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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

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That is very vague. – Qiaochu Yuan Aug 7 '11 at 23:41
Sounds like a finitist's talk to me. – Asaf Karagila Aug 7 '11 at 23:43
It's just Wildberger at it again, not bad ideas, but it would be better if he didn't denigrate the work of others. – André Nicolas Aug 8 '11 at 1:17
@Theo, the basic idea (as I understand it) is to replace length and angle measure with quadrance and spread, where the quadrance of a line segment is the square of its length, and the spread of two crossing lines is the square of the sine of the angle they make. Wildberger shows that this makes many trig problems easier. – Gerry Myerson Aug 9 '11 at 7:05
@Gerry: Thanks for the summary. Yes, that's about as much as I was able to extract from his texts available on the web. Unfortunately, I'm so thoroughly brain-washed by the malicious guild of logicians that I'm not particularly interested in questions on (non-)existence of infinity and problems of ZF when I'm in the mood of learning some geometry... – t.b. Aug 9 '11 at 8:04
up vote 23 down vote accepted

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.

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+1 on that last paragraph!! :-) – Asaf Karagila Aug 8 '11 at 5:07
I'm somewhat sympathetic to him. The set of real numbers has the same cardinality as that of the set of all subsets of the set of natural numbers. But I wonder if an arbitrary infinite subset of the set of natural numbers exists. A concrete subset like the set of even numbers do exist. But an arbitrary infinite subset? You can't construct it in general. You just imagine it. It seems an illusion to me. – Makoto Kato Jul 4 at 23:03

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.

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You should re-print the answer here, since we aren't mind-readers. You can then also choose your own answer as the correct answer. – New Alexandria Oct 8 '12 at 4:52

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