Your math teacher should have known that all infinities are not equal.
There are infinite integers. There are also infinite rational numbers. But there are more rational numbers than integers. So the number of rational numbers is a greater infinity than the number of integers. But the number of rational numbers is an integer ...
And I think that line of reasoning starts to get into the crazy territory of “large cardinals” that the OP is all about.
There are exactly as many rational numbers as there are integers. Look up Cantor's diagonal argument.