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Definition: Settheoretic Definition of Order Relation for Natural Numbers
Let \(m,n\in\mathbb N\) be any given natural numbers, using their von Neumann settheoretic representation.
Then we call:
 \(m\) smaller or equal \(n\)  denoted by \(m \le n\), if and only if the settheoretic representation of \(m\) is a subset of the settheoretic representation of \(n\),
 \(m\) smaller \(n\)  denoted by \(m < n\), if and only if the settheoretic representation of \(m\) is a proper subset of the settheoretic representation of \(n\),
 \(m\) greater or equal \(n\)  denoted by \(m \ge n\), if and only if the settheoretic representation of \(n\) is a subset of the settheoretic representation of \(m\),
 \(m\) greater \(n\)  denoted by \(m > n\), if and only if the settheoretic representation of \(n\) is a proper subset of the settheoretic representation of \(m\).
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Definitions: 1
Proofs: 2
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References
Bibliography
 Hoffmann, Dirk W.: "Grenzen der Mathematik  Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011