\begin{definition}[Definition:Ordinal Sum]
Let $y$ be an ordinal.
Let $\sequence {A_x}$ be a sequence of ordinals.
The '''ordinal sum of $A_x$''' is denoted $\ds \sum_{x \mathop = 1}^y A_x$ and defined using Transfinite Recursion on $y$ as follows:
:$\ds \sum_{x \mathop = 1}^\O A_x = \O$
:$\ds \sum_{x \mathop = 1}^{z^+} A_x = \sum_{x \mathop = 1}^z \paren {A_x} + A_{z^+}$
:$\ds \sum_{x \mathop = 1}^y A_x = \bigcup_{z \mathop \in y} \paren {\sum_{x \mathop = 1}^z A_x}$ for limit ordinals $y$.
{{NoSources}}
Category:Definitions/Ordinal Arithmetic
\end{definition}