en.wikipedia.org/wiki/Cardinality
The word “cardinality” does not mean ONE thing when it comes to finite sets and ANOTHER when it comes to infinite sets. It represents the same thing in both cases: the number of elements in a set.
That’s not a definition of the word “cardinality”. At best, what you’re doing here is defining what the term “same cardinality” means. You’re merely stating that if there’s a bijection between two finite or infinite sets that it logically follows that the two sets have the same cardinality i.e. the same number of elements. I don’t argue against that. But that’s not a definition of the word “cardinality” and it’s certainly not in opposition to the definition that I put forward.
If (x) is greater than (y) then it logically follows that (x) is not equal to (y).
Similarly, if (x) is less than (y) then it logically follows that (x) is not equal to (y).
Thus, to say that (x) is at the same time 1) equal to (y), 2) greater than (y), and 3) less than (y) is to say that (x) is at the same time 1) equal to (y), 2) not equal to (y), and 3) not equal to (y). In other words, it is to say that (x) is both equal to and not equal to (y) which is a textbook example of logical contradiction.
Let’s take two infinite sets:
(A = {0, 1, 2, 3, \dotso}) and (B = {1, 2, 3, \dotso}).
The question we want to ask is:
Do they have the same cardinality i.e. do they have the same number of elements?
What you’re saying is, and what I agree with, is that if we can put the two sets in one-to-one correspondence that it logically follows that they are equal in size.
The problem is that we cannot logically derive the answer to this question. In other words, there is absolutely nothing so far (no stated premise) that let us logically derive the answer to the question “Can we put them in one-to-one correspodence?”. We cannot answer with “Yes” just as we cannot answer with “No”. The answer is indeterminate.
It’s akin to saying that (x) and (y) are two positive integers and asking whether their sum will also be a positive integer. The only answer we can give is “It may or may not be the case”. In other words, the answer is indeterminate.
But you are trying to tell us that just because we can DECLARE that they are equal in size that it’s a LOGICAL NECESSITY that they are equal in size.
And that’s non-sense.
You can DECLARE that they are equal in size. Like so:
0 ↔ 1
1 ↔ 2
2 ↔ 3
3 ↔ 4
etc
But you can also DECLARE that they are not equal in size. Like so:
0 ↔ -
1 ↔ 1
2 ↔ -
3 ↔ 2
4 ↔ -
etc
It’s an ARBITRARY DECISION and there’s absolutely NO SENSE in insisting that an arbitrary decision be treated as some kind of logical necessity.
Note that there is nothing wrong with arbitrary decisions so as long they don’t contradict prior statements.
But when people say that (B) is an infinite set produced by making a copy of (A) and removing (0) from it, this logically implies that the set (B) is smaller than set (A). In such a case, you are NOT free to declare them equal because that would be a logical contradiction.
But one always has the choice to avoid facing the fact that they contradicted themselves by redefining words. If you claim that (2 + 2 = 2) and someone comes along and proves you wrong, you can hide your shame by saying “Hehehe, what I meant by (2) is actually (0), it’s just a different language, you see, so I’m still right!” And if that new language of yours becomes the mainstream language at the time, you can also hide behind that by saying “(2) means (0), dude, everyone agrees with that! Stop making up your own definitions!” And to those who say that it’s either one or the other, you can say “People should stop saying that 2 + 2 = 2 is either true or false because we can always imagine a language in which it is false and a language in which it is true”. It’s a very useful thing to remember if you’re into avoiding the negative consequences of your choices.