I know the two-tailed test procedure, that's not the issue. I am wondering about the philosophy about it.
The null hypothesis $ \mu = X $ makes sense to me, we assume a certain value for the population parameter. Frequentist interpretation, the population parameter is unknown but fixed.
The alternative hypothesis $ \mu \ne X $ doesn't. If the population parameter is only slightly off from what we assume, then it is true. [Edited] If we interpret that statement probabilistically (as in Bayesian statistics), then it is almost sure it is true. There is just no point testing.
It only makes sense to me if the null hypothesis is $ \mu $ close to $ X $, and therefore the alternative hypothesis becomes $ \mu $ is not close to $ X $.
Am I making sense? Or am I mixing apples with oranges when I use frequentist/Bayesian interpretation the understand hypothesis testing?
Editing notes, this is what I wrote the [edited], thank Dave for correcting my mistake.
If we use the frequentist interpretation, then it is weird because there is a range of value such that $ \mu = X $ is true, but then $ \mu $ assume multiple values.
That statement is simply not true, see comments for what I learnt.