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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.

Also see