Definition:One

From ProofWiki
Jump to: navigation, search

Next

Definition

The immediate successor element of zero in the set of natural numbers $\N$ is called one and has the symbol $1$.


Naturally Ordered Semigroup

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $S^*$ be the zero complement of $S$.

By Zero Complement is Not Empty, $S^*$ is not empty.

Therefore, by axiom $(NO4)$, it has a smallest element for $\preceq$.


This smallest element is called one and denoted $1$.


Historical Note

The ancient Greeks did not consider $1$ to be a number.

According to the Pythagoreans, the number One ($1$) was the Generator of all Numbers: the omnipotent One.

It represented reason, for reason could generate only $1$ self-evident body of truth.


While a number, according to Euclid, was an aggregate of units, a unit was not considered to be an aggregate of itself.

The much-quoted statement of Jakob Köbel might as well be repeated here:

Wherefrom thou understandest that $1$ is no number but it is a generatrix beginning and foundation for all other numbers.
-- $1537$

illustrating that this mindset still held sway as late as the $16$th century.


The ancient Greeks considered $1$ as both odd and even by fallacious reasoning.


Also see