# On using different notations for the same objects

Historically, in set theory we use two different notations to refer set theoretically same objects $\aleph_{\alpha}$ and $\omega_{\alpha}$. The folklore justification of this dual notation is that we use $\aleph_{\alpha}$ and $\omega_{\alpha}$ when we are dealing with cardinal and ordinal arithmetic respectively in order to avoid any confusion.

This method of using notations seems strange because one can define several structures (arithmetics) on a given set/class (in this case we are working in the proper class $Card$) and then using these different structures we can look at its objects from essentially different point of views. One should note that this difference comes from the structure not the objects in these structural contexts.

In very few set theory texts I saw an alternative approach which uses a common notation for both ordinal and cardinal numbers and emphasizes on their different arithmetics by means of using different notations for cardinal and ordinal arithmetic. For example: $\omega_1 +\omega_0$ means ordinal sum and $\omega_1\oplus\omega_0$ means cardinal sum.

As an example from ordinary mathematics consider three different summation operators $+$, $\oplus$, $\boxplus$ on the set of natural numbers $\mathbb{N}$. In the first approach we should use three different notations $\mathsf{n}$, $\mathfrak{n}$, $\mathcal{n}$ to refer natural numbers with respect to the context and in the second approach we just need to use different symbols $+$, $\oplus$, $\boxplus$ on $\mathbb{N}$ and show each natural number in a uniform way.

Question: Which approach is better to teach? What would be (dis)advantages of each method in my particular example in set theory?

For instance, I can't think of a situation where you would add a cardinal number to an ordinal number, so using regular addition (+) for both situations should be sufficient. Another example is finite direct products and finite direct sums which in this sense the same, but in linear algebra can get confused with inner direct products as they use the same notation ($\oplus$).