Chapter 4 - Preserving Large Cardinals

So far we have used the Radin forcing to change cofinalities: by Theorem 3.3.2, if $\mathrm{length}(\vec{V})=\delta <{\kappa }^{+}$ then $\kappa$ is singularized in the extension. For applications to cardinal arithmetic one needs the opposite: force with ${\mathbb{R}}_{\vec{V}}$ while keeping $\kappa$ regular, or even measurable. A basic theme in such applications is to arrange a particular pattern of the power function over the Radin club $C_G$ (sometimes adding Cohen subsets or collapsing cardinals in between its points) and then to cut the universe at $\kappa$ — constructions of this type were used by Foreman–Woodin [12], Cummings [9] and Merimovich [40]. In this chapter we isolate the two hypotheses on $\vec{V}$ that make this possible: a repeat point (preserving measurability) and large cofinality of $\mathrm{length}(\vec{V})$ (preserving regularity). Throughout, $\vec{V}\in \mathcal{A}$ is a $j$-sequence with $\kappa (\vec{V})=\kappa$, as supplied by Lemma 1.5.1, and $G\subseteq {\mathbb{R}}_{\vec{V}}$ is generic.

4.1 Repeat points

Definition 4.1.1 (Gitik, Definition 5.14).

An ordinal $\gamma <\mathrm{length}(\vec{V})$ is called a repeat point for $\vec{V}$ if for every $\delta$ with $\gamma \le \delta <\mathrm{length}(\vec{V})$ and every $A\in U(\delta )$, there is a ${\delta }^{\prime }<\gamma$ such that $A\in U({\delta }^{\prime })$. Equivalently, $\bigcup \vec{V}=\bigcup \vec{V}\upharpoonright \gamma$: the measures from $\gamma$ on produce no new sets.

The point of the definition is that below a repeat point, the forcings ${\mathbb{R}}_{\vec{V}}$ and ${\mathbb{R}}_{\vec{V}\upharpoonright \gamma }$ are essentially the same: the conditions coincide up to replacing the top pair $⟨\kappa ,\vec{V}⟩$ by $⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩$, and the families of measure-one sets $\bigcap \vec{V}$ and $\bigcap \vec{V}\upharpoonright \gamma$ agree. So a generic $G\subseteq {\mathbb{R}}_{\vec{V}}$ may be viewed as a generic subset of ${\mathbb{R}}_{\vec{V}\upharpoonright \gamma }$.

Proposition 4.1.2 (Gitik).

Suppose $2^{\kappa }={\kappa }^{+}$ and $\mathrm{length}(\vec{V})={\kappa }^{++}$. Then the set of repeat points for $\vec{V}$ is unbounded in ${\kappa }^{++}$ (indeed contains a club between ${\kappa }^{+}$ and ${\kappa }^{++}$). Consequently, $\vec{V}\upharpoonright \alpha$ has a repeat point for unboundedly many $\alpha <{\kappa }^{++}$.

Proof. The family $\bigcup \vec{V}={\bigcup }_{\delta <{\kappa }^{++}}U(\delta )$ consists of subsets of $V_{\kappa }$, and there are only $2^{\kappa }={\kappa }^{+}$ many of these. Given $\beta <{\kappa }^{++}$, each $A\in {\bigcup }_{\delta \ge \beta }U(\delta )$ appears for the first time at some stage ${\delta }_A$ with $\beta \le {\delta }_A<{\kappa }^{++}$; set $\gamma =\sup \{{\delta }_A+1\mid A\in {\bigcup }_{\delta \ge \beta }U(\delta )\}$. Then $\gamma \ge \beta$ and $\gamma <{\kappa }^{++}$, since $\mathrm{cf}({\kappa }^{++})={\kappa }^{++}>{\kappa }^{+}$, and by construction no measure $U(\delta )$ with $\delta \ge \gamma$ contains a set absent from all $U({\delta }^{\prime })$, ${\delta }^{\prime }<\gamma$. So $\gamma$ is a repeat point above $\beta$. Applying this to the tails $\vec{V}\upharpoonright \alpha$ gives the second assertion. $■$

4.2 Preserving measurability

Theorem 4.2.1 (Gitik, Theorem 5.15).

If $\gamma$ is a repeat point for $\vec{V}$ and $G\subseteq {\mathbb{R}}_{\vec{V}}$ is generic, then $\kappa$ remains measurable in $V[G]$.

Proof. Let $j:V\to M$ be a constructing embedding for $\vec{V}$, with $\mathrm{crit}(j)=\kappa$. As observed above, we may view $G$ as a generic subset of ${\mathbb{R}}_{\vec{V}\upharpoonright \gamma }$. Work in $V[G]$ and define, for $\dot{X}$ an ${\mathbb{R}}_{\vec{V}}$-name of a subset of $\kappa$: $\dot{X}[G]\in F⟺\text{for some }⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in G,\text{ in }M\text{ there is }B\in \bigcap j(\vec{V})\text{ with}$ $⟨d_1,\dots ,d_n,⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,A⟩,⟨j(\kappa ),j(\vec{V})⟩,B⟩{⊩}_{{\mathbb{R}}_{j(\vec{V})}}\check{\kappa }\in j(\dot{X}).$ Thus $F$ measures $\dot{X}[G]$ by forcing ”$\kappa$ enters $j(\dot{X})$” inside $M$, using a condition of ${\mathbb{R}}_{j(\vec{V})}$ obtained by inserting the pair $⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩$ with its block $A$ below the top pair $⟨j(\kappa ),j(\vec{V})⟩$.

$F$ is well defined. Suppose some $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in G$ forces $\dot{X}=\dot{Y}$. Then in $M$, $⟨d_1,\dots ,d_n,⟨j(\kappa ),j(\vec{V})⟩,j(A)⟩⊩j(\dot{X})=j(\dot{Y}).$ Since $A\in \bigcap \vec{V}$, in particular $A\in U(\gamma )$, so $⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩\in j(A)$ by the definition of the $j$-sequence; also $j(A)\cap V_{\kappa }=A$. Hence the triple $⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,A⟩$ is addible to this condition, and for every $B\in \bigcap j(\vec{V})$, $⟨d_1,\dots ,d_n,⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,A⟩,⟨j(\kappa ),j(\vec{V})⟩,B\cap j(A)⟩⊩\check{\kappa }\in j(\dot{X})\wedge j(\dot{X})=j(\dot{Y}),$ so the definition of $F$ does not depend on the choice of the name or of the condition in $G$.

Exercise 4.2.2.

Show that $F$ is a $\kappa$-complete ultrafilter on $\kappa$ extending $U(0)$. (Hint: ultrafilterness uses genericity — some condition in $G$ decides $\dot{X}$ — together with the well-definedness computation; $\kappa$-completeness uses that the relevant filters are $\kappa$-complete and that direct extensions decide bounded meets.)

Normality of $F$. Suppose $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in G$ forces "$\{\nu <\kappa \mid \dot{f}(\nu )<\nu \}\in \dot{F}$". Then in $M$, for some $B\in \bigcap j(\vec{V})$, $⟨d_1,\dots ,d_n,⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,A⟩,⟨j(\kappa ),j(\vec{V})⟩,B⟩⊩j(\dot{f})(\check{\kappa })<\check{\kappa }.$ Working in $M$, construct $B^{\prime }\in \bigcap j(\vec{V})$ by a diagonal intersection (in the sense of Definition 1.3.1) over the $\kappa$ many relevant lower parts, so that: whenever for some $\nu <\kappa$ a condition $⟨x_1,\dots ,x_{\ell },⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,C⟩,⟨j(\kappa ),j(\vec{V})⟩,E⟩$ forces $j(\dot{f})(\check{\kappa })=\check{\nu }$, then the same condition with $E$ replaced by $B^{\prime }$ forces it as well.

Back in $V$, let $D=\left\{⟨x_1,\dots ,x_{\ell },⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,C⟩⟩\in {\mathbb{R}}_{\vec{V}\upharpoonright \gamma }|\text{for some }\nu <\kappa ,⟨x_1,\dots ,x_{\ell },⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,C⟩,⟨j(\kappa ),j(\vec{V})⟩,B^{\prime }⟩⊩j(\dot{f})(\check{\kappa })=\check{\nu }\right\}.$ We claim $D$ is dense in ${\mathbb{R}}_{\vec{V}\upharpoonright \gamma }$ below $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$. Given any stronger condition $p$, the condition $p^{⌢}⟨⟨j(\kappa ),j(\vec{V})⟩,B^{\prime }⟩$ of ${\mathbb{R}}_{j(\vec{V})}$ (computed in $M$) is stronger than the displayed condition above, hence forces $j(\dot{f})(\check{\kappa })<\check{\kappa }$; some stronger condition $⟨x_1,\dots ,x_{\ell },⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,C⟩,⟨j(\kappa ),j(\vec{V})⟩,E⟩$ decides the value, say $\check{\nu }$. By the uniform property of $B^{\prime }$, replacing $E$ by $B^{\prime }$ still decides $\check{\nu }$, so $⟨x_1,\dots ,x_{\ell },⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,C⟩⟩\in D$ and is stronger than $p$.

Pick $⟨e_1,\dots ,e_m,⟨\kappa ,\vec{V}⟩,A^{\prime }⟩\in G\cap D$ below $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$. There is $\delta <\kappa$ such that $⟨e_1,\dots ,e_m,⟨⟨\kappa ,\vec{V}\upharpoonright \gamma ⟩,A^{\prime }⟩,⟨j(\kappa ),j(\vec{V})⟩,B^{\prime }⟩⊩j(\dot{f})(\check{\kappa })=\check{\delta },$ and then $\{\nu <\kappa \mid \dot{f}(\nu )=\delta \}\in F$ by the definition of $F$. Hence $F$ is normal, and $\kappa$ is measurable in $V[G]$. $■$

Remark (Gitik). The ultrafilter $F$ defined above extends $U(0)$, but its ultrapower embedding does not extend that of $U(0)$; instead it extends a certain iterated ultrapower embedding using ultrafilters of $\mathrm{Ult}(V,U(0))$ between $\kappa$ and $i_{U(0)}(\kappa )$. Similar arguments show that the degree of strongness, and even of supercompactness, of the embedding $j$ can be preserved. Note the contrast with Chapter 3: preserving measurability costs a repeat point, which by Lemma 1.5.1 and Proposition 4.1.2 requires a long sequence ($\mathrm{length}(\vec{V})={\kappa }^{++}$ under $2^{\kappa }={\kappa }^{+}$), while changing the cofinality of $\kappa$ to ${\aleph }_0$ already works at length $2$.

4.3 Fat trees

For regularity we need a strengthening of the Prikry property (Lemma 2.4.4) in the spirit of the tree formulation of Prikry forcing: to land in a dense open set, one should only have to specify which measures are used and measure-one sets in them — not which points are picked.

Definition 4.3.1 (Gitik, Definition 5.16).

Let $\vec{F}$ be a sequence of ultrafilters over $V_{\nu }$ for some $\nu \le \kappa$. A tree $T\subseteq [V_{\nu }{]}^{\le n}$ with $n<\omega$ levels is called $\vec{F}$-fat iff (1) for every $⟨{\nu }_1,\dots ,{\nu }_k⟩\in T$, $\kappa ({\nu }_1)<\kappa ({\nu }_2)<\cdots <\kappa ({\nu }_k)$; (2) for every non-maximal $⟨{\nu }_1,\dots ,{\nu }_k⟩\in T$ ($k<n$), there is an $\alpha <\mathrm{length}(\vec{F})$ such that ${\mathrm{Suc}}_T(⟨{\nu }_1,\dots ,{\nu }_k⟩)\in F(\alpha )$.

Definition 4.3.2 (Gitik).

Let $T$ be $\vec{F}$-fat and let $\eta =⟨\eta (1),\dots ,\eta (n)⟩$ be a maximal branch of $T$. A sequence $\vec{A}=⟨\vec{A}(1),\dots ,\vec{A}(n)⟩\in [V_{\nu }{]}^n$ is called a sequence of $\eta$-measure one if for every $i$ with $\eta (i)$ of the form $⟨{\tau }_i,{\vec{G}}_{{\tau }_i}⟩$ we have $\vec{A}(i)\in \bigcap {\vec{G}}_{{\tau }_i}$.

Let $p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in {\mathbb{R}}_{\vec{V}}$, and write $⟨⟨{\nu }_{n+1},{\vec{F}}_{n+1}⟩,A_{n+1}⟩$ for the top pair $⟨⟨\kappa ,\vec{V}⟩,A⟩$. Let $1\le i_1<\cdots <i_m\le n+1$ be indices with $d_{i_k}=⟨⟨{\nu }_{i_k},{\vec{F}}_{i_k}⟩,A_{i_k}⟩$, and for each $k$ let $T_k\subseteq [V_{{\nu }_{i_k}}{]}^{n_k}$ be an ${\vec{F}}_{i_k}$-fat tree, ${\eta }_k$ a maximal branch of $T_k$, and ${\vec{A}}_k$ a sequence of ${\eta }_k$-measure one. We denote by $p^{⌢}⟨{\eta }_1,{\vec{A}}_1{⟩}^{⌢}{\cdots }^{⌢}⟨{\eta }_m,{\vec{A}}_m⟩$ the condition obtained from $p$ by inserting, for each $k$, the sequence ${\eta }_k$ between $d_{i_k-1}$ and $d_{i_k}$ — an ordinal point ${\eta }_k(j)$ is inserted as an ordinal, a pair ${\eta }_k(j)=⟨{\tau }_j,{\vec{G}}_{{\tau }_j}⟩$ as the triple $⟨{\tau }_j,{\vec{G}}_{{\tau }_j},{\vec{A}}_k(j)⟩$ — and then shrinking all blocks in the unique minimal way required by Definition 2.2.1 (each block is cut down to avoid $V_{\kappa (t)+1}$ for the preceding entry $t$).

Lemma 4.3.3 (Gitik, Lemma 5.17).

Let $D$ be a dense open subset of ${\mathbb{R}}_{\vec{V}}$ and $p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in {\mathbb{R}}_{\vec{V}}$. Then there are $p^{\ast }=⟨d_1^{\ast },\dots ,d_n^{\ast },⟨\kappa ,\vec{V}⟩,A^{\ast }⟩{\le }^{\ast }p$; indices $1\le i_1<\cdots <i_m\le n+1$; and for $1\le k\le m$, ${\vec{F}}_{i_k}$-fat trees $T_k\subseteq [V_{{\nu }_{i_k}}{]}^{n_k}$, such that: for every sequence $⟨{\eta }_k\mid 1\le k\le m⟩$ with ${\eta }_k$ a maximal branch of $T_k$, there are sequences ${\vec{A}}_k$ of ${\eta }_k$-measure one with $p^{\ast }{}^{⌢}⟨{\eta }_1,{\vec{A}}_1{⟩}^{⌢}{\cdots }^{⌢}⟨{\eta }_m,{\vec{A}}_m⟩\in D.$

Remark (Gitik, Remark 5.18). Roughly, the meaning is this: in order to get into $D$ we need to specify certain ultrafilters $U(\alpha )$ (or $F(\alpha )$, below $\kappa$) and measure-one sets in them; then any choice of points from these sets puts us into $D$.

Proof. The proof is a refinement of that of Lemma 2.4.4, and we use the notation set up there ($\tilde{A}(\vec{d})$, $A(\alpha ,\vec{d})$, $A(\alpha )$, diagonal intersections). We treat the case $p=⟨⟨\kappa ,\vec{V}⟩,A⟩$; the general case is obtained by applying the same argument inside the block of each triple of the stem, as in the reduction in Lemma 2.4.4. So we need $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,A^{\ast }⟩{\le }^{\ast }p$ and a $\vec{V}$-fat tree $T$ of some finite height such that for every maximal branch $\eta$ of $T$ there is a sequence $\vec{A}$ of $\eta$-measure one with $p^{\ast }{}^{⌢}⟨\eta ,\vec{A}⟩\in D$.

If $p$ already has a direct extension in $D$, take such an extension and $T=\{⟨⟩\}$. Assume this is not the case. For each finite sequence $\vec{d}$ with ${\vec{d}}^{⌢}p\in {\mathbb{R}}_{\vec{V}}$, split $\tilde{A}(\vec{d})$ into two parts according to whether a one-step extension ${\vec{d}}^{⌢}{d}^{⌢}p$ has a direct extension in $D$: $A_0(\vec{d})=\{d\in \tilde{A}(\vec{d})\mid {\vec{d}}^{⌢}{d}^{⌢}p\text{ has a direct extension in }D\},A_1(\vec{d})=\tilde{A}(\vec{d})\setminus A_0(\vec{d});$ for $d=⟨\nu ,{\vec{F}}_{\nu }⟩$ “direct extension” includes the freedom to shrink the block of the new triple, i.e. ${\vec{d}}^{⌢}⟨\nu ,{\vec{F}}_{\nu },b_d{⟩}^{⌢}⟨⟨\kappa ,\vec{V}⟩,B_d⟩\in D$ for some $b_d\in \bigcap {\vec{F}}_{\nu }$ and $B_d\in \bigcap \vec{V}$. For each $\alpha <\mathrm{length}(\vec{V})$ choose the side $i_{\alpha }\le 1$ with $A_{i_{\alpha }}(\vec{d})\in U(\alpha )$, set $A(\alpha ,\vec{d})=A_{i_{\alpha }}(\vec{d})$, and let $A(\alpha )$ be the diagonal intersection; set $A_1=\bigcup \{A(\alpha )\mid \alpha <\mathrm{length}(\vec{V})\}$ and $p_1=⟨⟨\kappa ,\vec{V}⟩,A_1⟩$. As in Lemma 2.4.4 we get: $(\star {)}_1\text{if }{p}_1\le q=⟨e_0,\dots ,e_m,⟨\kappa ,\vec{V}⟩,B⟩\in D,\text{ then there is }\alpha <\mathrm{length}(\vec{V})\text{ such that for every }{e}_m^{\prime }\in A(\alpha )\setminus V_{\kappa (e_{m-1})+1},⟨e_0,\dots ,e_{m-1},e_m^{\prime },⟨\kappa ,\vec{V}⟩,A_1⟩\text{ has a direct extension in }D.$

If now for some $d\in A_1$ the condition $d^{⌢}{p}_1$ has a direct extension in $D$, we finish at height $1$: fix $\alpha$ with $d\in A(\alpha )$, for each $d^{\prime }\in A(\alpha )$ fix a direct extension $⟨{\tilde{d}}^{\prime },⟨\kappa ,\vec{V}⟩,B_{d^{\prime }}⟩\in D$ of $d^{\prime ⌢}{p}_1$ (where ${\tilde{d}}^{\prime }=d^{\prime }$ for ordinals and ${\tilde{d}}^{\prime }=⟨\nu ,{\vec{F}}_{\nu },b_{d^{\prime }}⟩$ for pairs), and let $A^{\ast }$ be the diagonal intersection $\{e\in A_1\mid \forall d^{\prime }\in V_{\kappa (e)}(e\in B_{d^{\prime }})\}\in \bigcap \vec{V}$; then $A^{\ast }\setminus V_{\kappa (d^{\prime })+1}\subseteq B_{d^{\prime }}$ for each $d^{\prime }$, so by openness $⟨{\tilde{d}}^{\prime },⟨\kappa ,\vec{V}⟩,A^{\ast }\setminus V_{\kappa (d^{\prime })+1}⟩\in D$ for all $d^{\prime }\in A(\alpha )\cap A^{\ast }$, and we take $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,A^{\ast }⟩$ with $T$ the one-level tree whose level is $A(\alpha )\cap A^{\ast }$.

Otherwise we pass to two-step extensions: replacing $A$ by $A_1$, define $\tilde{A}(\vec{d})$ as above and let $A_0(\vec{d})$ consist of those $d$ for which there are $\alpha (\vec{d})<\mathrm{length}(\vec{V})$ and a set $C(\vec{d})\subseteq \tilde{A}(\vec{d})\setminus V_{\kappa (d)+1}$, $C(\vec{d})\in U(\alpha (\vec{d}))$, such that for every $c\in C(\vec{d})$ the condition ${\vec{d}}^{⌢}{d}^{⌢}{c}^{⌢}{p}_1$ has a direct extension in $D$. Define the $A(\alpha ,\vec{d})$‘s, $A(\alpha )$‘s, $A_2$ and $p_2$ as before. If for some $d_1,d_2\in A_2$ a direct extension of ${d_1}^{⌢}{d_2}^{⌢}{p}_2$ is in $D$, then by $(\star {)}_1$ there is $\beta <\mathrm{length}(\vec{V})$ such that for every $d_2^{\prime }\in A_1(\beta )\setminus V_{\kappa (d_1)+1}$ the condition ${d_1}^{⌢}{d_2^{\prime }}^{⌢}{p}_2$ has a direct extension in $D$; and then for $\alpha$ with $d_1\in A(\alpha )$ we have $d_1\in A_0(⟨⟩)$, so every $d_1^{\prime }\in A(\alpha )$ has the same property for some ${\beta }^{\prime }$. We finish with a two-level tree, as above.

Continue in the same fashion. At stage $n$ we have sets $A_n(\alpha )\in U(\alpha )$, $A_n={\bigcup }_{\alpha }{A}_n(\alpha )$, $p_n=⟨⟨\kappa ,\vec{V}⟩,A_n⟩$, and the $n$-dimensional version of $(\star {)}_1$: $(\star {)}_n\text{if }{p}_n\le q=⟨e_0,\dots ,e_{m-1},d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,B⟩\in D,\text{ then there is an }n\text{-level }\vec{V}\text{-fat tree }{T}_q\text{ such that for every maximal branch }\eta \text{ of }{T}_q\text{ there are a sequence }\vec{A}\text{ of }\eta \text{-measure one and }{B}_{\eta }\in \bigcap \vec{V}\text{ with }⟨e_0,\dots ,e_{m-1}{⟩}^{⌢}⟨\eta ,\vec{A}{⟩}^{⌢}⟨⟨\kappa ,\vec{V}⟩,B_{\eta }⟩\in D.$ If for some $d_1,\dots ,d_n\in A_n$ a direct extension $q$ of $⟨d_1,\dots ,d_n{⟩}^{⌢}{p}_n$ is in $D$, we finish: take $T=T_q$ from $(\star {)}_n$ and let $A^{\ast }=\{e\in A_n\mid \forall \eta \in V_{\kappa (e)}(e\in B_{\eta })\}$ (the diagonal intersection of the $B_{\eta }$‘s), so that $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,A^{\ast }⟩$ works by openness of $D$.

Suppose, towards a contradiction, that the process does not stop at any $n<\omega$. Set $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,{\bigcap }_{n<\omega }{A}_n⟩;$ then $p^{\ast }{\le }^{\ast }p$ (the intersection is in $\bigcap \vec{V}$ by $\kappa$-completeness), and by our assumption no direct extension of $p^{\ast }$ lies in $D$. Pick $q=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,B⟩\le p^{\ast }$ with $q\in D$. Then $q{\le }^{\ast }⟨d_1,\dots ,d_n{⟩}^{⌢}{p}_n$, since the stem entries of $q$ lie in $A^{\ast }\subseteq A_n$. But by the construction at stage $n$, the existence of such a $q$ means the process was supposed to stop at stage $n$. Contradiction. $■$

4.4 Preserving regularity

Theorem 4.4.1 (Gitik, Theorem 5.19).

If $\mathrm{cf}(\mathrm{length}(\vec{V}))\ge {\kappa }^{+}$, then $\kappa$ remains regular (and hence inaccessible) in $V^{{\mathbb{R}}_{\vec{V}}}$.

Proof. Cardinals are preserved by Theorem 2.4.5, and $\kappa$ is a strong limit in $V$, so it is enough to show that $\kappa$ is not singularized. Suppose $\delta <\kappa$ and $\dot{f}$ is an ${\mathbb{R}}_{\vec{V}}$-name such that the weakest condition forces $\dot{f}:\check{\delta }\to \check{\kappa }$. Let $t=⟨{\mu }_1,\dots ,{\mu }_s,⟨\kappa ,\vec{V}⟩,E⟩\in {\mathbb{R}}_{\vec{V}}$; we find a $p\le t$ forcing ”$\mathrm{ran}\dot{f}$ is bounded in $\check{\kappa }$”. For $\xi <\delta$ consider $D_{\xi }=\{p\in {\mathbb{R}}_{\vec{V}}\mid \text{for some }d\in V_{\kappa }\setminus V_{{\mu }_s+1}\text{ appearing in }p,p⊩\dot{f}(\check{\xi })<\check{\kappa }(d)\}.$ Clearly $D_{\xi }$ is dense (decide the value of $\dot{f}(\check{\xi })$ and append an ordinal above it).

For every $\vec{d}=⟨d_1,\dots ,d_n⟩\in V_{\kappa }$ with ${\vec{d}}^{⌢}⟨⟨\kappa ,\vec{V}⟩,V_{\kappa }\setminus V_{\kappa (d_n)+1}⟩\in {\mathbb{R}}_{\vec{V}}$, apply Lemma 4.3.3 to this condition and $D_{\xi }$; we are interested only in the last tree $T_m$, and only when $i_m=n+1$, i.e. the tree hanging at the top pair. This $T_m$ is a $\vec{V}$-fat tree of some finite height; denote it by $T(\xi ,\vec{d})$. For every non-maximal node $\eta$ of $T(\xi ,\vec{d})$ there is $\alpha (\eta )<\mathrm{length}(\vec{V})$ with $\mathrm{Suc}(\eta )\in U(\alpha (\eta ))$; set $\alpha (\vec{d})=\bigcup \{\alpha (\eta )\mid \eta \in T(\xi ,\vec{d})\text{ non-maximal}\}.$ Since $T(\xi ,\vec{d})\subseteq V_{\kappa }$ has at most $\kappa$ many nodes and $\mathrm{cf}(\mathrm{length}(\vec{V}))\ge {\kappa }^{+}$, we have $\alpha (\vec{d})<\mathrm{length}(\vec{V})$. Then $\alpha (\xi )=\bigcup \{\alpha (\vec{d})\mid \vec{d}\in V_{\kappa }\}<\mathrm{length}(\vec{V})$ (there are $\kappa$ many $\vec{d}$‘s), and finally pick $\alpha <\mathrm{length}(\vec{V})$ above every $\alpha (\xi )$, $\xi <\delta$. Consider $B=\{⟨\nu ,{\vec{F}}_{\nu }⟩\in V_{\kappa }\mid \forall \xi <\delta \forall \vec{d}\in V_{\nu }(T(\xi ,\vec{d})\cap V_{\nu }\text{ is }{\vec{F}}_{\nu }\text{-fat})\}.$ Then $B\in U(\alpha )$: in $M$, at the pair $⟨\kappa ,\vec{V}\upharpoonright \alpha ⟩$, the condition reads ”$j(T)(\xi ,\vec{d})\cap V_{\kappa }$ is $(\vec{V}\upharpoonright \alpha )$-fat” for all $\xi <\delta$ and $\vec{d}\in V_{\kappa }$; but $j$ fixes $V_{\kappa }$ pointwise, so $j(T)(\xi ,\vec{d})\cap V_{\kappa }=T(\xi ,\vec{d})$, and $T(\xi ,\vec{d})$ is $\vec{V}$-fat with successor measures taken from $\{U(\beta )\mid \beta <\alpha (\xi )\}$, all of which lie in $\vec{V}\upharpoonright \alpha$ since $\alpha (\xi )<\alpha$.

For every $\xi <\delta$, let $A_{\xi }^{\ast }\in \bigcap \vec{V}$ be the set given by Lemma 4.3.3 applied to $D_{\xi }$ and $t$, and let $A^{\ast }={\bigcap }_{\xi <\delta }{A}_{\xi }^{\ast }\in \bigcap \vec{V}$.

Claim. Let $p\le ⟨{\mu }_1,\dots ,{\mu }_s,⟨\kappa ,\vec{V}⟩,A^{\ast }\setminus V_{{\mu }_s+1}⟩$ and suppose some $⟨\nu ,{\vec{F}}_{\nu }⟩\in B\setminus V_{{\mu }_s+1}$ appears in $p$. Then $p⊩\forall \xi <\check{\delta }(\dot{f}(\xi )<\check{\nu })$.

Proof of Claim. Suppose not: let $p$ be as above with $p⊩\dot{f}(\check{\xi })\ge \check{\nu }$ for some $\xi <\delta$, and write $p=⟨d_1,\dots ,d_{\ell },⟨⟨\nu ,{\vec{F}}_{\nu }⟩,a_{\nu }⟩,d_{\ell +2},\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩.$ Set $p^{\prime }=⟨d_1,\dots ,d_{\ell },⟨\kappa ,\vec{V}⟩,A^{\ast }\setminus V_{\kappa (d_{\ell })+1}⟩$. By the definition of $B$, the tree $T(\xi ,⟨d_1,\dots ,d_{\ell }⟩)\cap V_{\nu }$ is ${\vec{F}}_{\nu }$-fat. Since $a_{\nu }\in \bigcap {\vec{F}}_{\nu }$, we can choose, level by level, a maximal branch $⟨f_1,\dots ,f_m⟩$ through this tree lying inside $a_{\nu }$: at each non-maximal node the successor set lies in some $F_{\nu }(\beta )$ and hence meets $a_{\nu }$. By Lemma 4.3.3 and the remark following it, there is $q\le p^{\prime }$ with $q\in D_{\xi }$ of the form $q=⟨e_1,\dots ,e_i,{\tilde{f}}_1,\dots ,{\tilde{f}}_m,⟨\kappa ,\vec{V}⟩,A^{\ast }\setminus V_{\kappa (f_m)+1}⟩,$ where $\kappa (e_i)=\kappa (d_{\ell })$ and each ${\tilde{f}}_j$ is $f_j$ (for ordinals) or $⟨f_j,b_j⟩$ for some $b_j$ (for pairs). Since $q\in D_{\xi }$, $q⊩\dot{f}(\check{\xi })<\check{\kappa }(f_m)$, and $\kappa (f_m)<\nu$ because $f_m\in a_{\nu }\subseteq V_{\nu }$.

On the other hand $q$ and $p$ are compatible: $p\le ⟨{\mu }_1,\dots ,{\mu }_s,⟨\kappa ,\vec{V}⟩,A^{\ast }\setminus V_{{\mu }_s+1}⟩$, so the entries $⟨⟨\nu ,{\vec{F}}_{\nu }⟩,a_{\nu }⟩,d_{\ell +2},\dots ,d_n$ of $p$ come from $A^{\ast }$ and are addible to $q$ above $f_m$; hence $⟨e_1,\dots ,e_i,{\tilde{f}}_1,\dots ,{\tilde{f}}_m,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,a_{\nu }\setminus V_{\kappa (f_m)+1}⟩,d_{\ell +2},\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$ is a common extension. But it must force both $\dot{f}(\check{\xi })<\check{\nu }$ (via $q$) and $\dot{f}(\check{\xi })\ge \check{\nu }$ (via $p$). Contradiction. $■$

Finally, every condition below $t$ can be extended to one as in the Claim: shrink the measure-one sets into $A^{\ast }$ and append a pair from $B\setminus V_{{\mu }_s+1}$ (possible since $B\in U(\alpha )$ and every set in $\bigcap \vec{V}$ meets $B$). Hence the conditions forcing ”$\mathrm{ran}\dot{f}$ is bounded in $\check{\kappa }$” are dense below $t$, and since $t$ was arbitrary, the weakest condition forces that $\dot{f}$ is not cofinal in $\kappa$. $■$

Corollary 4.4.2 (Gitik, Remark 5.20).

The converse of Theorem 4.4.1 fails: by Theorem 4.2.1, if $\vec{V}$ has a repeat point $\gamma$ then $\kappa$ remains measurable — hence regular — in $V[G]$, while e.g. a sequence of length $\gamma +1$ may have $\mathrm{cf}(\mathrm{length}(\vec{V}))={\aleph }_0<{\kappa }^{+}$.

Exercise 4.4.3.

Let $\mathrm{cf}(\mathrm{length}(\vec{V}))\ge {\kappa }^{+}$ and $G\subseteq {\mathbb{R}}_{\vec{V}}$ generic. Determine the order type of the Radin club $C_G$ in $V[G]$. (Hint: combine Lemma 3.1.2 with Theorem 4.4.1 — a club in a regular $\kappa$ must have order type $\kappa$; contrast with Theorem 3.3.2.)

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Notes

Definition 4.1.1, Theorem 4.2.1, Definition 4.3.1, Lemma 4.3.3 and Theorem 4.4.1 are Gitik’s Definition 5.14, Theorem 5.15, Definition 5.16, Lemma 5.17 and Theorem 5.19, respectively; Corollary 4.4.2 is his Remark 5.20, and the remark after Lemma 4.3.3 is his Remark 5.18. In Theorem 4.2.1 we have used factorization (Lemma 2.4.2) where Gitik phrases the density argument in terms of the forcing ${\mathbb{R}}_{j(\vec{V})}\setminus \kappa +1$; the content is the same. The observation that regularity preservation is the key to applications in cardinal arithmetic — arranging a pattern of the power function along $C_G$ and cutting the universe at $\kappa$ — is Gitik’s, with references to Foreman–Woodin [12], Cummings [9] and Merimovich [40]. In Chapter 5 we will see that all of these results have cheaper analogues: coherent sequences of measures allow the same conclusions from Mitchell-order hypotheses alone.

References

Main reference:

• Moti Gitik. Prikry-type forcings. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages 1351–1447. Springer, Dordrecht, 2010.

Numbering below follows the bibliography of Gitik’s chapter:

• James Cummings. A model in which GCH holds at successors but fails at limits. Transactions of the American Mathematical Society, 329(1):1–39, 1992.
• [12] Matthew Foreman and W. Hugh Woodin. The generalized continuum hypothesis can fail everywhere. Annals of Mathematics (2), 133(1):1–35, 1991.
• [40] Carmi Merimovich. A power function with a fixed finite gap everywhere. The Journal of Symbolic Logic, 72:361–417, 2007.