Earlier quoted context omitted.
That part you quoted was interesting to me too. I remember once re-reading the incompleteness theorems - where it talks about a "finite set of axioms", it seemed there may be a loophole if we can imagine a theoretically infinite set of axioms, as a way to approach completeness. Overall I really enjoyed this article, short interviews with mathematicians and philosophers on a topic I've often thought about.
Some people, when confronted with a Gödel's Incompleteness Theorems, think "I know, I’ll use a theoretically infinite set of axioms." Now they have aleph-nought problems.
"..How about infinity plus one! Or infinite infinities!" -- Every child who learns about infinity for the first time, but also serious mathematicians in philosophical struggle with the truth. And Cantor, may the angels soothe his troubled soul.