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