Naturally Ordered Semigroup Exists

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, 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.

Also defined as
Some sources accept as axiomatic the naturally ordered semigroup itself.