TITLE: How can someone reject a math result if everything has to be proved?
QUESTION [52 upvotes]: I'm reading a book on axiomatic set theory, classic Set Theory: For Guided Independent Study, and at the beginning of chapter 4 it says:

So far in this book we have given the impression that sets are needed to help explain the important number systems on which so much of mathematics (and the science that exploits mathematics) is based. Dedekind's construction of the real numbers, along with the associated axioms for the reals, completes the process of putting the calculus (and much more) on a rigorous footing.

and then it says:

It is important to realize that there are schools of mathematics that would reject 'standard' real analysis and, along with it, Dedekind's work.

How is it possible that "schools of mathematics" reject standard real analysis and Dedekind's work? I don't know if I'm misinterpreting things but, how can people reject a whole branch of mathematics if everything has to be proved to be called a theorem and cannot be disproved unless a logical mistake is found?
I've even watched this video in the past: https://www.youtube.com/watch?reload=9&v=jlnBo3APRlU and this guy, who's supposed to be a teacher, says that real numbers don't exist and that they are only rational numbers. I don't know if this is a related problem but how is this possible?

REPLY [3 votes]: I believe the important point to be that real analysis as most mathematics is like a house. It is built on a foundation of axioms that create the "floorplan" of the house. The foundation is then built upon and decorated.
The current foundation is not the only possibility. Indeed over time it has changed. And it is quite possible, almost inevitable, that it will change in the future. What happens when the foundation changes is that you tear down the whole house and start over again. It has happened over time.
Now, you might believe that tearing down the foundation is destructive, and it is. But you may reach a situation where the current foundation simply does not manage to handle all the requirements, and then it is time to rebuild from bottom up.
This, tearing up the foundations, is where the different schools are. You are, as a student, being "brain washed" to accept the current main stream thinking. This is in a way good - you need to know the current state in order to build on it and change it in the future, and could be very bad as it might hinder you from thinking the really new thoughts needed for the field to go forward.
To make an analogy, look at physics (simplified). There has been a change from Newton's mechanics to Einstein's relativity to quantum mechanics and the standard model. Each of these has demolished the foundation, but sort of kept the old one as a special case. The current Physics cannot fully describe the basis of some of the real world phenomenons, as example gravity. You might believe gravity to be well known by now, but do a little search for "quantum gravity" and you end up in an a place where the foundations are being currently under review.