In decision making theory in books the null hypothesis $H_0$ is always counterbalanced by the alternative hypothesis $H_1$. The significance level $\alpha$ is defined as the probability of the rejection region given $H_0$: $\alpha=P(R|H_0)$, whereas the power $1-\beta$ as the probability of the rejection region given $H_1$, $1-\beta=P(R|H_1)$. As far as I am understanding to design a good test therefore one has to diminish $\alpha$, keeping the power as big as possible, ideally $\alpha=0$ and $1-\beta=1$.
Question: suppose that neither $H_0$ or $H_1$ is true. That is, suppose neither of the models $H_0$ or $H_1$ is true. What control do we have on this situation under this setting?
For example even if $\alpha=0$ and $\beta=1$ if we reject the null hypothesis it seems to me that we can safely say that $H_0$ is not true, because $\alpha=0$ but we cannot conclude that $H_1$ is true since we do not have control over what is happening when neither $H_0$ or $H_1$ is true...
I have the impression that we have to suppose that at least one between $H_0$ or $H_1$ is true, that is the experimentalist must have an idea at the beginning of all the possible models that could be true (e.g. if $H_1$ is the negation of $H_0$ this is trivially satisfied...)... Am I reasoning well?