Recently I found this book by Woodin. In the introduction of it the author writes the following:

The main result of this book is the identification of a canonical model in which the

Continuum Hypothesis$(\mathsf{CH})$ is false (1). This model is canonical in the sense that Gödel’s constructible universe $L$ and its relativization to the reals, $L(\mathbb R)$, are canonical models though of course the assertion that $L(\mathbb R)$ is a canonical model is made in the context of large cardinals. Our claim is vague, nevertheless the model we identify can be characterized by its absoluteness properties. This model can also be characterized by certain homogeneity properties. From the point of view of forcing axioms it is the ultimate model at least as far as the subsets of $\omega_1$ are concerned. It is arguably acompletionof $P(\omega_1)$, the powerset of $\omega_1$.

Also, in the same part of the book Woodin states the following theorem:

(2) Suppose that there exists a proper class of Woodin cardinals. Then for each sentence $\phi$, either $\mathsf{ZFC}+(*)\vdash_{\Omega} " H(\omega_2)\vDash\phi",$ or $\mathsf{ZFC}+(*)\vdash_{\Omega} " H(\omega_2)\vDash\neg\phi".$

Here the symbol "$\vdash_{\Omega}$" has to do with $\Omega$-logic. $(*)$ is the axiom:

$\mathsf{AD}$ holds in $L(\mathbb R)$ and $L(P(\omega_1))$ is a $\mathbb P_{\max}$-generic extension of $L(\mathbb R)$.

Now, my questions are

Is there a "intuitive" explanation of the construction of what Woodin calls "canonical model of $\mathsf{\neg CH}$?, What does Woodin mean by "ultimate model at least as far as the subsets of $\omega_1$"?.

is there a "roadmap" to learn the construction of the canonical model for $\mathsf{\neg CH}$ and the proof and ideas behind theorem(1)?

Regarding the second question, my background in set theory is chapters $14-18$ of Jech's *Set Theory*; forcing, iterated forcing, some basic stuff about large cardinals, $0^\sharp$, Jensen's covering lemma, and some of $L[U]$. These results; namely (1) and (2), are, of course, in Woodin's book, however, the book is way too technical and doesn't tell what should be the background nor the "roadmap" towards these results.

I guess Paul Larson's *The Stationary Tower* along with Jech's book should give the necessary backround for the possible "roadmap".

Thanks