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December 14th, 2015, 02:14 PM   #11
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Originally Posted by Pengkuan View Post
Yes. There is always a rational between two. I thought that would prove that rational are discrete, but alas no. Between any two irrationals there is always one irrational too, so are irrational discrete? There is no answer.
You seem very muddled.

In any non-empty 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.
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December 14th, 2015, 02:17 PM   #12
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In fact all infinite sets have aleph 0 cardinality or larger. But we do not know if there is cardinality between aleph 0 and the reals.
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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