Definition:Ordinal Addition
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Definition
Let $x$ and $y$ be ordinals.
The operation of ordinal addition $x + y$ is defined using the Second Principle of Transfinite Recursion on $y$, as follows.
Base Case
When $y = \O$, define:
- $x + \O := x$
Inductive Case
For a successor ordinal $y^+$, define:
- $x + y^+ := \paren {x + y}^+$
Limit Case
Let $y$ be a limit ordinal. Then:
- $\ds x + y := \bigcup_{z \mathop \in y} \paren {x + z}$
Examples
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:
- $x + \paren {n + 1} = \paren {x + n}^+$
Also see
- Results about ordinal addition can be found here.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.1$