Why are the ordinal operations defined that way?

Question

If $\mu$ is a limit ordinal, then we define that $$\alpha + \mu = \sup_{\beta < \mu} (\alpha + \beta).$$

Why don't we similarly define that if $\lambda$ is a limit ordinal, then $$\lambda + \beta = \sup_{\alpha < \lambda} (\alpha + \beta)?$$

Or if $\lambda$ and $\mu$ are limit ordinals, then

$$\lambda + \mu = \sup_{\alpha < \lambda, \beta < \mu} (\alpha + \beta)?$$

The same goes for ordinal multiplication and exponentiation - why are they defined that way?

Answer 1

If you defined

$$\lambda + \beta = \sup\limits_{\alpha < \lambda} (\alpha + \beta)$$

then you would have $\lambda + \beta = \lambda$ for all $\beta \leqslant \lambda$. In particular, no limit ordinal would have a successor.

Answer 2

Ordinal addition is really just concatenation of orders, then computing the order type. Because concatenation of well-orders is a well-order, there is a unique ordinal isomorphic to this concatenation. So everything is well-defined.

At limit points we want to have continuity, in the sense that addition and supremums commute, after all if we fix $\alpha$, then concatenating longer and longer ordinals should have the same result (at the limit stage) as concatenating the limit itself.

Remember that we first have the order structure, and we want the arithmetical operations to work with that.