Let us start by clarifying this a bit.  I am aware of some proofs that irrationals/reals are uncountable.  My issue comes by way of some properties of the reals. These issues can be summed up by the combination of the following questions:


Is it true that between any two rationals one may find at least one irrational?
Is it true that between any two irrationals one may find at least one rational?
Why are the reals uncountable?


I've been talking with a friend about why the answer of these three questions can be the case when they somewhat seem to contradict each other.  I seek clarification on the subject.  Herein lies a summary of the discussion:

Person A:  By  way of Cantor Diagonalization it can be shown that the reals are uncountable.

Person B:  But is it not also the case that one may find at least one rational between any two irrationals and vice versa?

Person A:  That seems logical, I can't pose a counterexample... but why does that matter?

Person B:  Wouldn't that imply that for every irrational there is a corresponding rational?  And from this the Reals would be equivalent to 2 elements for every element of the rationals?

Person A:  That implies that the Reals are countable, but we have already shown that they weren't... where is the hole in our reasoning?

And so I pose it to you... where is the hole in our reasoning?

The problem is here: "Wouldn't that imply that for every irrational there is a corresponding rational? And from this the Reals would be equivalent to 2 elements for every element of the rationals?" The problem is that for many different pairs of irrationals you would be choosing the same rational in between. If you want to avoid this problem, you'd have to describe a procedure by means of which:


You uniquely specify for each pair of irrationals $a<b$ a rational $c$ in between.
(This is easy but non-trivial since there are infinitely many candidates for $c$.)
Different pairs get assigned different rationals.


Of course, task 2 is impossible, so there is no correspondence between irrationals and rationals.
