Proof that infinity does not come in different sizes My understanding is that infinities are undefined. They can not be defined by equations but are a mental concept. So the idea of using logic of some kind on them has problems.
Generally the problem physically exists in this form,
Brain; (Abstraction)
More specifically,
Brain; (Abstraction 1, infinite set 1)
Brain; (Abstraction 2, infinite set 2)
And a relation,
Brain; (Abstraction 3, the relation of sets)
The problem I see is that any element of an infinite set is a finite number and can be reached by finite means. So those numbers that are finite can be subject to logic because they are defined.
Infinity as undefined is off limits to logic.
If you introduce infinities into the elements then again you are using undefined terms.
I do see some logic in the OP arguments.
It seems possible to map a smaller infinity, one to one, on a larger infinity simply by freezing the larger infinity and letting the smaller one catch up.
Since we set imaginary parameters anything goes. This is not based on anything physical at all.