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