Spencinar #3: Saturated ideals (part 2)

Now we'll construct an $\aleph_2$-saturated ideals on $\omega_1$. The construction is originally due to Kunen.

The idea is similar to the previous construction. We will look at the ideal induced by a large cardinal embedding $j$ after a collapsing the large cardinal $\kappa$ to become $\omega_1$. We know the quotient algebra is equivalent to the image of the collapse, so we hope that this is $\kappa^+$-c.c. To do this requires a huge cardinal, which gives a situation where where $j(\kappa)$ is also the amount of closure of the target model in $V$. We want to collapse to turn $j(\kappa)$ into $\kappa^+$, but in order to have a master condition to extend the embedding to the extension by this collapse, we cannot use the Levy collapse. Instead, we will use the Silver collapse. The first poset collapsing $\kappa$ to become $\omega_1$ must absorb the Silver collapse and so also cannot be the Levy collapse: we will construct a poset which will fulfill this purpose by design.

(Drinking game: consume a designated amount of an alcoholic beverage every time you read the word "collapse" in this post...)

Definition: The Silver collapse $S(\kappa,\lambda)$ is the forcing whose elements are partial functions $p: \kappa\times \lambda\rightarrow\kappa$ with domain of size $\le \kappa$ so that there is some $\eta<\kappa$ so that the domain is contained in $\eta\times \lambda$, and for all $(\alpha,\beta)$ in the domain, $p(\alpha,\beta)<\beta$.

Like the Levy collapse $\mathrm{Col}(\kappa,<\lambda)$, the Silver collapse adds surjections from $\kappa$ to everything less than $\lambda$ with uniformly bounded domain, but we are allowed to have some conditions with support of size equal to $\kappa$. It is $\lambda$-c.c., $<\kappa$-closed, and every regular cardinal of the ground model between $\kappa$ and $\lambda$ is collapsed with cofinality $\kappa$.

Start with a huge cardinal $\kappa$ and a huge embedding $j:V\rightarrow M$ with critical point $\kappa$ so that $j(\kappa)=\lambda$ and $M$ is closed under $\lambda$-sequences. We will define a "universal collapse" $\mathbb{P}$ with the $\kappa$-c.c. which will turn $\kappa$ into $\omega_1$, and then use the Silver collapse in the extension to turn $\lambda$ into $\omega_2$.

The crucial property of $\mathbb{P}$ is the following: if $\mathbb{Q}$ completely embeds into $\mathbb{P}$ and has inaccessible cardinality $\alpha$, then there is a complete embedding $i:\mathbb{Q}\ast S(\alpha,\kappa)\rightarrow \mathbb{P}$ extending the embedding of $\mathbb{Q}$. Thus, we are preparing $\mathbb{P}$ to absorb the Silver collapse.

The construction of $\mathbb{P}$ is an iteration with finite support starting with 0th stage $\mathrm{Col}(\omega,<\kappa)$, and at inaccessible stages $\alpha$, forcing with the Silver collapse of $V^\mathbb{Q}$ for some poset $\mathbb{Q}$ which completely embeds into $\mathbb{P}_\alpha$. By a suitable bookkeeping, we can achieve the crucial property. By the chain condition of the Silver collapse, it is easy to see that $\mathbb{P}$ has the $\kappa$-c.c.

Since $\mathbb{P}$ is $\kappa$-c.c., $\mathbb{P}$ completely embeds into $j(\mathbb{P})$ via $j$ (any maximal antichains of $\mathbb{P}$ have size $<\kappa$, so they are fixed by $j$ and remain maximal in $j(\mathbb{P})$).

So, letting $G$ be generic for $\mathbb{P}$ over $V$, we can force further to obtain $\hat{G}$ which is $V[G]$-generic for $j(\mathbb{P})$ and extend $j$ to $j^+:V[G]\rightarrow M[\hat{G}]$. Note that $M[\hat{G}]$ is $\lambda$-closed in $V[\hat{G}]$.

Therefore using the crucial property of $j(\mathbb{P})$, there is a complete embedding $i:\mathbb{P}\ast S(\kappa,\lambda)\rightarrow j(\mathbb{P})$ extending $j$ restricted to $\mathbb{P}$. Let $\mathbb{Q}$ be this Silver collapse, and let $H\in M[\hat{G}]$ be the generic for $\mathbb{Q}$ over $V[G]$ induced by $\hat{G}$ via the complete embedding (using closure of $M[\hat{G}]$. We can extend the embedding $j^+$ to $V[G\ast H]$, since $\bigcup j``H$ is a condition in $j(\mathbb{Q})$. This is where we used the Silver collapse--for the Levy collapse, $\bigcup j``H$ would have too large of a domain, it does not interact well with the huge embedding. The usual large cardinal techniques allow us to extend to $\hat{j}:V[G\ast H]\rightarrow M[\hat{G}\ast\hat{H}]$.

Now consider the $V[G\ast H]$-ultrafilter $U(\hat{j},\kappa)$ (defined in $V[\hat{G}\ast\hat{H}]$). We can use the $<\lambda$-closure of $j(\mathbb{Q})$ to find a $\tilde{U}$ in $V[\hat{G}]$ which is a $V[G\ast H]$-ultrafilter, normal, and $V[G\ast H]$-$\kappa$-complete. The construction works by taking a decreasing $\lambda$-sequence of conditions which together decide $\kappa\in \hat{j}(X)$ for every $X$ in $V[G\ast H]$, and additionally meeting dense sets which ensure that $V[G\ast H]$-sequences of fewer than $\kappa$-many of such $X$ which are forced in to be $U(\hat{j},\kappa)$ will also be in the ultrafilter.

In $V[G\ast H]$, $\kappa=\omega_1$ and $\lambda=\omega_2$. Let $I=\{X\subseteq \kappa: 1\Vdash X\in \dot{\tilde{U}}\}$. Then it can be checked that $I$ is normal and $\kappa$-complete, and since (using similar arguments as in Part 1) $j(\mathbb{P})/G\ast H$ is $\lambda$-c.c. and equivalent to the quotient algebra of $I$, $I$ is $\aleph_2$-saturated.