Here's a quote from the paper, which I think shows how confused Wildberger is about axioms:
"In ordinary mathematics, statements are either true, false, or they don't make sense. If you have an elaborate theory of 'hierarchies upon hierarchies of infinite sets', in which you cannot even in principle decide if there is something between the first and second 'infinity' on your list, there's a time to admit you are no longer doing mathematics."
Of course, the statement about not being able to decide if there is a cardinal between aleph_0 and aleph_1 is absurd, but leave that aside and look instead at the statement that there is a cardinal between aleph_0 and 2^aleph_0. How does this differ from the statement that the sum of the angles of a triangle cannot be determined in absolute geometry?
Suppose, when faced with the fact that not everyone found the parallel postulate to be intuituively true, the Greek geometers had simply removed it from consideration. They could then be attacked with the sneer that they were not doing mathematics at all, because they could not answer so basic a question as whether the sum of the angles of a triangle was less than, equal to, or greater than two right angles. Yet I think it is clear they *would* be doing geometry. For that matter a group theorist who cannot tell you if a generic group is abelian or nonabelian, since it might be either, is not failing to do mathematics.
The difference is that we've given up on the idea that there is a single correct geometry, but still feel (and that's nothing but intuition speaking) that there is a single true set theory, just as we think there is a single true number theory. But our intution is not strong enough to settle all the questions as to what this true set theory actually is. This is really a meta problem for mathematics, and not a question which can allow a person to conclude that set theorists are not mathematicians, any more than the existence of nonstandard models for first-order arithmetic would prove number theorists are not mathematicians. As to rigor, the mechanically verified proofs of the Mizar project are more rigorous than mere mortals like you, me, or Wildberger do, and they are based on set theory with (if needed) inacessible cardinals. The rigor argument is clearly therefore baloney, and Wildberger cannot seem to separate mathematics from metaphysics for the rest of it.