I majored in math and my biggest problem with this is that you don't get to "do" anything infinitely many times in the math that I'm used to. In discrete contexts where infinity is used, you instead can "do" something an unbounded but finite number of times. In a continuous setting you are allowed to pick an arbitrarily large (finite) number.

In that context the first quantity that you refer to above is nonsensical because you can't "increment infinitely many times".

Secondly, I'm not sure your construction is correct, since your Infinity+1 set cannot be a singleton (it must contain all the numbers less than Infinity).