Question

Question about the strange nature of the cardinality of the irrational numbers and the rational numbers:

It seems that intuition fails here. It's true that between every two distinct irrational numbers there exists a rational number. This almost makes it seem like they should have the same cardinality. It's been proven that they don't. So, does anyone have an intuitive way of explaining that? I think that perhaps the intuition just breaks down when it extends to the infinite like it does here, but I'd still like to try and search for a more satisfying explanation.

Answer 1

There is an easy and very intuitive explanation.
http://www.proofwiki.org/wiki/Real_Numbers_are_Uncountable#Cantor.27s_Diagonal_Argument

Answer 2

The problem is not that I don't agree that the real numbers are uncountable. What I am confused about is where the intuitive argument I described above breaks down: it's true that there is a rational number between any two given real numbers. But the order of the real numbers is greater than the order of rational numbers (not just twice as many, as you might guess by knowing that there is one rational between any two reals). This to me is very weird, but also true, so I want to figure out what goes wrong in this intuitive approach.

Thanks for the link though. I'd forgotten exactly how that argument goes.

Answer 3

rational no is countable and irrational no is uncountable

Answer 4

You have not read my question...

Answer 5

there is not ony one rational between any two reals, there are infinitely many of them. Same for oposit situation. I agree about the intuitive aproach failure.

Answer 6

"between every two distinct irrational numbers there exists a rational number"

remember the irrationals are uncountable so you can't get a hold on every single pair of irrational numbers to put a rational between them.