数理解析研究所講究録
第 1595 巻 2008 年 63-69

63

A NEW SATURATED FILTER
筑波大学・数学系

塩谷真弘

MASAHIRO SHIOYAINSTITUTE OF MATHEMATICSUNIVERSITY OF TSUKUBA
ABSTRACT. We construct a new model of ZFC in whicha saturated Mter.

$\omega_{1}$carries

1. INTRODUCTION
In the groundbreaking work [8] Kunen established

Theorem 1. If there is a huge cardinal, there is a forcing extension incames a saturated filter.
which$w_{1}$

See $[5, 6]$ for detailed expositions of Kunen’s proof.
In these notes we present a model as in Theorem 1 that can be
defined simply. This would make it easier to analyze the model indetail. Moreover the method of the proof is expected to work for otherproblems.

2. PRELIMINARIESWe refer the reader to [7] for background material. Throughout the
paper denotes a regular cardinal. By a filter on we mean a normal
one. We say that a filter on is saturated if it is -saturated.
Suppose that $P$ and are posets. We say that a map $\pi:Parrow Q$ isa projection if
is order-preserving, i.e. $p’\leq p$ implies $\pi(p’)\leq\pi(p)$ , andwith $\pi(p^{*})\leq q$ .
if $q\leq\pi(p)$ , then there is: $Parrow Q$ is a projection. Then it is easy to see
SuPpose that
$D$
that if
is dense open in $Q,$ $\pi^{-1}(D)$ is dense in $P$ . So if G C $P$ isgeneric, $\pi G$ generates a generic filter over . We say that a projection: $Parrow Q$ is total if ran is dense (or equivalently predense) in .
Note that a projection $\pi:Parrow Q$ is total if $\pi(1_{P})=1_{Q}$ .
$\kappa$$\kappa$$\kappa^{+}$$\kappa$$Q$$\bullet$$\pi$$p^{*}\leq p$$\bullet$

$\pi$

$Q$

$\pi$

$\pi$

$Q$

1991 Mathematics Subject Classification. $03E05,03E35,03E55$ .
Partially supported by JSPS Grant-in-Aid for Scientific Research No.19540112.

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Lemma 1. If there is a total projection from : $Parrow Q$ , thenbe completely embedded into $B(P)$ , the completion of $P$ .
$\pi$

Proof. Since

is total, we can define $e:Qarrow B(P)-\{0\}$ by$\pi$$e(q)= \sum\{p :It is easy to check that
Suppose that

$\mu$

$e$

$S_{i}$.

is a complete embedding.

$\prod\{S_{i}\mu :$\square$i\in I\}$for the -product of$\{S_{i} :$\mu$$\prod\{S_{i}\mu :in\pi(P)\leq q\}$is a cardinal and that $\{S_{i} : i\in I\}$ is a nonempty setof posets. We write$\prod\{S_{i}\mu :

can

$Q$

i\in I\}=\cup\{\prod_{i\in d}S_{i}$i\in I\}$, i.e.

: $d\in[I]<\mu\}$ .

is ordered by: $s’\leq s$ iff dom $s’\supset doms$ and $s’(i)\leq s(i)$i\in I\}$for every $i\in I$ .

3. MODIFYING THE SILVER COLLAPSEIn [9] Silver defined a variation of the Levy collapse, now called theSilver collapse. This section introduces a modification of the Silver
collapse and establishes its basic properties.
Suppose that is an inaccessible cardinal.
denotes the, whereset of all functions of the form :,, andis a set of -closed cardinalsof size$>\kappa$$\lambda$$S(\kappa, \lambda)$$\delta\cross darrow\lambda$$s$$\bullet\delta<\kappa$$\bullet$$d$$<\lambda$$\kappa$$\leq\kappa$

for every.
$\nu^{<\kappa}=\nu>\kappa$
Here a cardinal is -closed if.
is ordered by$s’\leqs$. Standard arguments show thatreverse inclusion:iff$\bullet$$s(\eta, \nu)<\nu$$(\eta, \nu)\in\delta\cross d$$\nu$$S(\kappa, \lambda)$$\kappa$$s’\supset s$

Lemma 2.

is -closed, has$S(\kappa, \lambda)$$\kappa$

Also note that if$\lambda- cc$hasand sizechange the class of -closed cardinals.
Here is the main result of this section:$P$$\kappa- cc$

and forces$\kappa$$\lambda=\kappa^{+}$, forcing with.

$P$

does not$\kappa$

Lemma 3. SuPpose that $P$ has
projection fromto$P\cross S(\kappa, \lambda)$

first coordinate.
Proof. Since has$\kappa- cc$

and size . Then there is a totalthat is the identity on the$\kappa$$P*\dot{S}(\kappa, \lambda)$

and size , if is a cardinal, there exist atrepresentatives from the P-names such that
most
. So
cardinal,
exactly
representatives from
if is a -closed
there exist
the P-names such that
Note
that
a
.
-closed cardinal has, then. Hence if is -closed andcofinality
for$P$$\kappa- cc$$\kappa$$\nu<\kappa$

$\nu$

$\nu$

$1\vdash P\dot{\tau}<\nu$$\dot{\tau}$$\nu$$\kappa$$|\vdash p\dot{\mathcal{T}}<\nu$$\dot{\tau}$$\geq\kappa$$\nu$$\kappa$$\kappa$$|\vdash p\dot{\mathcal{T}}<\nu$$|\vdash p\dot{\mathcal{T}}<\gamma$

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some

$\gamma<\nu$. Thus we can list as

for every -closed cardinal
if $\xi<\nu$ , then
, thenif
Define:by$|\vdash p\dot{\mathcal{T}}_{\zeta}<\nu$$|\vdash p\dot{\mathcal{T}}<\nu$$\bullet$a set of P-names so that\xi<\lambda\}$$\nu\leq\lambda$$\kappa$$\bullet$$\{\dot{\tau}_{\xi} :and$|\vdash p\dot{\mathcal{T}}=\dot{\tau}_{\xi}$for some $\xi<\nu$ .

$P\cross S(\kappa, \lambda)arrow P*\dot{S}(\kappa, \lambda)$$\pi$$\pi(p, s)=(p,\dot{s})$,as follows: Since $s\in$
is a P-name such that, there are and such that
dom $s=\delta\cross d$ ,and.
of size
is a set of -closed cardinals
Define a P-name so that $P$ forces
dom
and
.
for every
by $s(\eta, \nu)<\nu$ andfor every
Note that. Also remains a set of -closed cardinals
the choice of$P$.
after forcing with . Thus $P$ forceswhere$|\vdash P\dot{s}\in\dot{S}(\kappa, \lambda)$$\dot{s}$$S(\kappa, \lambda)$$d$$\delta$$\bullet$$\bullet$$\delta<\kappa$$\bullet$$d$$<\lambda$$\kappa$$\leq\kappa$$\dot{s}$$\dot{s}=\delta\cross d$$\bullet$$\bullet$$(\eta, \nu)\in\delta\cross d$$\dot{s}(\eta, \nu)=\dot{\tau}_{s(\eta,\nu)}$$(\eta, \nu)\in\delta\cross d$$|\vdash P\dot{s}(\eta, \nu)<\nu$$\{\dot{\tau}_{\xi} :$d$\xi<\lambda\}$$\kappa$$\dot{s}\in\dot{S}(\kappa, \lambda)$

Claim.

$\pi$

is a total projection.

Proof. Since$\pi(1_{P}, \emptyset)=(1_{P}, \emptyset)$tion. It is easy to see that, it remains to prove that$\pi$

is a projec-

is order-preserving.
.
SuPpose that $(p, s)\in P\cross S(\kappa, \lambda)$ and $(q, i)\leq\pi(p, s)$ in
such that $(p^{*},s^{*})\leq(p, s)$ andWe need to find$\pi(p^{*}, s^{*})\leq(q, i)$ . Let $p^{*}=q$ . It remains to givesuch that.
and.
, there are P-names and such that $P$
Since $P$ forces
forces,dom
and.
of size
is a set of -closed cardinals, there is. Since $P$ doessuch that
Since $P$ has
not change the class of -closed cardinals, there is a set d’ of -closed. Moreover sinceof size
cardinals
such that,and dom$\pi$$P*\dot{S}(\kappa, \lambda)$$(p^{*}, s^{*})\in P\cross S(\kappa, \lambda)$$s^{*}\in S(\kappa, \lambda)$$s^{*}\leq s$$\pi(p^{*}, s^{*})\leq$$(p^{*}, i)$$i\in S(\kappa, \lambda)^{P}$$\dot{\delta}$$\dot{d}$$i=\dot{\delta}\cross\dot{d}$$\bullet$$\dot{\delta}<\kappa$$\bullet$$\dot{d}$$\bullet$$<\lambda$$\kappa$$|\vdash P\dot{\delta}<\delta^{*}$$\delta^{*}<\kappa$$\kappa- cc$$\leq\kappa$$\kappa$$\kappa$$<\lambda$$\leq\kappa$$|\vdash Pi\in\dot{S}(\kappa, \lambda)$

there is a P-name$i^{*}$$|\vdash Pi^{*}:$$|\vdash Pd\subset d^{*}$$i=\dot{\delta}\cross\dot{d}\subset\delta^{*}\cross d^{*}$

such that$\delta^{*}xd^{*}arrow\lambda$

is in$\dot{S}(\kappa, \lambda)$

and$i^{u}\leq i$

.

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Since

,
in
dom
dom
dom . Hence by
we have.
so thatdom
Define :$s^{*}|doms=s$ anddom , thenifis the minimal
forces$(p^{*}, i)\leq(q, i)\leq\pi(p, s)$$p^{*}|\vdash p$$P*\dot{S}(\kappa, \lambda)$$i$$s\subset$$s\subset\delta^{*}\cross d^{*}$$i\subset\delta^{*}\cross d^{*}$$|\vdash P$, we have$\delta^{*}\cross d^{*}arrow\lambda$$s^{*}$$\bullet$$(\eta, \nu)\not\in$$\bullet$$s^{*}(\eta, \nu)$$s$$\dot{\tau}_{\xi}=i’(\eta, \nu)$$\xi$such that $P$.

. Note that this implies $s^{*}\leq s$ by
We claim thatand d’ is a set of -closed$s^{*}|doms=s$ . First recall that
. It remains to prove that $s^{*}(\eta, \nu)<\nu$ forof size
cardinals
every. If $(\eta, \nu)\in doms$ , then $s’(\eta, \nu)=s(\eta, \nu)<\nu$ by$(\eta,\nu)\not\in
doms$ , the conclusion follows $homI\vdash pi^{*}(\eta, \nu)<$. If.
and the choice of. Let $\pi(p^{*}, s^{*})=$
Finally we prove that $\pi(p^{*}, s^{*})\leq(p’, i)$ in. First recall that $P$. It suffices to show that
dom $s’=$ dom ’. It remains to prove thatforces dom$i=i$. First note that for every $(\eta, \nu)\in doms$
dom$s’\in S(\kappa, \lambda)$$\delta^{*}<\kappa$$<\lambda$$\kappa$$\leq\kappa$$(\eta, \nu)\in\delta^{*}\cross d^{*}$$s\in S(\kappa, \lambda)$$\{\dot{\tau}_{\xi} :$\nu$\xi<\lambda\}$$P*\dot{S}(\kappa, \lambda)$$p^{*}|\vdash P\dot{s}’\leq i$$(p’,\dot{s}^{*})$$i\subset\delta^{*}\cross d^{*}=$$\dot{s}$$p^{*}|\vdash p\dot{s}^{*}|$$P^{*}P\cdot$
and the third fromThe second equality follows from$(p^{*},i)\leq\pi(p, s)$. Next$s^{*}|doms=s$ ,$P$

forces that for every $(\eta, \nu)\in domi$ -dom$\dot{s}^{*}(\eta, \nu)=\dot{\tau}_{s(\eta,\nu)}=i^{*}(\eta, \nu)=i(\eta, \nu)$

To see the second equality, recall thatfor every.
follows from.

andforces dom’-dom . The third equality$i\subset\delta^{*}\cross d^{*}$$P$$(\eta, \nu)\in\delta^{*}\cross d$$\dot{\tau}_{s(\eta,\nu)}=i^{*}(\eta, \nu)$$s$$|\vdash Pi’\leq i$$\square$This completes the proof.

Remark 1. For a P-name$s$口for a poset let$\dot{S}$be the term space,$T(\dot{S})$orderediffin . It is known (and easy to see) thatby:
id: $P\cross T(\dot{S})arrow P*S$ is a total projection. See [2] for details. The
and size ,method of the proof of Lemma 3 shows that if $P$ hasi.e. the set of canonical representatives from$\dot{s}’\leq\dot{s}$$\{\dot{s} :|\vdash p\dot{s}\in\dot{S}\}$$\dot{S}$$|\vdash P\dot{s}’\leq\dot{s}$$\kappa- cc$$S(\kappa, \lambda)$$\kappa$

is isomorphic to{$\dot{s}\in T(\dot{S}(\kappa,$ $\lambda)):\exists\delta<\kappa\exists d\subset\lambda|\vdash p$

which is dense in$T(\dot{S}(\kappa, \lambda))$dom$\dot{s}=\delta\cross d$},.

Remark 2. The results in this section should be valid with the modified
Silver collapse replaced by a suitable modification of the Levy collapse.

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4. MAIN THEOREM
This section is devoted to a proof of

Theorem 2. Suppose that is huge and is a regular cardinal
andThen there is a forcing extension in whichsaturated filter.
$\kappa$$\mu$

Proof. Let $j:Varrow M$ be a huge embedding with critical point$\lambda=j(\kappa)$. Define$P= \prod^{\mu}\{S(\alpha, \kappa)$

It is easy to see that

Claim. $P$ has$\kappa- cc$$P\subset V_{\kappa}$:$\alpha\in\{\mu, \kappa)\cap Reg\}$$<\kappa$$\kappa ca7\tau ies$$\kappa=\mu^{+}$$\kappa$

.

a

and

.

is -closed and has size$\mu$$\kappa$.

.

. We need to find distinct

such that$\alpha\indoms\cap$ dom .
$s(\alpha)\cap
domt(\alpha)$
for every
agree on dom
andSince is inaccessible and $|doms|<\mu<\kappa$ for $s\in A$ , there issuch that {dom : $s\in B$ } forms a -system. Let be. For $s\in B$ andwith
the root. Since $|r|<\mu$ , there is.
and 1. Thenlet domsuch that
Since is inaccessible, there are $C\in[B]^{n}$ andfor every $s\in C$ andbe the root. Since $|r|<\mu$
forms a -system. Let. Then, there is
with
for
and
. Since is inaccessible, there isfor every
is constant on $D$ . Now
such that
it is easy to check that any two elements of $D$ are as desired.

Proof. Let$A\in[P]^{\kappa}$$s,$ $t\inA$$s(\alpha)$$t$$t(\alpha)$$\kappa$$B\in[A]^{\kappa}$$\Delta$$s$$r\subset\beta$$\beta<\kappa$$\alpha\in r$$r$$d_{\alpha}^{\delta}|\leq\alpha<\beta$$\delta_{\alpha}^{\delta}<\alpha<\beta$$s(\alpha)=\delta_{\alpha}^{s}\cross d_{\alpha}^{s}$$\langle\delta_{\alpha} :$\kappa$$\langle\delta_{\alpha}^{s} :\alpha\in r\rangle=\langle\delta_{\alpha} :$\{\bigcup_{\alpha\in r}\{\alpha\}\cross d_{\alpha}^{\epsilon} :\alpha\in r\rangle$$\Delta$\alpha\in r\rangle$

s\in C\}$$\bigcup_{\alpha\in r}\{\alpha\}\cross d_{\alpha}$$\alpha\in r$$d_{\alpha}\in[\kappa]<\beta$$\alpha\in r$$s(\alpha)(\delta_{\alpha}\cross d_{\alpha})\subset\gamma$$D\in[C]^{\kappa}$$\bigcup_{\alpha\in r}d_{\alpha}\subset\gamma$$\gamma<\kappa$$\kappa$$s\vdasharrow\langle s(\alpha)|(\delta_{\alpha}\cross d_{\alpha}) :\alpha\in r\rangle$$\square$

Since
Thus

$S(\mu, \kappa)$

can be completely embedded into $P,$

$P*\dot{S}(\kappa, \lambda)$

Claim.

forces

$P*\dot{S}(\kappa, \lambda)$

Proof. Since$\lambda=\kappa^{+}=\mu^{++}$.

forces$\kappa=\mu^{+}$.

forces that cames a saturated filter.

$\lambda M\subset M$$\kappa$, we have$\lambda)\mu$$S(\alpha,$$j(P)= \prod$

Define$P${$\varphi:j(P)arrow P\cross S(\kappa, \lambda)$:$\alpha\in[\mu,$$\lambda)\cap$

Reg}.

by

$\varphi(t)=(\langle t(\alpha)|(\alpha\cross\kappa):\alpha\in domt\cap\kappa\rangle,t(\kappa))$.

dom ) It is easy to check(It is understood that
ifthat is a total Projection. By Lemma 3 there is a total projectionthat is the identity on the first coordinate.
:$t(\kappa)=\emptyset$$\varphi$$\pi$$P\cross S(\kappa, \lambda)arrow P*S(\kappa, \lambda)$$\kappa\not\in$

$t.$

68
Let
be V-generic. Since is a projection,
generates$G\crossH$
a V-generic filter inthat has the form
. Since is a
projection, $\pi(G\cross H)$ generates a V-generic filter in. Sinceis the identity on the first coordinate, the generated filter has theform $G*K$ . Note that $jG=G\subset\overline{G}$ by. Hence we can extend$V[G]arrow
M[\overline{G}]$$j(P)$. SInceinto :has
and $\lambda MCM$ in, we havein.
The rest of the proof is essentially the same as that of Kunen, so wejust give an outline. In $V[G]$ let
list the set ofletnames for subsets of . In$\overline{G}\subset j(P)$$\varphi\overline{G}$$\varphi$$P\cross S(\kappa, \lambda)$$\pi$$P*\dot{S}(\kappa, \lambda)$$\pi$$P\subset V_{\kappa}$$j$$\lambda- cc$$V[\overline{G}]$

$j$

$V$

$\lambda M[\overline{G}]\subset M[\overline{G}]$$V[\overline{G}]$$\{X_{\xi} :$\kappa$\xi<\lambda\}$$S(\kappa, \lambda)-$$V[\overline{G}]$$s^{*}=\cup jK$.

The standard arguments show that
, we get a desending sequenceinsuch thatand each decides$s^{*}\in S(\lambda,j(\lambda))^{M[q}$$M[\overline{G}]$$V[\overline{G}]$$s_{0}\leq s^{*}$$\{s_{\xi}.

:$\kappa\in j(X_{\xi})$$s_{\xi}$. Since$\lambda M[\overline{G}]\subset$\xi<\lambda\}\subset S(\lambda,j(\lambda))^{M[\overline{q}}$. Then in$V[\overline{G}]$$U=\{(\dot{X}_{\xi})_{\kappa!S_{\xi}}|\vdash\kappa\in j(\dot{X}_{\xi})\}$

is a $V[G][K]$ -ultrafilter on . Since
can be completely
embedded into $B(j(P))$ and is arbitrary, there is a $j(P)/(G*K)-$name such that in $V[G][K]$$P*\dot{S}(\kappa, \lambda)$$\kappa$$\overline{G}$$\dot{U}$$j(P)/(G*K)|\vdash\dot{U}$is a $V[G][K]$ -ultrafilter on .
$\kappa$

Now in $V[G][K]$$F=\{X\subset\kappa:j(P)/(G*K)|\vdash X\in\dot{U}\}$is a filter on . It is easy to check thatdefines a complete embedding ffom
into $B(j(P)/(G*K))$ . Since$F$$j(P)/(G*K)$ hasis saturated.
$X \ovalbox{\tt\small REJECT}\mapsto\sum\{p :$\kappa$

p|\vdash X\in\dot{U}\}$$F^{+}$$\square$$\lambda- cc,$

This completes the proof.

口Remark 3. Suppose that is huge with target . Then it is easy to
see that is a Woodin cardinal. Refining a result of [4], Todor\v{c}evi\v{c}showed that if is Woodin, the Levy collapse Co1
produces a(see [1]). In contrast Co1
saturated filter ondoes not neces(see [3]). On the other hand thesarily produce a saturated filter on
Todor\v{c}evic result implies that the iteration Co1
produces a saturated filter on . It appears unknown, however, whether
produces a saturated filter on .
the iteration Co1$\lambda$$\kappa$$\lambda$$\lambda$$(\omega_{1)}\lambda)$$(\omega_{2}, \lambda)$$\omega_{1}$$\omega_{2}$$(\omega, \kappa)*Co1(\kappa, \lambda)$$\omega_{1}$$(\omega_{1}, \kappa)*Co1(\kappa, \lambda)$$\omega_{2}$69
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