Recently I read throught Jech's proof of the independence of the continuum hypothesis.
However there was something that bothered me a lot, the whole idea of the generic filter. Beginning the section of forcing in Jech's set theory the author stated that Cohen's original approach was to assume the existence of a countable transitive model of $\mathsf{ZFC}$, and then get a generic filter with respect to the required forcing conditions, then the author states that this cannot be done since such model canoot be proved to exists in $\mathsf{ZFC}$ unless $\mathsf{ZFC}$ is inconsistent, later stating that the usual way of working with forcing was postulating the existence of the generic filter for the required forcing notion.
I found Jech's approach really incomplete, so I searched on the internet and saw the following theorem:
If $\Lambda\subseteq ZFC$ is finite, then $\mathsf{ZFC}\vdash\exists M[M \text{ is transitive}\wedge|M|=\aleph_0\wedge M\models ZC\cup \Lambda].$
Then I argued as follows:
Suppose $\mathsf{ZFC}\vdash\mathsf{CH}$, then there exists some finite $\Lambda\subseteq\mathsf{ZFC}$ such that $\Lambda\vdash\mathsf{CH}$.Let $P$ be the notion of forcing such that for any generic $G$ of $P$ we have $M[G]\models2^{\aleph_0}\geq\aleph_2$. Let $G$ be generic over $M$; such $G$ exists as $M$ is countable.
Reading Jech's proof of the generic model theorem we can get a finite $\Omega\subseteq\mathsf{ZFC}$ such that $\Omega\supseteq\Lambda$ and if $M\models\Omega\cup\mathsf{ZC}$, then $M[G]\models\Lambda\cup\mathsf{ZC}$; the extension is required only for the different instances of the replacement axiom in $\Lambda$. Then clearly this is a contradiction as there exists such a countable $M$. Therefore $\mathsf{ZFC}\nvdash\mathsf{CH}$.
Is my approach correct?
