Discussion
There's no requirement for math as a whole to be logically consistent. Nor is there any requirement for all things in math to be provable.
Anyway, you can't argue that infinities don't exist by making an argument about the _effect_ of their existence. All you can do is argue that they can't exist in your constrained subset of math; they still exist in less constrained types of math.
We will have to agree to disagree then. I think logical consistency is the whole point of mathematics. If your system allows you to prove that A is not A then you can literally prove anything