Successor of Omega
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Theorem
- $\omega + 1 = \set {0, 1, 2, \ldots; \omega}$
where $\omega$ is the minimally inductive set and $\omega + 1$ is the successor of $\omega$.
Note the use of the semicolon; this is the notation for multipart infinite sets.
Proof
\(\ds \omega + 1\) | \(=\) | \(\ds \omega \cup \set {\omega}\) | Definition of Successor Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {0, 1, 2, \ldots} \cup \set \omega\) | Definition of Von Neumann Construction of Natural Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {0, 1, 2, \ldots; \omega}\) | Definition of Set Union |
$\blacksquare$
Comment
It is customary to use $\omega + 1$ rather than $\omega^+$ for transfinite arithmetic.
However, it needs to be borne in mind that this is not conventional natural number addition.
For example, $\omega + 1 \ne 1 + \omega$.
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Sources
- 1979: Irving M. Copi: Symbolic Logic (5th ed.) p. $205$
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- Weisstein, Eric W. "Ordinal Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrdinalNumber.html