# Definition:Ordinal Sum

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## Definition

Let $y$ be an ordinal.

Let $\left\langle{A_x}\right\rangle$ be a sequence of ordinals.

The **ordinal sum of $A_x$** is denoted $\displaystyle \sum_{x \mathop = 1}^y A_x$ and defined using Transfinite Recursion on $y$ as follows:

- $\displaystyle \sum_{x \mathop = 1}^\varnothing A_x = \varnothing$
- $\displaystyle \sum_{x \mathop = 1}^{z^+} A_x = \sum_{x \mathop = 1}^z \left({A_x}\right) + A_{z^+}$
- $\displaystyle \sum_{x \mathop = 1}^y A_x = \bigcup_{z \mathop \in y} \left({\sum_{x \mathop = 1}^z A_x}\right)$ for limit ordinals $y$.