Definition:Ordering on Natural Numbers/Von Neumann Construction

Definition

Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction.

The strict ordering of $\omega$ is the relation $<$ defined by:

$\forall m, n \in \omega: m < n \iff m \subsetneqq n$

The (weak) ordering of $\omega$ is the relation $\le$ defined by:

$\forall m, n \in \omega: m \le n \iff m \subseteq n$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Definition $5.1$