Let us consider a statistics model
$X_1,X_2,\cdots,X_n\sim^{i.i.d} f(x,\theta)$,
where $f(x,\theta)$ is a probability density function with a parameter $\theta$.
Let $\theta_0$ be a true value of $\theta$.
In the setting above, I often get confused about the difference between $\theta$ and $\theta_0$. For exmaple, why should we write \begin{align*} \left\{ \begin{array}{ll} H_0 : \theta=0, \\ H_1 : \theta\neq 0 \end{array} \right. \end{align*} instead of \begin{align*} \left\{ \begin{array}{ll} H_0 : \theta_0=0, \\ H_1 : \theta_0\neq 0. \end{array} \right. \end{align*} In fact, I have not seen the latter notation in books and papers.
My opinion
$\theta$ is an unknown generic parameter and not fixed. I understand $\theta$ like a variable (not a randam variable). On the otherhand, $\theta_0$ is fixed.
If $\theta_0=3$, we have $H_0: 3=0$ in the latter notation. It cannot make sense. Is my opinion correct?