Definition:Ordinal Sum

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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 the Second Principle of 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 $y$.