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Why do we say the set of irrational numbers is bigger than the set of rational numbers?

I know that there are such questions like this one here.

But after looking the answer they wrote that because rational numbers are countable but irrational numbers arn't.

Now why do we say the uncountable sets are bigger than countable ones?

Also it is better to post an answer that doesn't uses being countable or uncounatblity of sets.

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Taha Akbari
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    If the irrational numbers where boys and the rational number where girls and they where to dance in pairs, at least one boy would end up with no partner, but it is possible for all girls to have a partner – Asinomás Aug 01 '16 at 16:40
  • Well the answer to your question is "rationals are countable and irrationals aren't" so it's hard to avoid using those words. Even if you avoid saying those words, the point is the same. –  Aug 01 '16 at 16:43
  • Can you say what kind of answer you would find acceptable, rather than what kind you wouldn't? The central notion is countability and one-to-one correspondence, as in @CarryonSmiling's comment. – Brian Tung Aug 01 '16 at 16:44
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    For the same reason we say that infinite sets are bigger than finite sets. – Asaf Karagila Aug 01 '16 at 16:47
  • I can think of at least three senses in which the irrationals are bigger than the rationals: their cardinality is bigger, the rationals are topologically meager while the irrationals are not, and the rationals have Lebesgue measure zero while the irrationals have Lebesgue measure infinity. Is your question really about just comparison of infinite cardinalities? – Ian Aug 01 '16 at 16:49

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A countable set $C$ is smaller than an an uncountable set $U$ in the sense that there exists no surjective map $$ C\longrightarrow U. $$ I.e. there's no way to associate to an element of $C$ (e.g. a rational number) an element of $U$ (e.g. an irrational number) in such a way to cover all of $U$.

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