum “a New Kind of Science” of www.wolframscience.com

Abstract

Observing the existing relationships between the elementary operations of addition, multiplication (iteration of additions) and exponentiation (iteration of multiplications), a new operation (named incrementation) is defined, consistently with these laws and such that addition turns out to be an iteration of incrementations. Incrementation turns out to be consistent with Ackermann’s function. After defining the inverse operation of incrementation (named decrementation), we observe that R is not closed under it. So a new set of numbers is defined (named E, Escherian numbers), such that decrementation is closed on it. After defining the concept of pseudoorder (analogous to the order, but not transitive), addition and multiplication on E are analysed, and a correspondence between E and C is found. Finally, incrementation is extended to C, in such a way that decrementation is closed on C too. English keywords: hyper-operations, incrementation, zeration, Ackermann function, intransitive order, not transitive order, intransitive numbers, non transitive numbers, not transitive numbers, new number sets.

Pseudoorder of the Complex numbers