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).