Set of Natural Numbers Equals Union of its Successor
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Theorem
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$.
Then:
- $\bigcup \omega^+ = \omega$
Proof
We have that:
- $\omega^+ = \omega \cup \set \omega$
and so:
- $\omega \subseteq \omega^+$
$\Box$
By definition:
- $\bigcup \omega^+ = \set {x: \exists y \in \omega^+: x \in y}$
Thus:
- $x \in \bigcup \omega^+ \implies x \in \omega$
$\Box$
So by definition of set equality:
- $\bigcup \omega^+ = \omega$
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 3$ Derivation of the Peano postulates and other results: Exercise $3.3 \ \text {(b)}$