Explanation why irrationals are uncountable? I am a beginner and any help would be greatly appreciated.

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barene55d · Accepted Answer

Let&#039;s push your intuition a little further. Because there is a rational number between every two irrational numbers, perhaps your intuition tells you that maybe there should be an injective function       R    ∖      Q    ×      R    ∖      Q    →      Q   which sends every pair of irrational numbers   (  α  ,  β  ) to some rational number between them. Certainly if such a function existed then       R    ∖      Q   would be countable.But no such function can exist, and you can prove it by adapting Cantor&#039;s diagonal argument. Suppose that a function   f  :      R    ∖      Q    ×      R    ∖      Q    →      Q   of the sort described above exists, and let   A  ⊆      Q   denote the range of       f          −      1        (  A  ) (which we know to be       R    ∖      Q    ×      R    ∖      Q  ) is countable since   A is countable and   f is injective, so there is an enumeration   (      α    n    ,      β    n    ) of       f          −      1        (  A  ). Arrive at a contradiction by constructing   (  α  ,  β  ) such that the   nth digit of   α is different from the   nth digit of       α    n   (  β can be anything).To summarize the argument, the problem with your intuition that the order density of the rational numbers should imply the countability of the irrationals is that to &quot;fill the gap&quot; between every pair of irrational numbers with a rational number you have to reuse many rational numbers over and over.

Explanation why irrationals are uncountable? I am a beginner and any help would be greatly appreciated.

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