Disagreement on choice of null in popular hypothesis testing My question is about hypothesis testing. In hypothesis testing we want to put the hypothesis we want to prove in the alternative hypothesis since we fix the typeI error but cannot make any statements about the typeII error. Hence, we can't draw any conclusion if we reject the null. But why is the null hypothesis in two-sided Gauß test
$H_0: \mu = \mu_0$? Similarly, for all tests for normality, the nullhypothesis is $H_0: "sample comes from normal distribution". But why should we want to prove that some expectation is not a single number or does not have a specific distribution, that's not much helpful. I think we always test in the wrong direction.

We define the null hypothesis with equality because we need some null value to compute the test statistic and p-value. Consider the steps in computing the test statistic in a $t$-test.
$$
t = \dfrac{\bar x - \mu_0}{s/\sqrt{n}}
$$
In order to calculate this, we need some $\mu_0$ that we define in the null hypothesis, $H_0: \mu = \mu_0$.
If we define the null hypothesis as $H_0: \mu\ne\mu_0$, we cannot calculate the test statistic.
As you have noticed, this may be of dubious utility, and alternatives have been proposed that range from confidence intervals to equivalence testing (such as TOST) to Bayesian methods.
