[nyc111]

"What's irrefutably proven is that if you take this particular set of axioms, then these conclusions hold."

This is what I tried to say in my comment. It's the author who talks about the truth of the axioms. I'm objecting to his claim that we end up with "something we can know for sure". No. Your truth depends on your assumptions.

[Scarblac]

We do end up with something we know for sure: the whole proposition "if we take these axioms as true, then these statements hold."