# Definition:One

This page has been identified as a candidate for refactoring of medium complexity.In particular: Merge in Definition:One (Cardinal). Also refer to the unity of a ring or field.Until this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

It has been suggested that this page or section be merged into Definition:Unit (One).To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Mergeto}}` from the code. |

## 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 Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, $\struct {S^*, \circ, \preceq}$ 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.