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, 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