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).
Sorry, of course you're right on "Secondly". The right construction is ω, ω∪{ω}, ω∪{ω}∪{ω∪{ω}}...
For the first point, I went through the book long enough ago that I can't rebuild the proof here, but iirc the more rigorous idea is that you can construct a bijection between 1+ω and ω given the recipe I had above for how to represent numbers as sets, but you can't do it for ω+1, which is bijective with ω∪{ω}. The axiom of infinity declares that ω itself is a set, opening the door for transfinite numbers.
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).