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I'm in a lunch group at work of recreational math geeks and we came up with a question which we need help to resolve. I apologize in advance, if my explanation is not perfectly rigorous.

Given these two statements:

  • Between any two rational number X and Y there is an irrational number
  • Between any two irrational numbers X and Y there is a rational number
  • Rational numbers are countably infinite and irrationals are uncountably infinite

Does that imply that the real numbers always alternate between the rationals and irrationals (i.e. rational -> irrational -> rational -> irrational...etc) no matter how close X and Y get?

It seems like the vastly larger number of irrationals implies that there are two irrationals that don't have a rational between them.

What can help us resolve this misunderstanding on our part?

SteveED
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  • also between any two (ir)rational numbers is a(n) (ir)rational number – J. W. Tanner May 15 '19 at 20:29
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    @SteveED The set of real numbers is not countable... Rationals and irrationals don't "alternate" because you cannot have a numbered list of real numbers. You cannot take a given rational number and identity the next real number. What the results says is that the rationals are so well mixed within the set of reals that you cannot pick any interval, no matter how small, that does not have any rational number. – PierreCarre May 15 '19 at 20:40
  • One could well-order the real numbers, under ZFC. But I don't know how rationals fit into that picture. See https://math.stackexchange.com/questions/6501/is-there-a-known-well-ordering-of-the-reals?noredirect=1&lq=1. – twnly May 15 '19 at 20:45

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To get your alternating sequence idea, you seem to be leaning on the notion that there is one solitary rational between each pair of irrationals and vice versa. This is not the case.

In fact between any pair of irrationals we have not just one rational, but a countably infinite number of them, as well as an uncountably infinite number of irrationals. The same applies to any pair of rationals.

Looking at what’s going on between each pair therefore doesn’t really get you any closer to the sort of sequence you’re talking about than you were to start with when considering the whole real line. Either way you’re staring down the barrel of a countable infinity of rationals and an uncountable infinity of irrationals, which can’t be ordered in the way that you’re trying to do.

jst345
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