# Naturally Ordered Semigroup Exists

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## Theorem

There exists a Naturally Ordered Semigroup.

## Proof

We take as axiomatic the Zermelo-Fraenkel axioms.

From these, Existence of Minimal Infinite Successor Set is demonstrated.

This proves the existence of a minimal infinite successor set.

Then we have that the Minimal Infinite Successor Set forms Peano Structure.

It follows that the existence of a Peano structure depends upon the existence of such a minimal infinite successor set.

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

Hence the result.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 16$