Can ordinals multiplication be defined by recursion?

Question

I am trying to figure out weather ordinals multiplication be defined by transfinite recursion. It seems a bit problematic to me since multiplication is a function $f: \alpha \times \beta \rightarrow \gamma$.

What do you think?

Thank you,

Answer 1

It certainly can. Take any ordinal $\alpha$. Then we define $\alpha\times\beta$ by recursion as follows:

• $\alpha\cdot 0=0$
• $\alpha\cdot(\beta+1)=(\alpha\cdot\beta)+\beta$
• For limit $\beta,$ $\alpha\cdot\beta=\sup\{\alpha\cdot\gamma:\gamma<\beta\}$

We can also define ordinal addition and exponentiation either recursively or explicitly, depending on our preference, and on which properties are made simpler to prove.