# Canonical model for $\neg\mathsf{CH}$ and $\Omega$-logic

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 a completion of $P(\omega_1)$, 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

• In addition to the references you listed, the articles in the Handbook of set theory (Vol.3) by Koellner/Woodin and Larson probably offer the necessary background. Mar 28 '14 at 6:41

The canonical model that Woodin refers to is the $\mathbb{P}_{\mathrm{max}}$ extension of $L(\mathbb{R})$, assuming that $L(\mathbb{R})$ satisfies $\mathrm{AD}$ (although the context, the partial order and the model can all be varied to produce equally interesting models). It is the ultimate model for $\mathcal{P}(\omega_{1})$ in the sense that (assuming for instance that there are proper class many Woodin cardinals) every $\Omega$-consistent sentence of the form $H(\aleph_{2}) \models \phi$, where $\phi$ is $\Pi_{2}$ and may use parameters for the nonstationary ideal on $\omega_{1}$ and any fixed set of reals in $L(\mathbb{R})$, the $H(\aleph_{2})$ of the $\mathbb{P}_{\mathrm{max}}$ extension satisfies $\phi$ (the negation of $\mathrm{CH}$ is such a sentence). So (under the same large cardinal hypothesis) if it is possible to force $H(\aleph_{2})$ to satisfy $\phi$, then the $H(\aleph_{2})$ of the $\mathbb{P}_{\mathrm{max}}$ extension satisfies $\phi$. If you think of $\Pi_{2}$ sentences as asserting some sort of closure, then the $H(\aleph_{2})$ of the $\mathbb{P}_{\mathrm{max}}$ extension is complete (i.e., maximally closed) in this sense.
Conditions in $\mathbb{P}_{\mathrm{max}}$ have the form $\langle (M,I), a \rangle$, where $M$ is a countable transitive model of (a suitable fragement of) ZFC, $I \in M$ is in ideal on $\omega_{1}^{M}$ and $a$ is in $\mathcal{P}(\omega_{1})^{M}$ ($M$, $I$ and $a$ have certain additional properties which I'm skipping to streamline things). Given conditions $p= \langle (M,I), a \rangle$ and $q = \langle (N,J), b \rangle$, $p$ is stronger than $q$ if there exists a certain type of (non-elementary) embedding from $N$ to $M$ with critical point $\omega_{1}^{N}$ (I hope this isn't all too vague to be meaningful). The key property of the $\mathbb{P}_{\mathrm{max}}$ extension of $L(\mathbb{R})$ (which satisfies $V = L(\mathcal{P}(\omega_{1}))$) is that the $H(\aleph_{2})$ of the extension is the direct limit of the structures $H(\aleph_{2})$ of the conditions in the generic filter. In effect (modulo an iterability hypothesis), it is the direct limit of the structures $H(\aleph_{2})$ of all countable transitive models of ZFC containing Woodin cardinals (and satisfying weaker hypotheses that take longer to state).
• For canonicity there is also the fact that the theory of the model is forcing invariant, under the assumption of smallish large cardinals in $V$. Jul 10 '14 at 4:36