Recently I stumbled upon the website of Prof. N. J. Wildberger, who has an written a thought-provoking article about set theory and the real numbers. In his opinion, real numbers are actually "a joke". An excerpt, where he talks about the definition of the real numbers as equvalence classes of Cauchy secquences:
(...) But here is a very important point: we are not obliged, in modern mathematics, to actually have a rule or algorithm that specifies the sequence. In other words, 'arbitrary' sequences are allowed, as long as they have the Cauchy convergence property. This removes the obligation to specify concretely the objects which you are talking about. Sequences generated by algorithms can be specified by those algorithms, but what possibly could it mean to discuss a 'sequence' which is not generated by such a finite rule? Such an object would contain an 'infinite amount' of information, and there are no concrete examples of such things in the known universe. This is metaphysics masquerading as mathematics. (...)
For me this actually makes some sense. Why are opinions like this so rarely seen? Shouldn't we take Wildberger's arguments seriously? Are there really flaws in the foundations of mathematics?
EDIT #1: I think this question is not a duplicate (see my statement in the comments section).
EDIT #2: To clarify my question a bit: If I understand Wildberger correctly, he sees a problem in handling objects that contain an infinite amount of information (because no human being can fully understand such objects), like equivalence classes of Cauchy sequences. (He does not have a problem with infinite sets in general as I remember a video lecture of him where he says that he believes in the rational numbers). My question is: Can this (talking about objects that contain an infinite amount of information) be problematic for the science of mathematics? Or is it just a "believe-it-or-not-thing" like the Axiom Of Choice? I'm not looking for an answer like "Wildberger is right/wrong", but rather for a simple explaination of the key points, as I have the feeling of lacking a bit of advanced knowledge that is required to fully understand this topic, or to fully understand Wildbergers arguments. I've read the answers to the question this one is a possible duplicate of, but as far as I've understood them (I did not understand everything), they do not adress my question.