Modern Mathematics having serious problems with Real Numbers? While watching this N. Wildberger video, at 12:34 it is mentioned that Modern Mathematics has serious problems with real numbers and that Mathematicians are aware of it.
Can anyone point to what are the problems that he is refering to?
Thank you
 A: If you know about countable and uncountable infinities, consider the following problem: 
Is there a subset of the reals whose cardinality is strictly between that of the integers and that of the reals? 
Cantor's Continuum Hypothesis says the answer is "No". Godel and Cohen proved that one can neither prove nor disprove the Continuum Hypothesis on the basis of the usual axioms of set theory (ZFC). Some people consider this a serious problem; if we really know what the reals are, we should be able to decide whether or not there's a set bigger than the integers but smaller than the reals. Other people shrug their shoulders and get on with doing mathematics. 
If you don't know about countable and uncountable infinities and such, the above won't mean much to you, but then you have some very nice experiences waiting for you. 
A: Just realised the same video is also available via a general compilation page.
My question is in first comments, and is answered by the presenter. 
Googling "Wildberger set theory" brings up the refrences I was after.
A: The biggest problem with real numbers is the fact that almost all of them cannot be defined, because by definition of the concept of definition, all definitions must be finite. However, the set of numbers having a finite definition cannot have more elements than the set of all finite strings, and the set of finite strings is countable. Because the set of real numbers is not countable, almost all reals are not definable:
For almost all real numbers, there exists no finite definition.
π can be defined as the area of the unit circle. That is a pretty short definition. Even though π is not an algebraic number (a root of a polynomial of finite degree with integer coefficients, which is also a set of countably many elements) and therefore a transcendental number, it is a definable number, i.e. a finite definition exists. π is also a computable number. A real number r is computable when there exists a computation procedure defined by a finite string such that for any positive rational number p, however small, that procedure can be executed in a finite amount of time to find a rational approximation q of that real number r such that:
|r-q| < p.
Every computable number is defined by the finite computation procedure, therefore every computable real is definable. There exist definable numbers that are not computable. A stronger formulation of non-computability is non-approachability. In the following article a real number is defined that is shown to be not-approachable by any approximating procedure in the form of a Turing Machine.
https://arxiv.org/pdf/1003.0480.pdf
