How can a set be both countable and "dense"?

Question

• A countable set is one that can be put on a one-to-one correspondence with the natural numbers. This means that you can explicitly write down a table with the ordered Naturals in one column, and each member of the countable set in the other column. You literally "count" them.

• On the other hand, an ordered set can be such that, for any two members a and b, you can find another member c that is between them. I'm not sure what the name for this property is, so I call it "dense". (If there is another term for this please clarify).

At first glance, it seems to me that both those properties are incompatible. Edit: Why I think so: If I try to enumerate them, I would always fail at the second step: $0; 1; .. $ no wait. I forgot $0.1$. So I start over: $0; 0.1; .. $ no wait, I forgot $0,01$. So I start over: $0; 0.001; .. $no wait... etc. This won't happen with the naturals: $0; 1; 2; 3; .. $, since they are not "dense".

Yet, the set of the Rationals satisfies both. It is clear that it is "dense". And this is a way to count the Rationals. I have read it and I understand it. But I still can't wrap my mind about how can a set be countable and "dense" at the same time. To me it seems contradictory. Could someone explain it to me in layman terms?

Edit 2: As the comments have shown me, my confusion was that, I stated that it was contradictory to have a set that was dense and also had "an always-increasing bijection with the naturals". But I confused that last property, with "Countable", and that was wrong. I could ask a replacement question, if that's possible: Is it correct to state that a set that is "dense", cannot have "an always-increasing bijection with the naturals"?

Answer 1

You are trying to make the bijection order preserving ,which as you have shown is not possible it is however possible to come up with a bijection which is not monotonic as mentioned in the link .

Answer 2

This should feel surprising, because it is. This is one of the first hints we encounter that infinity is a somewhat hairier concept than we might have originally hoped.

The strategy you propose for enumerating the rationals is, indeed, doomed to failure. But that is not the only such strategy - as you should know from reading the proof of the countability of the rationals. I suggest "drawing out" a successful counting strategy (e.g. the one from the proof) on a number line - this should give you some intuition about how density is reached in a countable number of steps.