If you add even infinity and odd infinity you are filling in the void spaces between numbers.
But with that stupid number line thingy If you count as far as you can in a positive direction you count to infinity. If you count as far as you can with negative numbers you can count to infinity.
If you add them both together you get either double infinity or zero which is the opposite of infinity.
That got me sent to the office.
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.