Wildberger says that the axioms of set theory basically are ad hoc, constructed first by Cantor to model the real numbers, then found paradoxical. Next, axioms are presented that (as far as I understand) allow the same set theoretic functions of Cantor to mostly remain in tact but with consistency.

Wildberger's arguments are that we outsourced the foundations and specification of axioms to fulfill a higher level (less fundamental) goal, i.e. giving the real numbers a strong base logically. He also claims that if we were more careful with our axioms math could be more vibrant and less convoluted.

Early in the video he suggests that natural numbers form a better base for mathematics, and that set theory should be part of combinatorics, which itself floats up from natural numbers.

Discuss.