Math Help - On the Density of Q in R and the Uncountability of R, Countability of Q

1. 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.

2. 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.

3. This question has driven many crazy including the best of us.
Look at this page.

4. While I wouldn't consider this an argument for how Q is dense in R, I would say that thinking about how the real numbers are constructed should help you think about the issue. In particular, I recommend reading Dedekind's work on constructing the real numbers. The simple idea is that if we think about the "rational number line" we will always find gaps that are not filled by rationals (e.g., there is no rational whose square is 2). The argument is to show that the are many, many such gaps. Therefore, while your intuition regarding how we can always find a rational between two reals, you have to realize that you can find many more reals in that interval than you can rational numbers. The crux of the issue is whether or not we find for every real in between two reals, another rational. The answer is no. I say no because we will find more reals for every rational in that interval. Thinking about it in those terms, if we "drill down" in the interval, finding a rational q in that interval, and then finding a q* one one side of the split q causes of that interval, and so on, you will get more than just q, q*, q**, ... in terms of reals. For each you will get an uncountable many more numbers. This may appear question begging, because in a very strong way it is. How do we know we will find uncountably many more in those intervals? It comes back to the idea about that Dedekind construction of the reals. The stuff talked about in Plato's link is also significant. The arguments regarding countability are separate from density, but no doubt cardinality is important to discussing density. I do not believe you can make the implication you suggest. Density does not imply cardinality, at least not in the sense of defining it as is required when comparing card(Q) and card(R).

5. Originally Posted by bryangoodrich
While I wouldn't consider this an argument for how Q is dense in R, I would say that thinking about how the real numbers are constructed should help you think about the issue. In particular, I recommend reading Dedekind's work on constructing the real numbers. The simple idea is that if we think about the "rational number line" we will always find gaps that are not filled by rationals (e.g., there is no rational whose square is 2). The argument is to show that the are many, many such gaps. Therefore, while your intuition regarding how we can always find a rational between two reals, you have to realize that you can find many more reals in that interval than you can rational numbers. The crux of the issue is whether or not we find for every real in between two reals, another rational. The answer is no. I say no because we will find more reals for every rational in that interval. Thinking about it in those terms, if we "drill down" in the interval, finding a rational q in that interval, and then finding a q* one one side of the split q causes of that interval, and so on, you will get more than just q, q*, q**, ... in terms of reals. For each you will get an uncountable many more numbers. This may appear question begging, because in a very strong way it is. How do we know we will find uncountably many more in those intervals? It comes back to the idea about that Dedekind construction of the reals. The stuff talked about in Plato's link is also significant. The arguments regarding countability are separate from density, but no doubt cardinality is important to discussing density. I do not believe you can make the implication you suggest. Density does not imply cardinality, at least not in the sense of defining it as is required when comparing card(Q) and card(R).
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 $\text{card}(\mathbb{R})\leqslant 2^{2^{\aleph_0}}$.

6. Thanks for the link. I tried not to downplay the interconnection between density and cardinality too much. I figured "but no doubt cardinality is important to discussing density" would say as little as possible since I didn't want to say too much!