I don’t understand, and I hope it’s just bad writing.

Certainly you can build a branch of mathematics without an axiom of infinity, and that’s fine, it’s math over finite sets.

However, an axiom of infinity is independent, it doesn’t contradict anything in standard formalizations, and so it doesn’t make sense to say “infinity is wrong”.

He may think the axiom of infinity isn’t satisfied by our real physical world, but that’s not a math question! There’s nothing logically inconsistent about infinite sets nor their axiomatizations.

> But in the late 1800s, Georg Cantor and other mathematicians showed that the infinite really can exist.

I think, as I understand it, the objection is this. The proposition that infinity is "real", and there are actually infinite (not just very many) things.