Is this a consequence of the uncountability of a set? Just a random curiosity I developed now:
Does the uncountability of a set mean that we can't find numbers $a$ and $b$ in $a<x<b$ being $a<b$ for which $x$ doesn't exist?
For example, in a countable set ($\mathbb{N}$ for example), there is $a=2$ and $b=3$, $x$ does not exist. But for $\mathbb{R}$, I guess it's impossible to find $a$'s and $b$'s for which $x$ wouldn't exist.
I just had the idea when I was looking at some properties of the reals in an analysis book, could it be a plausible definition for a uncontable set of numbers? Perhaps it's true and it may be obvious but in the near past I made some mistakes by presuming something as obvious - I'm afraid of assuming this.
 A: Uncountability of a set isn't related to the notion of ordering of the elements in the set. Consider the complex numbers as an example. There is no ordering (which makes it an ordered field, at least) and still it is an uncountable set. So your question really assumes that there is an ordering within of a specific type. As such it is ill-framed.
A: For any uncountable limit ordinal $\alpha$ and any $x\in\alpha$, we have $x+1\in\alpha$. But there is no $y$ such that $x<y<x+1$.
A: The set $A=[0,1]\cup[2,3]$ is uncountable, and $1\in A$ and $2\in A$, but there is no $a\in A$ such that $1<a<2$. In the other direction, $\Bbb Q$ is countable, and whenever $p,q\in\Bbb Q$ with $p<q$, there is an $r\in\Bbb Q\cap(p,q)$.
Let $\langle X,\le\rangle$ be a linearly ordered set. $X$ is a dense linear order if it has the property that you’re interested in: for any $x,y\in X$ with $x<y$ there is a $z\in X\cap(x,y)$. $\Bbb Q$ is a countable dense linear order. (In fact, up to isomorphism it’s the only countable dense linear order without endpoints.) $\Bbb R$ is an uncountable dense linear order. The middle-thirds Cantor set $C$ is an uncountable linear order that is not dense: the endpoints of each removed interval are adjacent in $C$. And $\Bbb Z$ is a countable linear order that is very far from being dense.
