Ordinal Addition/Examples

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Examples of Ordinal Addition

Ordinal Addition by One

Let $x$ be an ordinal.

Let $x^+$ denote the successor of $x$.

Let $1$ denote (ordinal) one, the successor of the zero ordinal $\O$.


Then:

$x + 1 = x^+$

where $+$ denotes ordinal addition.


Ordinal Addition by Two

Let $2$ denote the successor of the ordinal $1$.


Then:

$x + 2 = x^{++}$


Ordinal Addition by Natural Number

Let $n$ be a natural number.


Then:

$x + \paren {n + 1} = \paren {x + n}^+$