Showing the relative consistency of $\neg CH$ using inner models

Question

This question showed up when I transitioned from studying constructibility and inner models of $\mathrm{ZF}$ to studying forcing. Is there a reason why we don't use inner models to show the (relative) consistency of $\neg \mathrm{CH}$ the way we use them to show the relative consistency of $\mathrm{CH}$? I have a wild guess, that is: if we could do so, the falsehood of $ \mathrm{CH}$, being a $\Pi_1$-statement, would trickle down to the minimal inner model, $\mathrm{L}$, by downward absoluteness while on the other hand we know that $ \mathrm{CH}$ holds in $\mathrm{L}$. I'm not sure this makes sense, or even if it is true that we can't build inner models of $\mathrm{ZFC} + \neg \mathrm{CH}$. (I'm not even sure $\neg \mathrm{CH}$ is $\Pi_1$...).

Answer 1

The problem is that it is consistent that $L$ is the only inner model, so you won't find any inner models that you can prove refute V=L (and hence CH). Things are less clear if we add large cardinal axioms so that this is no longer the case, but even if it were possible then, it would be suboptimal in terms of consistency strength.

No, $\neg$CH is not $\Pi_1.$ CH does say a certain bijection exists, but the objects it exists on ($\omega_1$ and $P(\omega)$) are $\Delta_2$ and $\Pi_1$ respectively so CH is $\Sigma_2.$ (As Hanul Jeon notes you can clinch this by observing that $\neg CH$ can be forced over $L$, so truth of CH isn’t preserved outward.)