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December 14th, 2015, 02:14 PM  #11  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra  Quote:
In any nonempty interval there are infinitely many ($\aleph_0$) rationals and infinitely many ($\mathfrak c$) reals. It doesn't matter whether the endpoints are rational, irrational or one of each.  
December 14th, 2015, 02:17 PM  #12 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,355 Thanks: 2469 Math Focus: Mainly analysis and algebra  Yes, this is very basic stuff. You should also be aware that we cannot prove that $\mathfrak c=\aleph_1$. That is: it has been proved that we cannot prove it. It is formally undecidable.


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cardinality, continuous, continuum, continuum hypothesis, hypothesis, set 
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