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.

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