# Definition of ordinal multiplication

The ordinal multiplication $$\cdot$$ can be defined recursively via ordinal addition $$+$$ for any ordinal $$\alpha$$ as follows:

• $$\alpha\cdot 0=0$$.
• $$\alpha\cdot (\beta+1)=\alpha\cdot \beta+\alpha$$ for any ordinal $$\beta$$.
• $$\alpha\cdot \lambda=\bigcup \{\alpha\cdot\beta: \beta\in \lambda\}$$ for any limit ordinal $$\lambda$$.

According to this widely accepted definition, $$\alpha\cdot \beta$$ can be informally interpreted as the "length" of the concatenation of $$\beta$$ copies of $$\alpha$$. For instance,

• $$\omega\cdot 2$$ is the length of the concatenation of 2 copies of $$\omega$$, which is strictly larger than $$\omega$$, and
• $$2\cdot \omega$$ is the length of the concatenation of $$\omega$$ copies of $$2=\{0,1\}$$, which is still equal to $$\omega$$.

Since $$\cdot$$ is not commutative as can be seen from the above example, $$\alpha\cdot \beta$$ cannot be interpreted as the "length" of the concatenation of $$\alpha$$ copies of $$\beta$$. However, I personally think that "the length of $$\alpha$$ copies of $$\beta$$" would have been a more natural way to informally interpret $$\alpha\cdot\beta$$. Therefore, I wonder if there are any historical reasons for this widely accepted definition instead of defining $$\cdot$$ the other way around, i.e., for any ordinal $$\beta$$

• $$0\cdot \beta=0$$.
• $$(\alpha+1)\cdot \beta=\alpha\cdot \beta+\beta$$ for any ordinal $$\alpha$$.
• $$\lambda\cdot \beta=\bigcup \{\alpha\cdot\beta: \alpha\in \lambda\}$$ for any limit ordinal $$\lambda$$.
• I think it best to write "sup" and not "$\bigcup$" in there. In case we do ordinals by some method other than that of von Neumann. Jun 5, 2021 at 16:50

Why? My guess: When defining $$\alpha + \beta$$ and $$\alpha^\beta$$ we do it by induction in the second variable. So to be consistent, do it that way also for the definition of $$\alpha\beta$$.