[ludston]

Kurt Godel kind of threw this line of reasoning into the bin unfortunately. No system can be both complete and consistent, therefore the authors statement that the set theory he is studying is the basis of all mathematics as well as consistent is probably false.

[mymoomin]

The article is right to say that set theory can serve as a foundation for almost all other mathematics, and you're also right to say that no reasonably-complex consistent system of axioms can be complete. The resolution to this is that if you ground something (let's say topology) in e.g. ZFC (the most commonly used system of axioms for set theory) then incompleteness in ZFC maps to incompleteness in topology. Here's an example https://en.wikipedia.org/wiki/Moore_space_(topology)#Normal_... .

There are other foundations, some of which are based on things other than set theory (category theory, type theory), but they're usually equivalent to ZFC ± a few axioms, because you can embed those other foundations in some kind of set theory, and embed set theory in the other foundations.