User blog:Edwin Shade/A Question Concerning Cardinal Infinities

So I accept the idea that you can count an indefinite number of objects and then count past them, using ordinal infinities, because it makes sense to me. But I have a problem with the idea of infinities literally "bigger" than other infinities.

Specifically, I have an issue with Cantor's diagonal argument. It asserts that the set of real numbers is uncountable and hence "bigger" than the set of whole numbers because if you tried to list out the real numbers, you would always find a real number not in that list, and hence you can't list them.

This logic seems flawed in my opinion, for two reasons. The first is that we could say the same thing about the set of integers. Write out a list of whole numbers. Then multiply together together all the numbers in this list and add 1. You will always end up with a new whole number that wasn't in the list, hence aren't the whole numbers also uncountable ?

Also, you would need an infinitely long list to enumerate both the reals and the integers, so does it really make sense to distinguish the size of these sets just by the way they are constructed ? Eventually you get infinitely long lists, so how can you say anything meaningful about these lists other than that they go on forever ?

I also fail to see how taking the power set of the reals yields a higher type of cardinal infinity than the reals itself. This just seems like adding one to infinity to me, it's still equal to infinity in my opinion.

Well, I trust mathematics hasn't put their trust in something false for almost a century, so I'm going to listen to what you have to say, but I might have some questions, because I honestly don't understand if the idea of cardinal infinities is something people invented, or whether it can be derived from the basic laws of math.