- "There are infinite real numbers and infinite integers, but that does not mean that there are equal numbers of real numbers and integers"

If they are not equal they are different, and this is a comparison.

- "I say you cannot say anything about the size of sets with infinite numbers of elements except that they are infinite. You cannot compare infinities."

In the first sentence you compared the size of the 2 sets asserting that it is not equal. If you cannot compare the "size" of the 2 sets, you cannot state neither that they are different, nor that they are equal.

But cardinality

(https://en.wikipedia.org/wiki/Cardinality#:~:text=In%20mathematics%2C%20the%20cardinality%20of,has%20a%20cardinality%20of%203.)

provides a way to compare the "size" of sets with infinite numbers. And it never states that the cardinality of integers is infinity and the cardinality of real numbers is infinity + something. It just states that the cardinality of Integers is strictly less than the one of real numbers.

I keep saying that you treat infinity as a number because it seems to me (but I may have just misunderstood) that you logically do:

- |N| (cardinality of integers) = ∞

- |R| (cardinality of reals) = ∞

so you assign to cardinality this special number infinity and than conclude that |N| and |R| are not comparable. But since infinity is a concept, not a number, cardinality has to deal with the infinity concept and cannot assign a number (not even infinity) to the size. But it can define relations between them. Or, more in details, it can prove that |N| < |R| after having defined the operators =, <= and < which applies to the elements of the sets and not to the number of elements in the sets.