Definition:Ordinal Multiplication
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Definition
Let $x$ and $y$ be ordinals.
The operation of ordinal multiplication $x \times y$ is defined using the Second Principle of Transfinite Recursion as follows:
Base Case
| \(\ds x \times \O\) | \(=\) | \(\ds \O\) | if $y = \O$ |
Inductive Case
| \(\ds x \times z^+\) | \(=\) | \(\ds \paren {x \times z} + x\) | if $y$ is the successor of some ordinal $z$ |
Limit Case
| \(\ds x \times y\) | \(=\) | \(\ds \bigcup_{z \mathop \in y} \paren {x \times z}\) | if $y$ is a limit ordinal |
Also denoted as
The operation $x \times y$ is also seen denoted as $x \cdot y$ or, commonly, as $x y$.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 8.15$