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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?

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In my experience, I use what ever notation is most simple, which doesn't get mixed up with other notation being used concurrently, and follows whichever text you choose to use for the course. If we were to not reuse notation, I think mathematics itself would collapse under the weight of it's own alphabet. One thing to always do is to define each operation and if there are other notations that can be used, then list those as well with emphasis on the one you choose to use. I strongly encourage people to avoid being pedantic with their notation, I think it distracts from our goals.

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 ($\oplus$).

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    $\begingroup$ +1 for I strongly encourage people to avoid being pedantic with their notation, I think it distracts from our goals. Related to this is something that has irritated me at times, this being someone who is overly dogmatic with personal views of "what's correct" in mathematics (especially to students) in cases where their views come from such a narrow point of view that they were not even aware of its narrowness, such as is illustrated by this semi-rant. $\endgroup$ – Dave L Renfro Aug 21 '14 at 16:14
  • $\begingroup$ BTW, the person I was responding to is actually fairly well-informed, and I actually wasn't overly bothered with what he said, but his post gave me an opportunity to launch into an essay on this pet peeve of mine. $\endgroup$ – Dave L Renfro Aug 21 '14 at 16:16
  • $\begingroup$ In looking at my comments again, it occurs to me that "the person I was responding to" might not be entirely clear. The reference is to the person I was responding to in the 23 April 2008 ap-calculus post at Math Forum, the post you get when you follow the embedded link in "this semi-rant". $\endgroup$ – Dave L Renfro Aug 21 '14 at 18:36

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