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

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