Chapter 2 - The Radin Forcing

We now define the forcing itself. Throughout this chapter, $\vec{V}=⟨\kappa ,U(0),U(1),\dots ,U(\alpha ),\cdots \mid \alpha <\mathrm{length}(\vec{V})⟩$ is a measure sequence in $\mathcal{A}$ with $\kappa (\vec{V})=\kappa$ and $\mathrm{length}(\vec{V})\ge 1$. (Such sequences are supplied by Lemma 1.5.1 under an extender hypothesis, and by Definition 1.6.3 in length $\ge 2$.) The forcing ${\mathbb{R}}_{\vec{V}}$ carries two orders $\le$ and ${\le }^{\ast }$: as with Prikry forcing, $\le$ is used to force, while ${\le }^{\ast }$ provides closure. Convention on the order. We write $p\le q$ when $p$ is stronger than $q$; this is the reverse of Gitik’s convention.

Convention. We identify an ordinal $\nu$ with the trivial sequence $⟨\nu ⟩$; with this convention $\mathcal{A}$ contains the ordinals, and members of measure-one sets are either ordinals or pairs $⟨\nu ,{\vec{F}}_{\nu }⟩$. Recall the notation $\bigcap \vec{F}$ (Definition 1.4.4) and $\kappa (d)$: for $d$ an ordinal $\nu$, a pair $⟨\nu ,\vec{F}⟩$, or a triple $⟨\nu ,\vec{F},B⟩$, we write $\kappa (d)=\nu$. For a triple $d=⟨\nu ,\vec{F},B⟩$ and a set $A$, we write $d\in A$ to mean that the first two coordinates satisfy $⟨\nu ,\vec{F}⟩\in A$. Finally, recall from Chapter 1 that every measure occurring in a measure sequence is an ultrafilter on the corresponding $V_{\nu }$; when below we say that $F$ is “a normal measure on $\nu$”, this is always shorthand for a normal measure on $V_{\nu }$ (typically concentrating on ordinals).

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2.1 Two warm-up cases to keep in mind

Before the formal definition, let us describe informally what a condition should look like in the two simplest cases; the notation here is temporary, and everything is defined formally in Section 2.2.

The case $\mathrm{length}(\vec{V})=1$. Here $\vec{V}=⟨\kappa ,U(0)⟩$ with $U(0)$ a normal measure on $V_{\kappa }$ concentrating on ordinals. To singularize $\kappa$ we build an $\omega$-sequence: a condition should consist of a finite increasing sequence of ordinals below $\kappa$ — the part of the sequence already chosen — together with a set $A\in U(0)$ from which all future points are to be picked. This is exactly the Prikry forcing with $U(0)$; see Proposition 2.3.1.

The case $\mathrm{length}(\vec{V})=2$. Here $\vec{V}=⟨\kappa ,U(0),U(1)⟩$, where $U(1)$ concentrates on pairs $⟨\nu ,F⟩$ with $F$ a normal measure on $V_{\nu }$. We would like both measures to participate in generating the generic sequence. A condition carries a measure-one set $A\in U(0)\cap U(1)$, and extending it means picking a point $a\in A$ and shrinking $A$. If $a$ is an ordinal, it is simply appended, as in the Prikry case. If $a=⟨\nu ,F⟩$, then $a$ can be appended only when $A\cap V_{\nu }\in F$; it then enters the stem together with its own block $B_{\nu }\in F$, $B_{\nu }\subseteq A\cap V_{\nu }$, from which all future points below $\nu$ must be chosen. Thus $⟨\nu ,F⟩$ behaves autonomously and starts producing its own Prikry sequence for $F$, while above $\nu$ the process continues as before. Generically this yields a cofinal sequence of order type ${\omega }^2$: $\omega$ many blocks, each of order type $\omega$. The details are in Example 2.3.2.

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2.2 The definition of ${\mathbb{R}}_{\vec{V}}$

Definition 2.2.1 (Gitik, Definition 5.2).

Let ${\mathbb{R}}_{\vec{V}}$ be the set of all finite sequences $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$ such that (1) $A\in \bigcap \vec{V}$ and $A\subseteq \mathcal{A}$; (2) $A\cap V_{\kappa (d_n)+1}=\varnothing$; (3) for every $m$ with $1\le m\le n$, either (3a) $d_m$ is an ordinal, or (3b) $d_m=⟨\nu ,{\vec{F}}_{\nu },A_{\nu }⟩$ for some ${\vec{F}}_{\nu }\in \mathcal{A}$, $A_{\nu }\subseteq \mathcal{A}$, and $A_{\nu }\in \bigcap {\vec{F}}_{\nu }$; (4) for every $1\le i<j\le n$, (4a) $\kappa (d_i)<\kappa (d_j)$, and (4b) if $d_j=⟨\nu ,{\vec{F}}_{\nu },A_{\nu }⟩$ then $A_{\nu }\cap V_{\kappa (d_i)+1}=\varnothing$.

The sequence $⟨d_1,\dots ,d_n⟩$ is called the stem of the condition, $⟨\kappa ,\vec{V}⟩$ the top, and $A$ the measure-one set. Each $d_m$ of the form $⟨\nu ,{\vec{F}}_{\nu },A_{\nu }⟩$ will give rise to the Radin forcing ${\mathbb{R}}_{{\vec{F}}_{\nu }}$, with ${\vec{F}}_{\nu }$ playing the role of $\vec{V}$.

Definition 2.2.2 (Gitik, Definition 5.3).

Let $p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$ and $q=⟨e_1,\dots ,e_m,⟨\kappa ,\vec{V}⟩,B⟩$ be in ${\mathbb{R}}_{\vec{V}}$. We say that $p$ is stronger than $q$ and write $p\le q$ iff (1) $A\subseteq B$; (2) $n\ge m$; (3) there are $1\le i_1<i_2<\cdots <i_m\le n$ such that for every $1\le k\le m$, either (3a) $e_k=d_{i_k}$, or (3b) $e_k=⟨\nu ,{\vec{F}}_{\nu },B_{\nu }⟩$ and $d_{i_k}=⟨\nu ,{\vec{F}}_{\nu },C_{\nu }⟩$ with $C_{\nu }\subseteq B_{\nu }$; (4) with $i_1,\dots ,i_m$ as in (3), the following holds for every $j$ with $1\le j\le n$ and $j\notin \{i_1,\dots ,i_m\}$: (4a) if $j>i_m$, then $d_j\in B$, or $d_j=⟨\nu ,{\vec{F}}_{\nu },C_{\nu }⟩$ with $⟨\nu ,{\vec{F}}_{\nu }⟩\in B$ and $C_{\nu }\subseteq B\cap V_{\nu }$; (4b) if $j<i_m$, then for the least $k$ with $j<i_k$, $e_k$ is of the form $⟨\nu ,{\vec{F}}_{\nu },B_{\nu }⟩$, and (i) if $d_j$ is an ordinal then $d_j\in B_{\nu }$; (ii) if $d_j=⟨\rho ,\vec{T},S⟩$ then $⟨\rho ,\vec{T}⟩\in B_{\nu }$ and $S\subseteq B_{\nu }$.

Remark. Clause (4) governs the newly added entries of the stem: (4a) says that entries appended above the last old entry come from the measure-one set $B$ of the top, while (4b) says that an entry inserted before $e_k$ must come from the block $B_{\nu }$ of the next old entry above it. In particular, an insertion is possible only into the block of a triple. (Recall that our $\le$ is the reverse of Gitik’s: for him $p\le q$ means that $q$ is stronger.)

Definition 2.2.3 (Gitik, Definition 5.4).

Let $p,q\in {\mathbb{R}}_{\vec{V}}$ be as above. We say that $p$ is a direct extension of $q$ and write $p{\le }^{\ast }q$ iff $p\le q$ and $n=m$.

Exercise 2.2.4.

Show that $\le$ and ${\le }^{\ast }$ are partial orders on ${\mathbb{R}}_{\vec{V}}$, and that ${\le }^{\ast }\subseteq \le$. Show also that $p{\le }^{\ast }q$ iff $p$ is obtained from $q$ by shrinking the measure-one set and the blocks of the triples of the stem (in particular, the stems have the same length and the same first coordinates).

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2.3 Two examples

Proposition 2.3.1 (Length one is Prikry forcing).

Let $\kappa$ be measurable and $U$ a normal measure on $V_{\kappa }$ concentrating on ordinals; set $\vec{V}=⟨\kappa ,U⟩$ (then $\vec{V}\in \mathcal{A}$ by Example 1.6.1(d)). Let $1^{\prime }=⟨⟨\kappa ,\vec{V}⟩,\kappa ⟩$ be the empty-stem condition whose measure-one set consists of all ordinals below $\kappa$. Then $⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩⟼⟨⟨\kappa (d_1),\dots ,\kappa (d_n)⟩,A⟩$ is an isomorphism of $\{p\in {\mathbb{R}}_{\vec{V}}\mid p\le 1^{\prime }\}$ onto the Prikry forcing ${\mathbb{P}}_U$, with ${\le }^{\ast }$ corresponding to the Prikry direct extension.

Proof. Every $p\le 1^{\prime }$ has an ordinal-only stem and $A\subseteq \kappa$: by clause (4a) each stem entry must lie in the measure-one set $\kappa$ of $1^{\prime }$, and $\kappa$ contains no pairs. Conversely, every Prikry condition is the image of such a $p$, and clause (2) of Definition 2.2.1 matches the requirement that the measure-one set lie above the maximum of the stem. The map is therefore a bijection, and it respects the orders: by clause (4b), an entry inserted below the last old entry must come from the block of a triple among the old entries — and there are none — so stems are only end-extended, with new entries taken from the old measure-one set and a shrunken final set. This is exactly the Prikry order. $■$

Remark. The condition $1^{\prime }$ is not the weakest element of ${\mathbb{R}}_{\vec{V}}$: the condition $⟨⟨\kappa ,\vec{V}⟩,\mathcal{A}\cap V_{\kappa }⟩$ is strictly weaker, and below it there are conditions whose stems contain genuine triples $⟨\nu ,{\vec{F}}_{\nu },A_{\nu }⟩$ coming from smaller measure sequences; such conditions are incompatible with $1^{\prime }$. By Lemma 2.4.2 below, the forcing below such a condition factors into a smaller Radin forcing times a Prikry tail. So ${\mathbb{R}}_{⟨\kappa ,U⟩}$ consists of the Prikry part $\{p\mid p\le 1^{\prime }\}$ together with side copies of smaller Radin forcings sitting below measurable cardinals $\nu <\kappa$.

Example 2.3.2 (Length two, from a μ-measurable cardinal).

Let $\kappa$ be μ-measurable (Definition 1.6.3), witnessed by $j:V\to M$, and let $\vec{V}=⟨\kappa ,U(0),U(1)⟩$ be the derived sequence of length $2$. Then $\vec{V}\in \mathcal{A}$: indeed, $U(1)$ concentrates on pairs $⟨\nu ,F⟩$ with $\nu$ measurable and $F$ a normal measure on $V_{\nu }$ (Exercise 1.6.4), and every such pair is a measure sequence of length $1$, hence belongs to $\mathcal{A}$ vacuously (Example 1.6.1(d)); so $\mathcal{A}\cap V_{\kappa }\in U(1)$, while $U(0)$ concentrates on ordinals.

A condition $p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩$ therefore consists of a finite stem of ordinals and triples $⟨\nu ,F,A_{\nu }⟩$ (with $A_{\nu }\in F$), a top $⟨\kappa ,\vec{V}⟩$, and $A\in U(0)\cap U(1)$, which we may take to consist of ordinals and such pairs. Let us analyze the one-step extensions of $⟨⟨\kappa ,\vec{V}⟩,A⟩$. An ordinal $a\in A$ is appended freely. A pair $⟨\nu ,F⟩\in A$ can be appended only when $A\cap V_{\nu }\in F$; this is a measure-one phenomenon: the set $X_A=\{⟨{\nu }^{\prime },F^{\prime }⟩\mid A\cap V_{{\nu }^{\prime }}\in F^{\prime }\}$ lies in $U(1)$, since in $M$ we have $⟨\kappa ,U(0)⟩\in j(X_A)$ iff $j(A)\cap V_{\kappa }\in U(0)$, and $j(A)\cap V_{\kappa }=A\in U(0)$. When $A\cap V_{\nu }\in F$, the triple enters the stem with a block $B_{\nu }\in F$, $B_{\nu }\subseteq A\cap V_{\nu }$, and from then on all points below $\nu$ must come from $B_{\nu }$: the pair starts its own Prikry sequence for $F$. Above $\nu$ the process continues with the remaining part of $A$.

Let $G$ be generic and let $C_G=\{\kappa (d)<\kappa \mid \exists p\in G(d\text{ appears in the stem of }p)\}$ be the Radin club (studied systematically in the next chapter). The shape of the generic object is as follows.

Proposition 2.3.3.

In the situation of Example 2.3.2: (a) $C_G$ is a closed unbounded subset of $\kappa$; (b) a final segment of $C_G$ has order type ${\omega }^2$; (c) the limit points of $C_G$ are exactly those $\nu$ for which a pair $⟨\nu ,F_{\nu }⟩$ appears in the stem of some $p\in G$; each such $\nu$ is measurable in $V$, and the points of $C_G$ between two consecutive limit points form a Prikry sequence for the corresponding $F_{\nu }$.

In particular $V[G]\models \mathrm{cf}(\kappa )=\omega$: length $2$ still singularizes $\kappa$ to cofinality $\omega$. Changing the cofinality to an uncountable value requires longer sequences and is the subject of the next chapter, where (a)–(c) are proved in full generality.

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2.4 Basic structural properties

We turn to the four lemmas on which the whole theory rests: the chain condition, factorization, closure, and the Prikry property; combined, they give cardinal preservation.

Lemma 2.4.1 (Gitik, Lemma 5.5).

$⟨{\mathbb{R}}_{\vec{V}},\le ⟩$ satisfies the ${\kappa }^{+}$-c.c.

Proof. Two conditions with the same stem and the same top are compatible: intersect the measure-one sets and the blocks of the stem triples; the intersections remain measure one by completeness of the relevant filters. A stem is a finite sequence of elements of $V_{\kappa }$, and $|V_{\kappa }|=\kappa$ since $\kappa$ is inaccessible, so there are at most $\kappa$ stems. Hence every antichain has size at most $\kappa$. $■$

For the next two lemmas we need some notation. Let $p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩\in {\mathbb{R}}_{\vec{V}}$ and suppose that for some $m$ with $1\le m\le n$, $d_m=⟨{\nu }_m,{\vec{V}}_m,A_m⟩$ is a triple. Set $p^{\le m}=⟨d_1,\dots ,d_m⟩\text{and}{p}^{>m}=⟨d_{m+1},\dots ,d_n,⟨\kappa ,\vec{V}⟩,A⟩.$ Then $p^{\le m}\in {\mathbb{R}}_{{\vec{V}}_m}$ and $p^{>m}\in {\mathbb{R}}_{\vec{V}}$. For $\vec{W}\in \mathcal{A}$ and $q\in {\mathbb{R}}_{\vec{W}}$, write ${\mathbb{R}}_{\vec{W}}/q=\{r\in {\mathbb{R}}_{\vec{W}}\mid r\le q\}$.

Lemma 2.4.2 (Gitik, Lemma 5.6; Factorization).

${\mathbb{R}}_{\vec{V}}/p\simeq {\mathbb{R}}_{{\vec{V}}_m}/p^{\le m}\times {\mathbb{R}}_{\vec{V}}/p^{>m}$.

Proof. Map $r\in {\mathbb{R}}_{\vec{V}}/p$ to $(r^{\le m},r^{>m})$. By clause (4b) of Definition 2.2.2, every entry of $r$ inserted below ${\nu }_m$ comes from the blocks of the triples among $d_1,\dots ,d_m$, while by clause (4a) every entry above ${\nu }_m$ comes from the blocks of $d_{m+1},\dots ,d_n$ or from the top measure-one set; hence the map is a bijection between the cones. That it respects both $\le$ and ${\le }^{\ast }$ is immediate from the definition of the order. $■$

Lemma 2.4.3 (Gitik, Lemma 5.7).

$⟨{\mathbb{R}}_{\vec{V}}/p^{>m},{\le }^{\ast }⟩$ is ${\nu }_m$-closed.

Proof. Let $⟨r_{\xi }\mid \xi <\mu ⟩$, $\mu <{\nu }_m$, be a chain in the cone with $r_{\eta }{\le }^{\ast }{r}_{\xi }$ for $\xi <\eta$. Direct extensions do not change the length or the first coordinates of the stem, so only the top measure-one set and the blocks of the stem triples shrink. All of these belong to filters that are $\lambda$-complete for some $\lambda >{\nu }_m$: the top filter $\bigcap \vec{V}$ is $\kappa$-complete, and each stem triple $⟨\nu ,{\vec{F}}_{\nu },B_{\nu }⟩$ occurring in $p^{>m}$ has $\nu >{\nu }_m$ and $B_{\nu }\in \bigcap {\vec{F}}_{\nu }$. The entrywise intersections therefore remain measure one, and the condition with the common stem and the intersected sets extends every $r_{\xi }$. $■$

The heart of the matter is the Prikry property. The new point, compared with Prikry forcing, is that a condition may be extended by picking elements from different measures $U(\alpha )$ of the sequence $\vec{V}$; one has to show that different choices cannot decide a statement differently — roughly, that one can pass from one choice of a measure to another while staying with compatible conditions.

More precisely, the proof below has exactly the same architecture as the proof of the Prikry property for Prikry forcing — split the possible one-step extensions into three parts according to whether they can be directly extended to decide $\sigma$ or $\neg \sigma$, shrink once by a diagonal intersection, take a decider of minimal length, and derive a contradiction by amalgamating two incompatible decisions — and the reader who knows that proof is encouraged to reconstruct this one. The changes are concentrated in three places:

• a one-step extension may pick its new point from any measure $U(\alpha )$ of the sequence, so the three-way split is made separately for each $\alpha$, and the shrunken measure-one set is a union $A^{\ast }={\bigcup }_{\alpha }A(\alpha )$, which lies in $\bigcap \vec{V}$ but in general in no single $U(\alpha )$;
• a new entry need not be an ordinal: a pair $⟨\nu ,{\vec{F}}_{\nu }⟩$ enters the stem together with its own block, so a direct extension deciding $\sigma$ shrinks blocks as well, and these blocks must be synchronized over the measure in use — this is the role of the sets $A^{<\alpha }$ and $A^{\prime }(\alpha )$ below;
• an extension may also insert points into the blocks of triples already present in the stem, so in the minimality argument the new entries of a $\neg \sigma$-extension can be scattered relative to the chosen measure; the sets $A^{\le \alpha }$, $A^{>\alpha }$ and the final three-case analysis locate them.

The diagonal intersections themselves are the usual ones (Definition 1.3.1), evaluated in $M$ using that $j$ fixes $V_{\kappa }$ pointwise.

Lemma 2.4.4 (Gitik, Lemma 5.8; Prikry property).

$⟨{\mathbb{R}}_{\vec{V}},\le ,{\le }^{\ast }⟩$ satisfies the Prikry property: for every $p\in {\mathbb{R}}_{\vec{V}}$ and every statement $\sigma$ of the forcing language there is $p^{\ast }{\le }^{\ast }p$ deciding $\sigma$.

Proof. We assume for simplicity that $p=⟨⟨\kappa ,\vec{V}⟩,A⟩$; the general case is obtained by applying the same argument to each block of the stem, using Lemma 2.4.2. Suppose, towards a contradiction, that no direct extension of $p$ decides $\sigma$.

Preparation: one-step extensions. For a finite sequence $\vec{d}=⟨d_1,\dots ,d_n⟩$ from $V_{\kappa }$, write ${\vec{d}}^{⌢}p=⟨d_1,\dots ,d_n,⟨\kappa ,\vec{V}⟩,A\setminus V_{\kappa (d_n)+1}⟩$ whenever this is a condition. Let $\tilde{A}(\vec{d})=\{d\in A\mid d\text{ an ordinal and }{\vec{d}}^{⌢}{d}^{⌢}p\in {\mathbb{R}}_{\vec{V}},\text{ or }d=⟨\nu ,{\vec{F}}_{\nu }⟩\text{ and }{\vec{d}}^{⌢}⟨\nu ,{\vec{F}}_{\nu },A\cap V_{\nu }{⟩}^{⌢}p\in {\mathbb{R}}_{\vec{V}}\}.$ Then $\tilde{A}(\vec{d})\in \bigcap \vec{V}$. Split $\tilde{A}(\vec{d})$ into three parts: $A_0(\vec{d})$ consists of those $d\in \tilde{A}(\vec{d})$ for which the corresponding one-step extension has a direct extension forcing $\sigma$, i.e.

• if $d$ is an ordinal: for some $B_d$, $⟨{\vec{d}}^{⌢}d,⟨\kappa ,\vec{V}⟩,B_d⟩{\le }^{\ast }{\vec{d}}^{⌢}{d}^{⌢}p$ and $⟨{\vec{d}}^{⌢}d,⟨\kappa ,\vec{V}⟩,B_d⟩⊩\sigma$;
• if $d=⟨\nu ,{\vec{F}}_{\nu }⟩$: for some $B_d$ and $b_d$, $⟨{\vec{d}}^{⌢}⟨\nu ,{\vec{F}}_{\nu },b_d⟩,⟨\kappa ,\vec{V}⟩,B_d⟩{\le }^{\ast }{\vec{d}}^{⌢}⟨\nu ,{\vec{F}}_{\nu },A\cap V_{\nu }{⟩}^{⌢}p$ and it forces $\sigma$;

$A_1(\vec{d})$ is defined identically with $\neg \sigma$ in place of $\sigma$, and $A_2(\vec{d})=\tilde{A}(\vec{d})\setminus (A_0(\vec{d})\cup A_1(\vec{d}))$. Note that $A_0(\vec{d})\cap A_1(\vec{d})=\varnothing$: two direct extensions of the same condition are compatible, so no $d$ can admit both.

For each $\alpha <\mathrm{length}(\vec{V})$ choose $i_{\alpha }\le 2$ with $A_{i_{\alpha }}(\vec{d})\in U(\alpha )$ and set $A(\alpha ,\vec{d})=A_{i_{\alpha }}(\vec{d})$; if ${\vec{d}}^{⌢}p\notin {\mathbb{R}}_{\vec{V}}$, set $A(\alpha ,\vec{d})=A$. Now take the diagonal intersection (in the sense of Definition 1.3.1) $A(\alpha )=\{d\in A\mid \forall \vec{d}\in [V_{\kappa }{]}^{<\omega }({\max }_k\kappa (d_k)<\kappa (d)\Rightarrow d\in A(\alpha ,\vec{d}))\}.$

Claim 1. $A(\alpha )\in U(\alpha )$.

Proof. For every $\vec{d}\in [V_{\kappa }{]}^{<\omega }$ we have $A(\alpha ,\vec{d})\in U(\alpha )$, i.e. $⟨\kappa ,\vec{V}\upharpoonright \alpha ⟩\in j(A(\alpha ,\vec{d}))$ in $M$. Since $V_{\kappa }^M=V_{\kappa }$, the finite sequences $\vec{d}$ quantified over in $j(A(\alpha ))$ are exactly those of $V$; and $\kappa (⟨\kappa ,\vec{V}\upharpoonright \alpha ⟩)=\kappa$ exceeds ${\max }_k\kappa (d_k)$ for every such $\vec{d}$. Hence $⟨\kappa ,\vec{V}\upharpoonright \alpha ⟩\in j(A(\alpha ))$, which means $A(\alpha )\in U(\alpha )$. $⊣$

Set $A^{\ast }={\bigcup }_{\alpha <\mathrm{length}(\vec{V})}A(\alpha )$. Then $A^{\ast }\supseteq A(\beta )\in U(\beta )$ for every $\beta$, so $A^{\ast }\in \bigcap \vec{V}$ by upward closure. Put $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,A^{\ast }⟩{\le }^{\ast }p$.

Minimal length. By our assumption, no direct extension of $p^{\ast }$ decides $\sigma$. Pick an extension $⟨d_1,\dots ,d_{n+1},⟨\kappa ,\vec{V}⟩,B⟩\le p^{\ast }$ deciding $\sigma$ with $n$ as small as possible; say it forces $\sigma$. Pick $\alpha <\mathrm{length}(\vec{V})$ with $d_{n+1}\in A(\alpha )$ and write $\vec{d}=⟨d_1,\dots ,d_n⟩$. The decider is a direct extension of ${\vec{d}}^{⌢}{d}_{n+1}^{⌢}{p}^{\ast }$ forcing $\sigma$, so $d_{n+1}\in A_0(\vec{d})$; in particular $i_{\alpha }=0$ and $A(\alpha ,\vec{d})=A_0(\vec{d})$. Since $\kappa (d)>\kappa (d_n)$ implies $d\in A(\alpha ,\vec{d})$ for every $d\in A(\alpha )$, we conclude: $(\star )\text{for every }d\in A(\alpha )\setminus V_{\kappa (d_n)+1}\text{ there are }\tilde{d}\text{ and }{B}_d\text{ with }⟨{\vec{d}}^{⌢}\tilde{d},⟨\kappa ,\vec{V}⟩,B_d⟩{\le }^{\ast }{\vec{d}}^{⌢}{d}^{⌢}{p}^{\ast }\text{ forcing }\sigma ,$ where $\tilde{d}=d$ if $d$ is an ordinal, and $\tilde{d}=⟨\nu ,{\vec{F}}_{\nu },b_d⟩$ for some $b_d\in \bigcap {\vec{F}}_{\nu }$ if $d=⟨\nu ,{\vec{F}}_{\nu }⟩$.

We will find a set $C$ such that $⟨\vec{d},⟨\kappa ,\vec{V}⟩,C⟩\le p^{\ast }$ forces $\sigma$, contradicting the minimality of $n$.

Shrinking below $\alpha$. Suppose $\alpha >0$ (the case $\alpha =0$ is similar and slightly easier). Take the diagonal intersection of the $B_d$‘s: $B^{\ast }=\{e\in A^{\ast }\mid \forall d\in V_{\kappa (e)}(B_d\text{ is defined}\Rightarrow e\in B_d)\}.$

Claim 2. $B^{\ast }\in U(\beta )$ for every $\beta <\mathrm{length}(\vec{V})$.

Proof. For each $d$ with $B_d$ defined, $B_d\in \bigcap \vec{V}$, so in $M$, $⟨\kappa ,\vec{V}\upharpoonright \beta ⟩\in j(B_d)$. Since $V_{\kappa }^M=V_{\kappa }$, the quantification over $d\in V_{\kappa (⟨\kappa ,\vec{V}\upharpoonright \beta ⟩)}=V_{\kappa }$ in $j(B^{\ast })$ ranges exactly over the $d$‘s of $V$. Hence $⟨\kappa ,\vec{V}\upharpoonright \beta ⟩\in j(B^{\ast })$. $⊣$

Each $b_d$ lies in $\bigcap {\vec{F}}_{\nu }$ (it is the block of a legal condition). Consider $A^{<\alpha }=j(⟨b_d\mid d\in A(\alpha )⟩)(\vec{V}\upharpoonright \alpha ),$ the value of the function $d\mapsto b_d$ at the generic point $\vec{V}\upharpoonright \alpha$; this is defined since $A(\alpha )\in U(\alpha )$. By elementarity, $A^{<\alpha }\in U(\beta )$ for every $\beta <\alpha$. Next, the set $A^{\prime }(\alpha )=\{d\in A(\alpha )\mid A^{<\alpha }\cap V_{\kappa (d)}=b_d\}$ lies in $U(\alpha )$: indeed $j(A^{<\alpha })\cap V_{\kappa }=A^{<\alpha }=j(⟨b_d\mid d\in A(\alpha )⟩)(\vec{V}\upharpoonright \alpha )$, so $\vec{V}\upharpoonright \alpha \in j(A^{\prime }(\alpha ))$. Set $A^{\le \alpha }=(A^{<\alpha }\cup A^{\prime }(\alpha ))\cap A^{\ast }$; then $A^{\le \alpha }\in U(\beta )$ for every $\beta \le \alpha$.

Shrinking above $\alpha$. Consider $A^{>\alpha }=\{⟨\nu ,\vec{F}⟩\in A^{\ast }\mid \exists \xi <\mathrm{length}(\vec{F})(A^{\prime }(\alpha )\cap V_{\nu }\in F(\xi ))\}.$ Then $A^{>\alpha }\in U(\beta )$ for every $\beta$ with $\alpha <\beta <\mathrm{length}(\vec{V})$: in $M$, $j(A^{\prime }(\alpha ))\cap V_{\kappa }=A^{\prime }(\alpha )\in U(\alpha )$, so $\xi =\alpha <\beta$ witnesses $⟨\kappa ,\vec{V}\upharpoonright \beta ⟩\in j(A^{>\alpha })$. Finally set $A^{\ast \ast }=(A^{\le \alpha }\cup A^{>\alpha })\cap B^{\ast }.$ Then $A^{\ast \ast }\in \bigcap \vec{V}$: for $\beta \le \alpha$ it contains $A^{\le \alpha }\cap B^{\ast }\in U(\beta )$, and for $\beta >\alpha$ it contains $A^{>\alpha }\cap B^{\ast }\in U(\beta )$.

Three cases. Let $p^{\ast \ast }=⟨⟨\kappa ,\vec{V}⟩,A^{\ast \ast }⟩$ and $q={\vec{d}}^{⌢}{p}^{\ast \ast }$. By the minimality of $n$, no direct extension of $q$ decides $\sigma$. Pick $r\le q$ with $r⊩\neg \sigma$, say $r=⟨e_1,\dots ,e_m,⟨\kappa ,\vec{V}⟩,C⟩$. By Definition 2.2.2(3) there is $k\le m$ with $\kappa (d_n)=\kappa (e_k)$.

Case 1: $k=m$. Choose $d=⟨\nu ,{\vec{F}}_{\nu }⟩\in A(\alpha )\cap C$ with $\kappa (d)>\kappa (e_m)$ and $C\cap V_{\nu }\in \bigcap {\vec{F}}_{\nu }$ (possible: apply Lemma 1.4.5 to $A(\alpha )\cap C\in U(\alpha )$). Since $d\in A(\alpha )$ and $d\in B^{\ast }$‘s index set, $(\star )$ gives $b_d$ with $s=⟨d_1,\dots ,d_n,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,b_d⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\setminus V_{\kappa (d)+1}⟩⊩\sigma .$ Then $t=⟨e_1,\dots ,e_m,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,b_d\cap C⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\cap C\setminus V_{\kappa (d)+1}⟩$ extends both $s$ (the block $b_d\cap C\subseteq b_d$, the tail set $\subseteq A^{\ast \ast }\setminus V_{\kappa (d)+1}$) and $r$ (via $⟨\nu ,{\vec{F}}_{\nu }⟩\in C$, by clause (4a)). But $t$ extends a condition forcing $\sigma$ and a condition forcing $\neg \sigma$ — impossible.

Case 2: $k<m$ and $e_j\in A^{<\alpha }$ for all $k<j\le m$. Pick $d=⟨\nu ,{\vec{F}}_{\nu }⟩\in A^{\prime }(\alpha )\cap C$ with $\kappa (d)>\kappa (e_m)$ and $C\cap A^{<\alpha }\cap V_{\nu }\in \bigcap {\vec{F}}_{\nu }$ (Lemma 1.4.5 again). Since $d\in A^{\prime }(\alpha )$, we have $b_d=A^{<\alpha }\cap V_{\nu }$, so $s=⟨d_1,\dots ,d_n,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,A^{<\alpha }\cap V_{\nu }⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\setminus V_{\kappa (d)+1}⟩⊩\sigma .$ Consider $t=⟨e_1,\dots ,e_m,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,C\cap A^{<\alpha }\cap V_{\nu }⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\cap C\setminus V_{\kappa (d)+1}⟩$. Then $t\le s$: the entries $e_1,\dots ,e_k$ correspond to $d_1,\dots ,d_n$ (since $r\le q$), and the entries $e_j$ with $k<j\le m$ lie in $A^{<\alpha }\cap V_{\nu }=b_d$, hence are insertions into the block of $⟨⟨\nu ,{\vec{F}}_{\nu }⟩,A^{<\alpha }\cap V_{\nu }⟩$ permitted by clause (4b). Also $t\le r$ by clause (4a), since $⟨\nu ,{\vec{F}}_{\nu }⟩\in C$. Again this contradicts $r⊩\neg \sigma$.

Case 3: $k<m$ and some $e_j\notin A^{<\alpha }$. Let $j^{\ast }$ be the minimal such $j$. Since $e_{j^{\ast }}\in A^{\ast \ast }\subseteq A^{\le \alpha }\cup A^{>\alpha }$ and $e_{j^{\ast }}\notin A^{<\alpha }$, we have $e_{j^{\ast }}\in A^{\prime }(\alpha )\cup A^{>\alpha }$.

Subcase 3a: $e_{j^{\ast }}\in A^{\prime }(\alpha )$. Write $e_{j^{\ast }}=⟨⟨\nu ,{\vec{F}}_{\nu }⟩,E⟩$. Then $b_{e_{j^{\ast }}}=A^{<\alpha }\cap V_{\nu }$, so $s=⟨d_1,\dots ,d_n,⟨⟨\nu ,{\vec{F}}_{\nu }⟩,A^{<\alpha }\cap V_{\nu }⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\setminus V_{\nu +1}⟩⊩\sigma ,$ and $t=⟨e_1,\dots ,e_{j^{\ast }-1},⟨⟨\nu ,{\vec{F}}_{\nu }⟩,E\cap A^{<\alpha }⟩,⟨\kappa ,\vec{V}⟩,A^{\ast \ast }\setminus V_{\nu +1}⟩\le s$ by the minimality of $j^{\ast }$ (the entries $e_j$ with $k<j<j^{\ast }$ lie in $A^{<\alpha }$). But $t$ and $r$ are compatible: intersect the blocks and measure-one sets. This contradicts $r⊩\neg \sigma$.

Subcase 3b: $⟨\nu ,{\vec{F}}_{\nu }⟩\in A^{>\alpha }$, where $e_{j^{\ast }}=⟨⟨\nu ,{\vec{F}}_{\nu }⟩,E⟩$, $E\in \bigcap {\vec{F}}_{\nu }$. By the definition of $A^{>\alpha }$, there is $\xi <\mathrm{length}({\vec{F}}_{\nu })$ with $A^{\prime }(\alpha )\cap V_{\nu }\in F_{\nu }(\xi )$; hence $A^{\prime }(\alpha )\cap E\in F_{\nu }(\xi )$. Since ${\vec{F}}_{\nu }\in \mathcal{A}$, Lemma 1.4.5 provides $⟨\tau ,{\vec{G}}_{\tau }⟩\in (A^{\prime }(\alpha )\cap E)\setminus V_{\kappa (e_{j^{\ast }-1})+1}$ with $E\cap V_{\tau }\in \bigcap {\vec{G}}_{\tau }$. Extend $r$ by inserting $⟨⟨\tau ,{\vec{G}}_{\tau }⟩,E\cap V_{\tau }⟩$ into the block of $e_{j^{\ast }}$: the resulting $r^{\prime }\le r$ still forces $\neg \sigma$, but now the entry following $e_{j^{\ast }-1}$ comes from $A^{\prime }(\alpha )$, and we are back in Subcase 3a. Contradiction.

All cases are impossible; hence some direct extension of $p$ decides $\sigma$ after all. $■$

Combining the lemmas, we obtain the main preservation theorem.

Theorem 2.4.5 (Gitik, Theorem 5.9).

Let $G\subseteq {\mathbb{R}}_{\vec{V}}$ be generic. Then $V[G]$ is a cardinal-preserving extension of $V$.

Proof. By induction on $\kappa =\kappa (\vec{V})$. Fix $p\in {\mathbb{R}}_{\vec{V}}$ and a cardinal $\xi$; we show $\xi$ is preserved below $p$.

Case $\xi >\kappa$. By Lemma 2.4.1, ${\mathbb{R}}_{\vec{V}}$ has the ${\kappa }^{+}$-c.c., so all cardinals $\ge {\kappa }^{+}$ are preserved.

Case $\xi \le \kappa$. Suppose the stem of $p$ contains a triple $d_m=⟨{\nu }_m,{\vec{V}}_m,A_m⟩$ with ${\nu }_m<\xi$, and take the last such $m$. By Lemma 2.4.2, ${\mathbb{R}}_{\vec{V}}/p\simeq {\mathbb{R}}_{{\vec{V}}_m}/p^{\le m}\times {\mathbb{R}}_{\vec{V}}/p^{>m}.$ The first factor is a Radin forcing on ${\nu }_m<\kappa$ and preserves all cardinals by the induction hypothesis. In the second factor $\xi >{\nu }_m$, so we may argue with ${\mathbb{R}}_{\vec{V}}/p^{>m}$ in place of ${\mathbb{R}}_{\vec{V}}/p$. Hence we may assume that every triple in the stem of $p$ satisfies $\kappa (d_m)\ge \xi$; for simplicity of notation, assume the stem has no triples at all (the general case is identical, working above $\rho =\min \{\kappa (d_m)\mid d_m\text{ a triple}\}\ge \xi$).

It suffices to show that no new subsets of $\delta$ are added for any cardinal $\delta <\xi$: a collapse of $\xi$ to some $\delta <\xi$ would yield such a new subset (coding the collapsing well-order). So fix $\delta <\xi$ ($\delta <\kappa$) and shrink: $p^{\ast }=⟨⟨\kappa ,\vec{V}⟩,A\setminus V_{\delta +1}⟩{\le }^{\ast }p$. Every extension of $p^{\ast }$ has all its stem points above $\delta$, so by the completeness of the relevant filters, $⟨{\mathbb{R}}_{\vec{V}}/p^{\ast },{\le }^{\ast }⟩$ is ${\delta }^{+}$-closed. Now let $\dot{\tau }$ be a name with $p^{\ast }⊩\dot{\tau }\subseteq \check{\delta }$. Using the Prikry property (Lemma 2.4.4) successively and taking lower bounds by ${\delta }^{+}$-closure, we find a ${\le }^{\ast }$-chain $⟨p_{\gamma }\mid \gamma \le \delta ⟩$ with $p_0=p^{\ast }$ and $p_{\gamma +1}{\le }^{\ast }{p}_{\gamma }$ such that $p_{\gamma +1}$ decides $\dot{\tau }(\check{\gamma })$. Then $p_{\delta }$ forces $\dot{\tau }$ to equal a ground model subset of $\delta$. $■$

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Notes

The definition of ${\mathbb{R}}_{\vec{V}}$ follows Gitik’s Definitions 5.2–5.4, which in turn follow Woodin’s concrete approach to the forcing originally isolated axiomatically by Radin [48] and Mitchell (see also Cummings–Woodin [10]); we have reversed Gitik’s order convention, so that for us $p\le q$ means that $p$ is stronger than $q$. Lemmas 2.4.1–2.4.3 and Theorem 2.4.5 are Gitik’s Lemmas 5.5–5.7 and Theorem 5.9; Lemma 2.4.4 is Gitik’s Lemma 5.8, whose proof is the technical heart of the subject. Proposition 2.3.1 makes precise Gitik’s remark that the length-$1$ case “is the usual Prikry forcing”; the analysis of one-step extensions in Sections 2.1 and 2.3 is Gitik’s treatment of the case ${\alpha }^{\ast }=2$.

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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:

• [10] James Cummings and W. Hugh Woodin. A book on Radin forcing. In preparation.
• [48] Lon B. Radin. Adding closed cofinal sequences to large cardinals. Annals of Mathematical Logic, 22(3):243–261, 1982.