I´m a bit puzzled by the non-ctm (countable transitive model) approach to forcing as described in Kunen (see [here](https://i.imgur.com/ikppOQK.png)). Kunen calls this "forcing over the universe". I can see how the proofs of (a), (b) and (c) wold go (although I haven't yet sat down to work through them), and how, once we have them, we can get relative consistency results. However, I'm lost as to how one actually gets $\mathbb{1}\Vdash^\* \neg $CH.

When looking at the proof involving ctms, we start with a ctm $M$, consider the poset (inside $M$) Add$(\aleph\_2,\omega)$, and see that in the generic extension we have $\aleph\_2$-many new reals, using the fact that this poset preserves cardinals since its ccc. This uses the definition of $\Vdash$ at several points, i.e. $p\Vdash \phi$ iff for every $M$-generic $G$ with $p\in G$, $M[G]\models\phi$ (with the required interpretations).

Now, if we never define $\Vdash$ nor talk about models, how can we get $\mathbb{1}\Vdash^\* \neg $CH? The definition of $p\Vdash^\*\phi$ is by recursion on the length of $\phi$, and unwinding something like $\neg$CH is not doable. Moreover, results like

> 
> Forcing with such or such poset preserves cardinals. 
> 
> 
> 

wouldn´t make sense if I don't have the models $M$ and $M[G]$ to talk about.

Plus, Kunen states it without further comment, so I assume there's something elementary which I' overlooking.

I know that the forcing over the universe approach is doable through Boolean valued models, but can we still make sense of the above without introducing them?