# 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. – Gerald Edgar Jun 5 at 16:50

## 1 Answer

If you look at Cantor's writing you will find the same idea. He first defined multiplication the way you suggest. But later he switched to the definition we use today. Reference:
Joseph W. Dauben. Georg Cantor, his mathematics and philosophy of the infinite. Harvard University Press, 1979

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 $$\alpha\beta$$.