[mopierotti]

I would say the number of items in a set, so by that logic the number of items in every type of infinite set would each seem to be infinity.

Maybe where I'm struggling is that I'm not familiar with why this notion of differently sized infinities is useful.

[jcranmer]

The clearest example is probably the diagonalization argument. Suppose you have a complete, infinite list of real numbers. You can construct a number by taking the i'th digit of the i'th number and changing it to a different digit. This number is not on the list, and yet it is still a real number.

There are four possible responses to this argument. The first is to accept that this means that there are infinite sets which have different fundamental properties (the "infinite" in a "real number has an infinite number of digits" can't be iterated the same way as the "infinite" in the "infinite number of real numbers"), and the way these differ is labeled the "size" of the infinite set. The second is to object to definition of a real number (which has other repercussions in other branches of mathematics). The third is to object to the ability to iterate over an infinite set (essentially, finitism). The final is to object to the idea of an infinite set in the first place (essentially, ultrafinitism).

The response to Cantor's proof of the uncountability of real numbers was basically for mathematicians to explore all the different responses, and ultimately, the first response is the one that is accepted by the majority of mathematicians, although some still work under models that object to the proof's correctness in some fashion.

[kccqzy]

Defining the number of items in a set requires the existence of natural numbers in the first place. (In typical set theory, people start with the existence of sets and then define natural numbers from sets.) And it doesn't help when dealing with sets that are as numerous as the natural numbers, or more.

That said it's not wrong to lump together all infinite sets and say their size is infinite. That's how third graders understand the size of a set anyways. It just isn't precise.

[throwawaymaths]

> the number of items in a set,

That's a good start. Now we need to precisely define "number of" and "in".

Suppose I put a few bonbons on a plate in front of you. How would you assign "the number" of items in the "set of bonbons on the plate in front of you"?