This is pop-sci drivel.

These aren't mathematicians who "don't believe in infinity". They're just trying to come up with yet another mathematical model of sets. This time one with one less axiom, the axiom allowing for infinite sets.

Infinity – the mathematical concept – clearly exists and no sane mathematician would ever deny that.

The sane, correct, way to write this article would be to say they're mathematicians trying to figure out how to build a useful mathematical framework without infinity. Title it "Mathematics Without Infinity" or something.

God damn journalists...