This might be very odd question but hope be still worthy...
$\mathbb{Q}$ is known as countable, as it is a union of countable sets $\mathbb Q_n, n\in \mathbb N$, where $$ \mathbb Q_n = \left\{{a\over b}: a, b\in \mathbb Z, b\ne 0, a+b=n \right\} $$
Hence we can enumerate rational numbers as a sequence $\left\{q_n\right\}\vert_{n\in \mathbb N} $ by giving them an index in an total order(order of ascending in magnitude). Here is the problem looking for. Think of two consecutive rational numbers $q_i$ and $q_{i+1}$ for some $i$. Then, there must be no other rationals between them, but I can make one, for instance, the average $(q_i+q_{i+1})/2$. So, re-index it(ackward process I would say, but just for curious), like infinitely many times to overcome this contradictory things, until the moment that there is no other rationals between them. Is it possible though? The assertion I've made should not make a postulation that $\mathbb Q$ is not a countable set. Then how can I explain that it's still possible to give an index to each rational numbers?
What is wrong with my logic? There should be a fatal error, but couldn't find it. Really appreciate for your help.