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?
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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.
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.