# On the Density of Q in R and the Uncountability of R, Countability of Q

• Jun 2nd 2011, 04:29 PM
Relmiw
On the Density of Q in R and the Uncountability of R, Countability of Q
I just finished my first course in Analysis as an undergraduate student and I've got something that I can't quite reconcile in my mind. Maybe someone can help me understand how this is possible?

We learned that regarding the set of real numbers (R) and the set of rational numbers (Q), Q is dense in R. That is, for every two real numbers, there exists a rational number between them.

We also learned R is uncountable, and Q is countable. That is, there are more real numbers than there are natural numbers.

It seems to me that the Density of Q in R implies that Q has the same cardinality as R (or has an even larger cardinality). My thinking is that if we have r1 and r2, there is a q1 between them. If we have r1, r2, r3, r4, then there must at least exist q1, q2, and q3 between them. (In fact, there could also be q4, q5, ..., qn and not violate the density of Q in R).

This has been driving me crazy lately, I can't figure out how all of these statements can be true.
• Jun 2nd 2011, 05:28 PM
Spoonwood
Density and Cardinality
Density has to do with the order structure of Q and R. If r and q are real numbers such that r $\leqslant$ q, then there exists a rational number j such that r $\leqslant$ j $\leqslant$ q. Actually, it seems clear to me that there exist "infinitely many" rationals between any two reals. We can just take (r*+q*)/2, where r* and q* indicate rationals such that r $\leqslant$ r* $\leqslant$ q* $\leqslant$ q. I've digressed, but this should make it clear that the density property concerns the ordering of the rationals and reals which talks about how the rationals and reals relate to other rationals and reals. In other words, the order property concerns elements of the sets. Countability and un-countability concern the entire sets themselves, not relations between their elements.
• Jun 2nd 2011, 05:30 PM
Plato
This question has driven many crazy including the best of us.
• Jun 2nd 2011, 08:59 PM
bryangoodrich
The above said is correct, but density and cardinality do have important interplays. There are much more genera theorems (see here for example) but you know for example that $\text{card}(\mathbb{R})\leqslant 2^{2^{\aleph_0}}$.