Density of $\mathbb{Q}$ in $\mathbb{R}$ and countability I don't understand why the density of $\mathbb{Q}$ in $\mathbb{R}$ does not contradict the fact that $\mathbb{Q}$ is countable and $\mathbb{R}$ is not.
There is always a rational number $r$ between two irrational $i,j$ numbers, why does this not imply that there are as many rationals as irrationals? I mean we could just take for every irrational $i$ the nearest greater rational $r$ and count it this way? By choosing the nearest greater rational, we have a condition that there is only one rational for one irrational.
I don't understand why this does not contradict fact that $\mathbb{R}$ is uncountable. Is there a good intuitive explanation why we can not?
(Please not just a proof why $\mathbb{R}$ is uncountable and $\mathbb{Q}$ is countable)
 A: You write:

By choosing the nearest greater rational,

In response to Ian's observation that there is never such a rational, you further write:

between every irrational number there is a rational number. Why does this not imply that there should theoretically exist always a nearest greater rational

Let me address this issue specifically, since I think this is the main source of your confusion.
The rationals are dense in the reals - this means that between any two distinct real numbers, there is a rational number. This immediately means that there is never a "nearest greater rational": if $r$ is a real number and $q>r$ is rational, then by density there is some $t\in (r, q)$ which is rational. Intuitively, the rationals let us walk "closer and closer" to any given real without ever reaching it.
To see a concrete example of this, think about $\pi$. There's a natural sequence of rational numbers greater than $\pi$ which "walks towards" $\pi$; namely, $$4, \quad3.2, \quad3.15, \quad3.142, ...$$
Clearly there is no term in this sequence which is closer to $\pi$ than any other term: no matter how close the $n$th term is, the $(n+1)$th term is closer still. In fact, it can be shown that this sequence walks as close to $\pi$ as possible: if $q>\pi$ is rational,then there is some term in the sequence above which is between $\pi$ and $q$.

This shows that your specific attempt to build a bijection between $\mathbb{Q}$ and $\mathbb{R}$ won't work. Cantor's theorem meanwhile proves that no such bijection exists at all, but it often leaves doubt in the reader's mind that there isn't a contradiction somewhere in the very way we talk about sizes of infinity. So you may still ask: is there a snappy reason why you shouldn't believe that the density of $\mathbb{Q}$ in $\mathbb{R}$ means that they have the same cardinality? My response to that is twofold. 
First, Cantor's argument is intuitive; it's just that it appeals to an intuition you may not have yet developed. Intuition, in my opinion, is learned more often than not, and the trick to understand something initially unintuitive is to find the right vantage point to let it convince you. For Cantor's theorem (and in general, in my  opinion) I think the right approach is in terms of games. You and I are playing the following game: I bring you a list of real numbers, and you try to produce a real not on the list. If the reals are countable, then I clearly have a winning strategy; conversely, if you have a winning strategy, then the reals must be uncountable.
Cantor's argument, then, is describing a winning strategy for you. By contrast, your intuition that density implies largeness is vaguely hinting at the reasonability of a winning strategy for me. The clarity of Cantor's strategy balanced against the vagueness of the density strategy should suggest not only that the former is reasonable but also that the latter may not be as well justified as you think.
Following this doubt takes us to my second response, and this is a point about infinite sets in general: reasoning about infinite sets from a set-theoretic perspective is fundamentally different from thinking of them "geometrically." E.g. Galileo observed that two line segments have the same cardinality, regardless of their length, thus divorcing ideas about "measurement" in the geometric sense and cardinality (= measurement in the set-theoretic sense). This is just a more involved instance of the same thing: that when you're talking about cardinality, you can only safely draw intuition from ideas about actual functions - surjections, injections, and bijections - and not from the geometric nature of the sets you're analyzing. Indeed, note that while density suggests to you (and plenty of other people) that the reals are countable, it also is the very thing that most quickly kills your argument (and all similar arguments)!
That is, the right response to an incorrect intuition is often a correct meta-intuition explaining why the original intuition was not to be trusted; and in this case the meta-intuition is that set theory and geometry look at the world in fundamentally different ways. The way the rationals "sit inside" the real line tells us essentially nothing from a set-theoretic perspective (except that there are infinitely many rationals, of course).
