# Definition: Set-theoretic Definition of Order Relation for Natural Numbers

Let $$m,n\in\mathbb N$$ be any given natural numbers, using their von Neumann set-theoretic representation. Then we call:

• $$m$$ smaller or equal $$n$$ - denoted by $$m \le n$$, if and only if the set-theoretic representation of $$m$$ is a subset of the set-theoretic representation of $$n$$,
• $$m$$ smaller $$n$$ - denoted by $$m < n$$, if and only if the set-theoretic representation of $$m$$ is a proper subset of the set-theoretic representation of $$n$$,
• $$m$$ greater or equal $$n$$ - denoted by $$m \ge n$$, if and only if the set-theoretic representation of $$n$$ is a subset of the set-theoretic representation of $$m$$,
• $$m$$ greater $$n$$ - denoted by $$m > n$$, if and only if the set-theoretic representation of $$n$$ is a proper subset of the set-theoretic representation of $$m$$.

Definitions: 1
Proofs: 2

Github: ### References

#### Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011