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I do understand the proofs, I've read, for the countability of $\mathbb{Q}$ and the uncountability of $\mathbb{R}$, but I don't intuitively understand how the set of irrational numbers can be uncountable, and is therefore, intuitively speaking, a lot larger than the set of all rational numbers, yet at the same time for all real numbers there exists a rational number such that the rational number can be put between any two real numbers. If this could be done for all real numbers, how can there be way more irrational numbers than rational numbers?

eager2learn
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    Because there are also irrational numbers between any two real numbers. A lot more than rational numbers. – Daniel Fischer Feb 14 '14 at 14:40
  • But $\mathbb{I}\subseteq \mathbb{R}$, so any irrational number is also a real number, isn't it? – eager2learn Feb 14 '14 at 14:41
  • The density criterion you're talking about simply doesn't say anything informative about the cardinality of the sets (aside from "they're both infinite.") – rschwieb Feb 14 '14 at 14:42
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    Your intuition is that because of the facts quoted the numbers are alternatingly rational, irrational, rational, irrational, ... Such intuition is wrong because of the fact that there is no such thing as consecutive reals. – Hagen von Eitzen Feb 14 '14 at 14:42
  • @eager2learn Yes, rational and irrational numbers (in this context) are by definition real numbers. All real numbers are either one or the other. – rschwieb Feb 14 '14 at 14:43
  • @eager2learn here's another way to look at it: between any two irrational numbers, there are countably many rational numbers. BUT between any two rational numbers there are uncountably many irrational numbers. – rschwieb Feb 14 '14 at 14:45
  • @HagenvonEitzen ah ok I didn't think of that. Thanks, that basically answers my question. – eager2learn Feb 14 '14 at 14:45
  • Consider also the set of algebraic numbers, i.e., those numbers that are solutions to polynomials in $\Bbb Q[x]$. This set of rationals and irrationals is also countable, which means that only the set of transcendental numbers is uncountable (amongst the reals). – abiessu Feb 14 '14 at 14:54

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