# Naturally Ordered Semigroup Exists

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## Theorem

Let the Zermelo-Fraenkel axioms be accepted as axiomatic.

Then there exists a **naturally ordered semigroup**.

## Proof

We take as axiomatic the Zermelo-Fraenkel axioms.

From these, Minimally Inductive Set Exists is demonstrated.

This proves the existence of a minimally inductive set.

Then we have that the Minimally Inductive Set forms Peano Structure.

It follows that the existence of a Peano structure depends upon the existence of such a minimally inductive set.

Then we have that a Naturally Ordered Semigroup forms Peano Structure.

Hence the result.

$\blacksquare$

## Also defined as

Some sources accept as axiomatic the **naturally ordered semigroup** itself.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers