If you search his podcast “Patterson in Pursuit” for “infinit” there are several episodes where he goes into more depth including interviews with mathematicians who are unable to make a coherent argument about how infinite sets can exist
Discussion
There is no logical proof that an infinite set can be well defined except by circular reasoning. It’s an approximation at best
Infinite *series
As per the article, the paradox supposes that “It is necessarily impossible to complete an infinite set of tasks.”
In physical space, that may be impossible. Modern physics simply does not know.
In mathematical logic, it's certainly possible to have a system where an infinite set of tasks is accomplished. That's just a matter of definitions; math can conjure up all kinds of things.
It is not possible to have a logically consistent system where such a thing is possible. This was a big topic of debate in mathematics in the late 1800s when NonEuclidean geometries prompted mathematicians to examine the foundation mathematical more closely. They discovered that the concept of a limit was poorly defined. Unfortunately set theory won out https://youtu.be/HeQX2HjkcNo
That video is not really about this topic but it covers some of the history and is interesting non the less. Wikipedia has a page about it https://en.m.wikipedia.org/wiki/Actual_infinity
I would argue the Banach-Tarski Paradox proves that actual infinity cannot exist even as a logical or mathematical construct
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