# Definition:One

## 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.

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.