Ordinal Addition/Examples/Ordinal Addition by Natural Number
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Theorem
Let $x$ be an ordinal.
Let $x^+$ denote the successor of $x$.
Let $n$ be a natural number.
Then:
- $x + \paren {n + 1} = \paren {x + n}^+$
where $+$ denotes ordinal addition.
Proof
\(\ds x + \paren {n + 1}\) | \(=\) | \(\ds x + n^+\) | Ordinal Addition by One | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + n}^+\) | Definition of Ordinal Addition |
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 3$ Some ordinals