I do enjoy how people think “If you don’t agree with the standard treatment of infinities, you must not understand them.”
Demonstrating a few more cases of conceptually incoherent cases of “infinity” does not help your point. I do not agree with the “set theoretic use” of infinity, either — in fact, historically speaking, that one has caused more harm than in calculus.
There are no “different sizes of infinity”, either — that stems from Cantor’s paradoxical mathematics, which I’ll devote another entire piece to. I am fully aware of the standard arguments for “different sizes of infinite sets”, but as with the convergence of infinite series, it’s wrong and conceptually contradictory.
An “infinite set” is not a coherent idea — much less an “infinite amount of infinite sizes of infinite sets”. To preempt your response, yes, I am aware that rejecting the paradoxes in set theory means the foundations of mathematics need to be revisited. They do. (I am partial to Russell’s Logicism for the foundations of mathematics).