Definition:Ordinal Sum

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


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


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