Definition:Ordinal Sum

Definition
Let $y$ be an ordinal and let $\langle A_x \rangle$ be a sequence of ordinals.


 * $\displaystyle \sum_{x \mathop = 1}^y A_x$ shall be defined using Transfinite Recursion on $y$


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