数理解析研究所講究録
第 1595 巻 2008 年 47-62

47Partial stationary reflection in$\mathcal{P}_{\omega_{1}}\omega_{2}$Hiroshi Sakai

Graduate School of Information Science
Nagoya University
Abstract

For a stationary
and $k=0,1$ , let $SR_{k}(S^{*})$ denote theprinciple that every stationaryreflects to some ordinal inof cofinality . We show that if ZFC is consistent then ZFC together with
$SR_{k}(S^{*})$ is also consistent for both $k=0,1$ .
$S^{*}\subseteq \mathcal{P}_{w_{1}}w_{2}$$S\subseteq S^{*}$$\omega_{2}\backslash \omega_{1}$$\omega_{k}$$\exists S^{*},$

1

Introduction

In this paper we consider the consistency of the following partial stationary:reflection principle in$\mathcal{P}_{w_{1}}w_{2}$Deflnltion 1.1. For a stationary
the following pninciple;
For every stationarythat cf
and$\alpha=\omega_{k}$$S\subseteq S^{*}$$S\cap \mathcal{P}_{w_{1}}a$$S^{*}\subseteq P_{w_{1}}w_{2}$and $k=0,1$ let $SR_{k}(S^{*})$ denotethere exists an ordinal
is stationary in$\mathcal{P}_{w_{1}}\alpha$.

$\alpha\in w_{2}\backslash w_{1}$such, which is $oRen$ caUedthat the stationary reflection principle in
there
the we&reflection principle, states that for every stationary
stationary.
principle.
Let
existswithdenote this$\bm{t}dSR_{k}(S^{*})$ is apartial version ofis strengthening of$R\epsilon call$$\mathcal{P}_{w_{1}}w_{2}$$S\subseteq \mathcal{P}_{w_{1}}w_{2}$$\alpha\in w_{2}\backslash w_{1}$$SR(\mathcal{P}_{w_{1}}w_{2}),$$SR_{k}(\mathcal{P}_{w_{1}}w_{2})$$SR_{k}(\mathcal{P}_{w_{1}}w_{2})$$SR(\mathcal{P}_{w_{1}}\omega_{2})$$S\cap \mathcal{P}_{w_{1}}\alpha$.

known that if awe&ly compact cardinal is L\’evy coUapsed to thenshowed that if $SR(P_{w_{1}}w_{2})$holds. On the other handandholds thenis weakly compact in L. Hence bothare equiconsistent with the weakly compact cardinal axiom. It seems to be an
or not.
open question whether
implies
two important facts are akeady iown. FirstAs for the consistency of$Foreman-Todor\check{c}evi\acute{c}[2]$
is not consisthat
it is essentially shown in
tent. Next it shown in K\"onig-Larson-Yoshinobu[4] that if $2^{w_{1}}=w_{2}$ then$SR_{0}(S^{*})$ does not hold for any stationary. As acoroUary of
together withthe latter, K\"onig-Larson-Yoshinobu[4] obtained that
$SR(P_{w_{1}}w_{2})$.
implies
It is

$weU$

$\omega_{2}$$Veli\check{c}kovi\acute{c}[8]$$SR_{1}(\mathcal{P}_{w_{1}}w_{2})$$SR_{1}(\mathcal{P}_{w_{1}}w_{2})$$w_{2}$$SR(\mathcal{P}_{w_{1}}w_{2})$$SR_{1}(\mathcal{P}_{w_{1}}w_{2})$$SR_{0}$$SR_{0}(\mathcal{P}_{w_{1}}w_{2})$$is$$S^{*}\subseteq \mathcal{P}_{\omega_{1}}w_{2}$$SR(\mathcal{P}_{w_{1}}\omega_{2})$$2^{w_{1}}=w_{2}$$SR(P_{\omega_{1}}w_{2})$48
But it remains to be unknown whether the existence of a stationary
holds is consistent or not. Here we give a positive
such that$S^{*}\subseteq$$SR_{0}(S^{*})$$\mathcal{P}_{\omega_{1}}w_{2}$answer:

Theorem 1.2. If ZFC is consistent then so is ZFC with the existence of aholds.
such thatstationary$SR_{0}(S^{*})$$S^{*}\subseteq \mathcal{P}_{\omega_{1}}\omega_{2}$In the above theorem note that we do not need any large cardinal for theconsistency of $SR_{0}(S^{*})$ for some “. We prove that this is also the case with
$SR_{1}(S^{*})$ :$S$

Theorem 1.3. If ZFC is consistent then so is ZFC unth the existence of aholds.
such that$SR_{1}(S^{*})$$stationa\eta S^{*}\subseteq \mathcal{P}_{w_{1}}\omega_{2}$This paper is devoted to the proof of the above theorems. We prove themin Section 5. In Section 2 we present our notation and basic facts used in thispaper. In Section 3 and 4 we present tools, developed by Shelah, which weuse in the proof of the above theorems. In Section 3 we review the iteration ofT-complete forcing notions, and in Section 4 we present a lemma on stationary.
subsets of$\mathcal{P}_{w_{1}}w_{2}$2Preliminaries

Here we present our notation and $ba8ic$ facts used in this paper. For thove which
are not presented below, consult Baumgartner [1], Jech [3] and ShelA [5].
can beThe notion of club, stationary and nonstationary subsets ofis stationary if and only iffound in [3]. We often use the fact that$x\inS$
which is closed under .
there exists
for every function :isto aifwe say that
and
For
.
stationary inwith $cf\alpha=w_{k}$ .
denotes the set of $aU$ limit ordinals
For $k=0$ , llet
In this paper we $exten8ively$ use structures and their elementary submodek.
is astructure of someis astructure then it means that
If we say that$e\varphi
ansion$ of astructure
is ancountable language. We say that astructure
is obtained $\theta omN$ by adding countable many functions, predicatesifand constants. We often use the fact that if is aregular uncountable cardinal
for every countable
thenand $M$ is an elementary submodel of$\mathcal{P}_{\kappa}\lambda$$S\subseteq \mathcal{P}_{\omega_{1}}\lambda$

$f$

$f$

$[\lambda]<warrow\lambda$$Sr\epsilon flects$$\alpha.\in w_{2}\backslash w_{1}$$S\subseteq \mathcal{P}_{w_{1}}w_{2}$$S\cap \mathcal{P}_{w_{1}}$$a$$\mathcal{P}_{w_{1}}\alpha$$\alpha\in w_{2}$$E_{k}^{2}$$\mathcal{M}$$\mathcal{M}$$\mathcal{M}$$\mathcal{N}$$\mathcal{M}$$\theta$$x\subseteq M$$\langle \mathcal{H}_{\theta}, \in\rangle$$x\in M$.

A $forcing$ notion denotes apartial ordering which have the greatest element

and whose universe is aset.
Let be aforcing notion and be acardinal. We say that has the -chain
if there axe no antichain in of cardinality . is said to becondition-distributive if for every countable family $D$ of dense open subsets of
only if the forcing extension by does
is dense in . is -distributive if
not add any countable sequence of ordinals.
is called
Next let be aforcing notion and $M$ be anonempty set.
$D\capM$
every
predense
is predense below for-generric condition ifan$\delta$$\mathbb{P}$$\delta$$\mathbb{P}$$\delta$$\mathbb{P}$$(\delta-c.c.)$$\mathbb{P}$$\mathbb{P},$$\omega$$\mathbb{P}$$\mathbb{P}$$w$$\bm{t}d$$\mathbb{P}$$p\in \mathbb{P}$$\mathbb{P}$$(M,\mathbb{P})$$p$$\cap \mathcal{D}$49$D\subseteq \mathbb{P}$with $D\in M$ . Moreover for a P-generic filter over$M[G]=\{\dot{a}_{G}|\dot{a}\in V^{P}\cap M\}$

where$\dot{a}_{G}$denotes the evaluation of

$V$

let

,

by $G$ . We use the folowing basic fact:$\dot{a}$Fact 2.1 (Shelah [5]). Let be a forcing notion, be a sufficiently large oegular). Let be the canonicalcadinal and $M$ be an elementary submodel of \langle
name for a -generic filter. Then the folloeuing hold:$\theta$$\mathbb{P}$$\dot{G}$$\mathcal{H}_{\theta},$$\in,\mathbb{P}$$\mathbb{P}$(1)$M[\dot{G}]\prec\langle \mathcal{H}_{\theta}^{V[6]}, \in\rangle’$$|\vdash r$(Hence if$\dot{c}_{0},\dot{c}_{1},$(2) If$p$$\dot{c}_{0},\dot{c}_{1},$$\ldots\rangle’.$$\ldots$are.

$\mathbb{P}$-names in $M$ then$M[\dot{G}]\prec(\mathcal{H}_{\theta}^{V[\delta]},$$|\vdash r$

$\in$

)

is an $(M, P)$ -generic condition then

$p1\vdash r$$M[\dot{G}]\cap V=M$“.

An iteration of forcing notions of length will be denoted as. Each is a forcing notion and each is a -name of a forcing.
consists of total functions on such that
notion.
as . We let denote a fixed -name of the greatest
We abbreviate, let supp $p:=\{\eta<\zeta|p(\eta)=i_{\eta}\}$ .
. For eachelement ofis called a countable support iteration if
An iteration \langleI$(\mathbb{P}_{\xi},\dot{\mathbb{Q}}_{\eta}|\xi\leq$$\zeta$$\dot{\mathbb{Q}}_{\eta}$$\zeta,\eta<\zeta\rangle$$\mathbb{P}_{\eta}$$\mathbb{P}_{\xi}$$p$$\mathbb{P}_{\xi}$$|\vdash r_{\xi}$$\dot{\mathbb{Q}}_{\eta}$$pr\eta|\vdash r_{\xi}(p(\eta)\in\dot{Q}_{\eta}’$$\xi$$i_{\eta}$$|\vdash\epsilon$$\mathbb{P}_{\eta}$$p\in \mathbb{P}_{\zeta}$$\mathbb{P}_{\xi},\dot{\mathbb{Q}}_{\eta}$$\xi\leq\zeta,$ $\eta<\zeta\rangle$

for every limit with cf
is the inverse limit of \langleI
direct limit for every with cf $\xi>w$ . Note that if (.
countable support iteration then $|suppp|\leq\omega$ for every$\mathbb{P}_{\xi’}$$\mathbb{P}_{\xi}$$\xi’<\xi\rangle$$\xi=\omega$$\xi$

and is theis a$\mathbb{P}_{\xi},\dot{\mathbb{Q}}_{\eta}|\xi\leq\zeta,\eta<\zeta\rangle$$\xi$$p\in \mathbb{P}_{\zeta}$

Iteration of T-complete forcing notions

3

Here we review the iteration of T-complete forcing notions, which was developed
by Shelah [5]. For the completeness of this paper we give the proof of almost
all lemmata.
We begin with the definition of T-completeness:
Deflnition 3.1 (Shelah). Let be a forcing notion and $M$ be a countable set.
-generic
with the following properties an
We call a sequence (sequence:$\mathbb{P}$$(M,\mathbb{P})$$p_{n}|n\in\omega\rangle$(i)$\langle p_{n}|n\in\omega\rangle$

is a descending sequ ence in(ii) For every dense open subset $D\in M$ of$\mathbb{P}$$\mathbb{P}$unth $p_{n}\in M$ for every$n\in\omega$.

there exists $n\in w$ utth $p_{n}\in D$ .

Here note that if is a lower bound of some $(M,P)$ -generic sequence then-generic condition.
is an
$p$

$p$

$(M,\mathbb{P})$andDeflnition 3.2 (Shelah). Let be a forcing notion, be an ordinalfollowing:the. We say that is T-complete if it satisfies
be a subset of$\lambda$$\mathbb{P}$$T$$\mathbb{P}$$\mathcal{P}_{w_{1}}\lambda$$\geq w_{1}$50
is a suff ciently large regular cardinal, and $M$ is a countableelementary submodel ofwith $M\cap\lambda\in T$ then every
sequence has a lower bound in P.

If$\theta$$\langle \mathcal{H}_{\theta}, \in, \mathbb{P},T\rangle$$(M, \mathbb{P})- gener\dot{\tau}c$

Below we present basics on T-complete forcing notions. As is the case withproperness, there are several slightly different definitions of T-completeness.
First we give one of them. The proof of the following is similar as that for

properness:

Lemma 3.3. Let be a forcing notion. Let be an ordinaland be asubset ofThen is T-complete if and only if it satisfies the follounng;$\mathbb{P}$$\mathcal{P}_{w_{1}}\lambda$.

$T$$\geq w_{1}$$\lambda$$\mathbb{P}$There exists a regular cardinal withand an $e\varphi ansion$$M$such that if is a countable $elementa\eta$of the structure
submodel ofwith $M\cap\lambda\in T$ then every $(M,P)$ -generic sequence
has a lower bound in P.
$\theta$$\mathcal{M}$$\mathbb{P},T\in \mathcal{H}_{\theta}$$\langle \mathcal{H}_{\theta}, \in\rangle$$\mathcal{M}$

It is easy to see that if
is stationary then T-completeness implies wdistributivity. Next we observe this:$T$

Lemma 3.4 (Shelah). Let be an ordinal
and
every
T-complete.
forcingThennotionisof$\lambda$$\geq\omega_{1}$$T$be a stationary subset$w- dist\dot{n}bu\hslash ve$$\mathcal{P}_{w_{1}}\lambda$.

is a T-complete forcing notions. Take an arbitrary family$\{D_{n}|n\in w\}$ of dense open subsets ofand an arbitrary. We must find.
which belongs to
Let be a sufficiently large regular cardinal. Because $T$ is stationaxy there
such that $\{p\}\cup\{D_{n}|$exists a countable elementary submodel $M$ of$n\in w\}\subseteq M$ and $M\cap\lambda\in T$ . Then we can take an-generic sequence$p_{0}\leq
p$
(
with
$T$
-completeness,there exists a lower bound. ThenByof$p^{*}\leq p$ andclearly.

Prvof Suppose that$\mathbb{P}$$p\in \mathbb{P}$$\mathbb{P}$$p^{*}\leq p$$\bigcap_{n\in w}D_{n}$$\theta$$\langle \mathcal{H}_{\theta}, \in,P, T\rangle$$p_{n}|n\in\omega\rangle$$(M,\mathbb{P})$.

$p^{*}$$\langle p_{n}|n\in w\rangle$$\square$$p^{*} \in\bigcap_{n\in w}D_{n}$T-completeness is preserved by countable support iterations:Lemma 3.5 (Shelah). Let be an rdinal and $T$ be a subset of. SupposeOn, is a countable support iteration ofthat
T-complete forcing notions. Thenis T-complete.
$\lambda$$0$$\mathcal{I}=\langle \mathbb{P}\epsilon, \mathbb{Q}_{\eta}|\xi\leq\zeta,\eta<\zeta\rangle,$$\mathcal{P}_{w_{1}}\lambda$$\zeta\in$$P_{\zeta}$

be a sufficiently large regular cardinal. Suppose that $M$ is a countable elementary submodel of (and thatis angeneric sequence. By Lemma 3.3 it suffices to show thathas alower bound. We use the following claim:

Prvof. Let$\theta$$\mathcal{H}_{\theta},$$\in,\mathcal{I},T\rangle$$\langle p_{n}|n\in\omega\rangle$$(M,P_{\zeta})-$$\langle p_{n}|n\in\omega\rangle$. Then-genenc
is an
Claim. Suppose that
. Thenseguence. Moreover suppose thata lower bound of
I
is an
is
-generic sequence, whereforces thatthe canonical name for -generic filter.
$\eta\in\zeta\cap M$$\langle p_{\mathfrak{n}}(\eta|n\in w\rangle$$p^{*}i_{8}$$p^{*}$$\langle p_{n}(\eta)|n\in\omega\rangle$$\bm{1}_{\eta}$$(M[\dot{G}_{\eta}],\dot{\mathbb{Q}}_{\eta})$$(M,\mathbb{P}_{\eta})$$\langle p_{n}r\eta$$n\in w\rangle$$\dot{G}_{\eta}$

51
is a
Proof of Claim. First we prove the former. ClearlyI. Take an arbitrary dense open subset $D\in M$
descending sequence in.
of . We must show that there exists $n\in w$ with$n\in w\rangle$$\langle p_{n}r\eta$$\mathbb{P}_{\eta}\cap M$$p_{n}(\eta\in D$$\mathbb{P}_{\eta}$

Note that the set
is dense open in
and belongs$M$
to . Then by the-genericity ofthere exists $n\in w$ with
$p_{n}\in D’$ . Then $p_{n}(\eta\in D$ for such.
.
Next we prove the latter. It suffices to show the genericity ofTake an arbitrary -nameof a dense open subset of . We show that
”
“
.
with
there exists
“
$D”$
$:=$
It is easy to see that the set“} is dense{I
open inand belongs to $M$ . Hence there exists
with $p_{n}\in D’’$ . Then““
(Claim).
and$D’$ $:=\{p\in \mathbb{P}_{\zeta}|p(\eta\in D\}$$\mathbb{P}_{\zeta}$$\langle p_{n}|n\in\omega\rangle$$(M, \mathbb{P}_{\zeta})$$n$$\langle p_{n}(\eta)|n\in\omega\rangle$$\dot{D}\in M$$P_{\eta}$$n\in\omega$$p^{*}|\vdash\eta$$\mathbb{Q}_{\eta}$$p_{n}(\eta)\in\dot{D}$$p\in \mathbb{P}_{\zeta}$$n\in\omega$$\mathbb{P}_{\zeta}$$p_{n}(\eta)\in\dot{D}$$p^{*}|\vdash\eta$$p(\eta)\in\dot{D}$$pr\eta|\vdash\eta$$p^{*}\leq p_{n}r\eta$$\square$. willUsing the above claim we construct a lower bound of
is a -name of a condition
be a function whose domain is and such that
. The following. By induction on
we chooseoffor each
are the induction hypotheses:$p^{*}(\eta)$$\zeta$(i)(i1)$p^{*}\lceil\eta|\vdash\eta p^{*}(\eta)$$p^{*}(\eta)=i_{\eta}$$P_{\eta}$$p^{*}(\eta)$$\eta<\zeta$$\eta<\zeta$$\dot{\mathbb{Q}}_{\eta}$

is a lower bound of (for every$\eta\in\zeta\backslash M$$p^{*}$$\langle p_{n}|n\in\omega\rangle$$p^{*}$$p_{n}(\eta)|n\in w\rangle$.

.

(ii) assures that supp is countable because $M$ is countable. In general note
has been chosen to satisfy the induction hypotheses forthat ifand$\eta’<\eta$is a lower bound of ($p_{n}r\eta|n\in w\rangle$ .
each
then
is
-generic condition because
is anNote also that-generic condition by Claim.
anhas beenand. Suppose thatNow we describe the choice of. Note. In this case letconstructed. First suppose also thatis a countable set belonging
for eachbecause suppthat supp. Hence $p_{n}(\eta)=i_{\eta}$ for each $n\in w$ , and thus
to $M$ andsatisfies the induction hypothesis (i).
be the canonical name for -generic
Next suppose that $\eta\in M$ . Let$p^{*}$$\eta\leq\zeta$$p^{*}(\eta’)$$p^{*}r\eta=\langle p^{*}(\eta’)|\eta’<\eta\rangle$$p^{*}\square \eta$$\langle p_{n}r\eta|n\in w\rangle$$(M,\mathbb{P}_{\eta})$$(M,\mathbb{P}_{\eta})$$p^{*}(\eta)=i_{\eta}$$\eta\not\in M$$p_{n}\subseteq M$$p^{*}r\eta$$\eta<\zeta$$p^{*}(\eta)$$n\in\omega$$p_{n}$$p^{*}(\eta)$$M\prec\langle \mathcal{H}_{\theta}, \in\rangle$$\dot{G}_{\eta}$$\mathbb{P}_{\eta}$

filter. Then note that$p^{*}r\eta|\vdash\eta((p_{n}(\eta)|n\in w\rangle$

is an$(M[\dot{G}_{\eta}],\dot{\mathbb{Q}}_{\eta})$-generic sequence”

by Claim. $Mor\infty ver$$p^{*}r\eta^{1\vdash}\eta$$M[\dot{G}_{\eta}]\prec\langle \mathcal{H}_{\theta}^{V[\delta_{\eta}]}, \in,\dot{\mathbb{Q}}_{\eta},T\rangle\wedge M[\dot{G}]\cap\lambda=M\cap\lambda\in T$’-generic. Hence $p^{*}|\eta$ forces
by Fact 2.1 and the fact that $p^{*}r\eta$ isbe. Let
has a lower bound by T-completeness ofthat. Clearly the inductionina -name of a lower bound of
hypotheses are satisfied.
. This completesof (Now we could construct a lower boundthe proof.
$(M,\mathbb{P}_{\eta})$$\langle p_{n}(\eta)|n\in w\rangle$$\mathbb{P}_{\eta}$$\dot{\mathbb{Q}}_{\eta}$$\langle p_{n}(\eta)|n\in w\rangle$$p^{*}$$p^{*}(\eta)$$\dot{\mathbb{Q}}_{\eta}$$p_{n}|n\in\omega\rangle$$\square$52
At the end of this section we present a condition for iterations to have
We use the following condition for forcing notions:Deflnition 3.6. A forcing notion
good:(i) Every$\mathbb{P}$(iii) For each$q$.

Unth the following properties is said to be

is a function such that $|p|=w$ and ran$p\in \mathbb{P}$(ii) $p\leq q$ in$\mathbb{P}$$w_{2^{-}}c.c$.
if and only if
if $pr$ (dom dom )$p\subseteq w_{1}$.

$p\supseteq q$$p\cap$$p,q\in \mathbb{P}$$q$$=q\lceil$(dom$p\cap$are compatible.

dom ) then
$q$

$p$

and

If satisfies the follounng additional condition then we say that is beuer:(iv) If a descending sequence (inhas a lower bound then$\mathbb{P}$$\mathbb{P}$$\bigcup_{n\in w}p_{n}\in \mathbb{P}$$p_{n}|n\in w\rangle$.

The standard argument using thethen goodness implies the:$\Delta$$P$-system lemma shows that if CH holds$w_{2}- c.c$Lemma 3.7. Every good forcing notion has the $(2^{w})^{+}- c.c$.
If CH holds andis stationary then a countable support iteration of T:
complete better forcing notions have the$T$$w_{2}- c.c$and be a stationary subset of.
Lemma 3.8. Let be an ordinalOn, is a countable uppon
Suppose that.
has theiteration of T-complete better forcing notions. Then$T$$\geq w_{1}$$\lambda$$\mathcal{I}=\langle \mathbb{P}_{\xi},\dot{\mathbb{Q}}_{\eta}|\xi\leq\zeta,\eta<\zeta\rangle,$$\mathcal{P}_{w_{1}}\lambda$$\zeta\in$$s$$\mathbb{P}_{\zeta}$

Proof. We may assume that

may also assume thatFirst we show that$i_{\eta}=\emptyset\vee$$(2^{\omega})^{+}- c.c$

. Weforces that dom
On for each. Outline of our proof is as follows:for each$q\subseteq$$\mathbb{P}_{\eta}$$q\in\dot{\mathbb{Q}}_{\eta}$$\eta<\zeta$$D$ $:=\{p\in \mathbb{P}_{\zeta}|\forall\eta<\zeta\exists q\in V, p(\eta)=\check{q}\}$is dense in . After that, we show that the forcing notion obtained by restricthas theing
to $D$ is good. This together with Lemma 3.7 implies that$\mathbb{P}_{\zeta}$$\mathbb{P}_{\zeta}$$\mathbb{P}_{\zeta}$$(2^{w})^{+}- c.c$.

. We
Now we start to show that $D$ is dense in . Ihke an arbitrary$D$findwhich is in .
Let be asufflciently large regular cardinal, and take acountable elementarytake sui $M$ because iswith $p_{0}\in M$ . We
submodel $M$ of
below . Our-generic sequence (stationary. Ako, t&e an. The construction of is based on
will be alower bound ofthat in the proof of Lemma 3.5.
of acondition of . The
we choose a -name
By induction on
induction hypotheses are the same as (i) and (ii) in the proof of Lemma 3.5.
has been constructed. Ifthen letand that
Suppose thatas in the Proof of Lemma 3.5. Then suppose that $\eta\in M$ . In
this case we claim the foUowing:$p_{0}\in \mathbb{P}_{\zeta}$$\mathbb{P}_{\zeta}$$p^{*}\leq p_{0}$$\theta$$p_{n}|n\in\omega\rangle$$(M,\mathbb{P}_{\zeta})$$\langle p_{n}|n\in w\rangle$$p^{*}$$\eta<\zeta$$\eta<\zeta$$p^{*}(\eta)=i_{\eta}=\emptyset\vee$$T$$C\bm{t}$$\langle \mathcal{H}_{\theta}, \in,\mathcal{I},T\rangle$$\mathbb{P}_{\eta}$$p^{*}r\eta$$p_{0}$$p^{*}$$p^{*}(\eta)$$\dot{\mathbb{Q}}_{\eta}$$\eta\not\in M$

53
Claim. For each$n\in\omega$

there exists $q_{n}\in V$ such that

Proof of Claim. Fix $n\in w$ . First note that$\mathbb{P}_{\eta}$and 3.5. Hence the set$p^{*}r\eta|\vdash\eta\phi_{n}(\eta)=q_{n^{z}}^{\vee}$.

is w-distributive by Lemma 3.4$B=\{p\in \mathbb{P}_{\eta}|\exists q\in V, p|\vdash\eta p_{n}(\eta)=\check{q}’\}$

is a denve open subset of . Moreover $B\in M$ .
Then there exists $m\in w$ with $p_{m}\in B$ by the-genericity ofI(See Claim in the proof of Lemma 3.5). Then $p^{*}r\eta\in B$ becauseisa lower bound of $\langle p_{m}|m\in w$ ). Therefore there exists $q_{n}\in V$ such thatforces that.
(Claim)$\mathbb{P}_{\eta}$$\langle p_{m}r\eta$$(M,\mathbb{P}_{\eta})$$m\in w\rangle$$p^{*}$$p^{*}\square \eta$$p_{n}(\eta)=q_{n}^{\vee}$$\square$be as in the above claim for each, and let be. Herethe same argument as in the proof of Lemma 3.5 shows thatforces thathas a lower bound in. Thenforces thatis a lower
bound of ($p_{n}(\eta)|n\in w\rangle$ by betterness of . Letbe .
Now we have constructed . It follows from the construction ofthat$D$$p^{*}\inD$
and. This completes the proof of the density of .
Below, for each $p\in D$ and each $\eta<\zeta$ , we letdenote $q\in V$ suchthat
. Note thatis a countable function from On toby thew-distributivity of .
For each $p\in D$ let be the partial function fromsuch that
On to

Let

$n\in\omega$$q_{n}$$q^{*}$$\bigcup_{n\in w}q_{n}$$p^{*}[\eta$$\langle p_{n}(\eta)|n\in w\rangle$$p^{*}[\eta$$\dot{Q}_{\eta}$$p^{*}(\eta)$$\dot{\mathbb{Q}}_{\eta}$$q^{*}\vee$$q*\vee$$p^{*}$$p^{*}$$p^{*}\leq p_{0}$$p(\eta)$$p(\eta)=\check{q}$$p(\eta)$$w_{1}$$\mathbb{P}_{\eta}$$\hat{p}$$\bullet$$\bullet$$\zeta\cross$$w_{1}$

dom $\hat{p}=\{(\eta, a)|a\in domp(\eta)\}$ ,for each$\hat{p}(\eta, a)=p(\eta)(a)$$\langle\eta, \alpha\rangle\in dom\hat{p}$.

Then let be the forcing notion
ordered by reverse inclusions.
It is easy to see that is good. Hence has the. It is also easy tocheck that is isomorphic to the forcing notion obtained by restricting
to$D$ .
Therefore. because $D$ is dense in .
has the口This completes the proof of the lemma.
$\hat{\mathbb{P}}$$\{p^{\wedge}|p\in D\}$$\hat{\mathbb{P}}$$\hat{\mathbb{P}}$$(2^{\omega})^{+}- c.c$$\hat{\mathbb{P}}$$\mathbb{P}_{\zeta}$$(2^{\omega})^{+}- c.c$$\mathbb{P}_{\zeta}$

4$\mathbb{P}_{\zeta}$

Sup depending stationary set

In the proof of Theorem 1.2 and 1.3 we use the $f_{0}g_{oW}ing$ lemma due to Shelah:

Lemma 4.1 (Shelah). Suppose that
subsets of . Then the set$\langle E_{i}|i<w_{1}\rangle$

is a sequence of stationary$E_{0}^{2}$

$T:=$

{

$x \in \mathcal{P}_{w_{1}}w_{2}|x\cap\omega_{1}\in w_{1}\wedge\sup x\not\in x$

is stationary in$\mathcal{P}_{w_{1}}w_{2}$A $supx\in E_{x\cap w_{1}}$ }.

Variants of this lemma are used in Shelah [6] and Shelah-Shioya [7] to obtainconsequences of the stationary refiection principle. Here we present the proof

54of the above lemma for the completeness of this paper. We use a two players’
game of length .
For : $[w_{2}]<warrow w_{2}$ and
let $D(f,i)$ be the following two players’ gameof length :$w$$i\in w_{1}$$f$$\omega$

In the n-th stage, first BAD chooses. G0OD wins if$a_{\mathfrak{n}}<\omega_{2}$and then G0OD chooses$\beta_{n}$with$a_{n}\leq\beta_{n}<\omega_{2}$$c1_{f}(i\cup\{\beta_{n}|n\in w\})\cap w_{1}=i$,where cl$f(x)$ denotes the closure of under . Otherwise BAD wins.
Note that $D(f, i)$ is an open game for BAD and thus it is determined. We
claim the following:
$f$

$x$

Lemma 4.2. For every :a unnning strategy in $D(f, i)$ .
$f$$[w_{2}]<\omegaarrow w_{2}$there exists

Proof. On the contrary, assume that is a function$f$$i\in\omega_{1}$such that G0OD has$hom[w_{2}]<\omega$to$w_{2}$and that

there are no $i\in w_{1}$ such that G0OD has a winning strategy in $D(f,i)$ . Thenfor BAD in $D(f, i)$ for every. Let
there exists a winning strategy$i\in\omega_{1}$$\sigma_{i}$$\vec{\sigma}:=\langle\sigma_{i}|i\in w_{1}\rangle$.

Let be a sufficiently large regular cardinal, and let $M$ be a countable. Note that $:=M\cap\omega_{1}\in w_{1}$ .
elementary submodel of (. Suppose that
By induction on $n\in w$ we takeso that$n\in w$ and thathas been taken. Then let$\theta$$\mathcal{H}_{\theta},$$\in,$$i^{*}$$f,\tilde{\sigma}\rangle$$\beta_{n}\in M$$a_{n},\beta_{n}\in\omega_{2}$$\langle a_{m},\beta_{m}|m<n\rangle$

Clearly$\tilde{\sigma}\rangle$$a_{n}$$:=$$\sigma_{i}\cdot(\langle\beta_{m}|m<n\rangle)$$\beta_{n}$$;=$$\sup\{\sigma_{i}(\langle\beta_{m}|m<n\rangle)|i\in\omega_{1}\}$$\alpha_{n}\leq\beta_{n}<w_{2}$.

I. Moreover

because$\beta_{n}\in M$$\{\beta_{m}|m<n\}\subseteq M\prec\langle \mathcal{H}_{\theta},$$\in$

is a sequence of moves in $D(f, i”)$ in which BAD has. Hence BAD wins with this moves.
played according to the winning strategy$f(i^{*}\cup\{\beta_{n}|n\inw\})\subseteq
M$ because $M$ is closed under
On the other hand cl
and $i^{*}\cup\{\beta_{n}|n\in w\}\subseteq M$ . Thus cl$f(i^{*}\cup\{\beta_{n}|n\in w\})\cap w_{1}=i^{*}$ , that is, G0OD.
wins with the moves
This is a contradiction.

Now \langle$a_{n},\beta_{n}$$n\in\omega\rangle$$\sigma_{i}\cdot$$f$$\langle a_{n}, \beta_{n}|n\in w\rangle$$\square$Now we can prove Lemma 4.1:

Proof of Lemma 4.1. Take an arbitrary function$f$: $[w_{2}]<warrow w_{2}$ . We findclosed under .
in
By Lemma 4.2 take $i”\in\omega$ such that G0OD has a winning strategy$M$$D(f,i^{*})$ . Let
be an uncountbe a sufficiently large regular cardinal, and let) such that
. Noteable elementary submodel of \langle$x^{*}\in T$$f$$\sigma^{*}$$\theta$$\mathcal{H}_{\theta},$that$w_{1}\subseteq M\cap\omega_{2}\in w_{2}$.

$\in,$ $f,\sigma^{*}$$M\cap\omega_{2}\in E_{i}\cdot\backslash w_{1}$55
Take an increasing sequence \langleI$:=\sigma^{*}((\alpha_{m}|m\leq n\rangle)\in M$ for each$\beta_{n}$converging to. Moreover let$n\in w\rangle$$\alpha_{n}$$n\in\omega$$x^{*}:=c1_{f}(i^{*}\cup\{\beta_{n}|n\in\omega\})$$M\cap\omega_{2}$, and let.

It uffices to show that $x^{*}\in T$ .
First note that
. On the other$M$$i”\cup\{\beta_{n}|n\inw\}\subseteq
M$
hand,becauseandis closed under . Hence$supx^{*}\leq M\cap\omega_{2}$ . Therefore $supx^{*}=M\cap\omega_{2}\in E_{i^{*}}$ . Moreover.
Note also thatis a sequence of moves in $D(f,i^{*})$ in which
G00D has played according to the winning strategy . Hence“.
$x^{*}\inT$,,
Therefore
is,
.
and
that
$s$

$supx^{*}\geq\sup_{\mathfrak{n}\in\omega}\beta_{n}\geq\sup_{n\in w}a_{n}=M\cap\omega_{2}$$x^{*}\subseteq M$$f$$supx^{*}\not\in x^{*}$$\langle a_{\mathfrak{n}},\beta_{n}|n\in w\rangle$$x^{*}\cap\omega_{1}=i$$\sigma^{*}$$x^{*}\cap\omega_{1}\in w_{1}$5$supx^{*}\not\in x^{*}$$supx^{*}\in E_{x^{*}\cap w_{1}}$$\square$

Proof of Theorem 1.2 and 1.3

Here we prove Theorem 1.2 and 1.3. In fact we prove slightly more.
To state our result we introduce the following subsets offor asequence:$\mathcal{P}_{w_{1}}w_{2}$$\square _{w_{1}^{-}}$$c=\sim\langle c_{\alpha}|a\in Lim\omega_{2}\rangle$$S_{0}^{c}arrow:=$

the set of all$x\in \mathcal{P}_{w_{1}}w_{2}$such that(i) $x\cap w_{1}\in w_{1}$ and. $\sup x\not\in x$ ,(ii) o.t. $c_{\sup x}<x\cap w_{1}$ ,(iii)$S_{1}^{\partial}$$;=$$c_{\sup x}\subseteq x$

the set of all.

$x\in \mathcal{P}_{w_{1}}w_{2}$such that(i) $x\cap w_{1}\in w_{1}$ and $8Upx\not\in x$ ,(ii) o.t. $c_{\sup x}=x\cap\omega_{1}$ ,(iii).
$c_{\sup x}\subseteq x$

The difference between
andis the property (ii) of their elements. As we
see in the following lemma, these sets have maximality properties with respect
to the stationary reflection. Note that the folowing lemma implies that (every
subsets of)and that (every
does not reflect to any ordinal in:subset of)does not reflect to any ordinal in$S_{0}^{\theta}$$S_{1}^{c}\sim$$E_{0}^{2}$$\mathcal{P}_{\omega_{1}}\omega_{2}\backslash S_{0}^{\partial}$$E_{1}^{2}$$\mathcal{P}_{w_{1}}w_{2}\backslash S_{1^{\vee}}^{c}$Lemma 5.1. Let
holds:$c’=\langle c_{\alpha}|.a\in Lim\omega_{2}$(J)$S_{0}^{\partial}\cap \mathcal{P}_{w_{1}}\alpha$contains a club in(2)$S_{1}^{l}\cap P_{w_{1}}\alpha$contains a club in

In particular both

$S_{0}^{\delta}$

and$S_{1}^{c}\sim are$$\mathcal{P}_{w_{1}}$

) be
$a$

$\mathcal{P}_{w_{1}}a$$a$$\coprod_{w_{1}}$forfor eve-sequence. Then the following$eve\eta\alpha\in E_{0}^{2}\backslash w_{1}$stationary in$\eta a\in E_{1}^{2}$$\mathcal{P}_{\omega_{1}}\omega_{2}$

.

.

.

56

Proof. (1) Suppose thatthe set of all

club in$\mathcal{P}_{w_{1}}a$$a$$x\in \mathcal{P}_{w_{1}}$, andsuch that$C\subseteq S_{0}^{c}\sim$(2) Suppose that
Let $C$ be the set of all$a\in E_{1}^{2}$and$supx=\beta_{x\cap w_{1}}\not\in x$.
club inWe claim that$a\in E_{0}^{2}\backslash \omega_{1}$. Note that o.t.
and o.t.

$c_{\alpha}\subseteq x$

.

. Let

is countable. Let $C$ be. Then $C$ is a$c_{\alpha}$$c_{\alpha}<x\cap\omega_{1}\in w_{1}$be the increasing enumeration of .
is a countable limit ordinal,
such that. Then it is easy to see that is a$\langle\beta_{i}|i<w_{1}\rangle$$c_{\alpha}$$x\cap w_{1}$$x\in \mathcal{P}_{\omega_{1}}\alpha$$C$$\{\beta_{i}|i\in x\cap w_{1}\}\subseteq x$$\mathcal{P}_{w_{1}}a$. Note that if$C\subseteq S_{1}^{\mathcal{E}}$$x\in C$then$c_{\sup x}=c_{\beta_{x\cap u_{1}}}=\{\beta_{i}|i\in x\cap w_{1}\}$. Hence if $x\in C$ thenby the coherency of

Therefore$C\subseteq S_{1^{\vee}}^{c}$$carrow$.

$c_{\sup x}\subseteq x$

and o.t.

$c_{\epsilon upx}=x\cap w_{1}$.

$\square$We prove the following:-sequence.
aholds. Let
Theorem 5.2. Assume that GCH and. w-distributive forcing extension in which
Then there exzsts an.
$k=0,1$
holds for both$c\sim be$$\square _{w_{1}}$$\square _{w_{1}}$$SR_{k}(S_{k}\mathfrak{h}$$\omega_{2}-c.c$andare absolute between the
In the above theorem note that both
ground model and the forcing extension because the extension preserves all
cardinais and adds no new countable subsets of ordinais.
The extension of the above theorem will be obtained by making all nonrenonstationary by a countable supportandflecting stationary subsets ofiteration of club shootings.
First we describe the club shooting used in each stage:$S_{0^{\wedge}}^{c}$$S_{1}^{\mathcal{E}}$$S_{1}^{\partial}$$S_{0}^{\delta}$

Deflnition 5.3. Let be a subset of
consisting of all such that$S$$\mathcal{P}_{\omega_{1}}w_{2}$. Then let$\mathbb{C}(S)$be the forcing notion

$p$

(i)

$p$

is a function ffom(ii) if $x\in S$ andfor some$p\in \mathbb{C}(S)$$x\subseteq d$$d\in \mathcal{P}_{w_{1}}w_{2}$we let$d_{p}$.

$d\cross d$then$p\leq q$

denote$x$to$w_{1}$

is not closed under

if and only if$d\in \mathcal{P}_{\omega_{1}}\omega_{2}$Below we present easy facts on
Lemma 5.4. Let(1) For every$S$(2) Suppose that $G$ is afunction from$\cup G$.

$(S)\mathbb{C}(S)$is better.

the set$\mathbb{C}(S)$$w_{2^{V}}\cross w_{2^{V}}$$p\supseteq q$$p$.

for each$\mathcal{P}_{w_{1}}\omega_{2}$.
is dense in-generic filter over V.
$w_{1^{V}}f$. For each:$\{p\in \mathbb{C}(S)|y\subseteq d_{p}\}$to$p,$ $q\in \mathbb{C}(S)$satisfying (i) and (ii) above.

$\mathbb{C}(S)$be a subset of$y\in \mathcal{P}_{\omega_{1}}\omega_{2}$,$\mathbb{C}(S)$.

is a total
and there are no $x\in S$ closed underThen$\cup G$

57
Proof. (1) Take an arbitrary$y\in \mathcal{P}_{\omega_{1}}\omega_{2}$

.
find
with
, and takeLet betodefined as follows:$p^{*}\leq p$$y\subseteq d_{p}\cdot$$d^{*}$$d_{p}\cup y$$\gamma\in w_{1}\backslash d^{*}$

and an arbitrary. Then let$p^{*}$. We must$p\in \mathbb{C}(S)$be a function $homd^{*}\cross d^{*}$$w_{1}$$p^{*}(a)=$$\{\begin{array}{ll}p(a) if a\in d_{p}\cross d_{p}\gamma \end{array}$otherwise

then is not closed under .
All we have to show is that if $x\in S$ and.
below and thatis a condition inThis implies that
$x\in
S$
. In
when
. First consider the case
Suppose thatand. Hence is not closed under
this case is not closed under because. In this case there
which extends . Next consider the case when.
, and thus
. ThenexistsTherefore is not closed under .
$x\subseteq d^{*}$$p^{*}$$p^{*}$$x$$y\subseteq d_{p}\cdot=d^{*}$$p$$\mathbb{C}(S)$$x\subseteq d_{p}$$x\subseteq d^{*}$$x$$p\in \mathbb{C}(S)$$p$$p^{*}$$x$$x\not\subset d_{p}$$p$$p^{*}(a)\not\in x$$p^{*}(a)=\gamma\not\in d^{*}\supseteq x$$a\in(x\cross x)\backslash (d_{p}\cross d_{p})$$p^{*}$$x$(2) Clear from (1).

satisfies the properties (i) and (ii) in Definition 3.6. We check(3) Clearlysatisfies (iii) and (iv).
thatFirst we check (iii). Suppose that $p,q\in \mathbb{C}(S)$ and that $pr$ (dom $p\cap domq$ )$qr$ (domof and .
dom ). We must find a common extension
be a function $hom$
.
let
Then
,
and t&e
Let d’ bedefined as follows:to$\mathbb{C}(S)$$\mathbb{C}(S)$$=$$p\cap$$p^{*}$$q$$d’\cross d^{*}$$q$$p^{*}$$\gamma\in w_{1}\backslash d^{*}$$d_{p}\cup d_{q}$$p$$w_{1}$$p^{*}(a)=$$\{\begin{array}{ll}p(a) if a\in d_{p}xd_{p}q(a) if a\in d_{q}xd_{q}\gamma \end{array}$otherwise

is wel-defined because and coincide on dom $p\cap domq$ . An we have tothen is not closed under .
show is that if $x\in S$ andthen the same argument as in. If
Suppose that $x\in S$ andthe proof of (1) shows that is not closed under and thus that is not closed
then is not closed under , and hence is notunder . Similarly, ifclosed under .
and an. In this case take an
andSo suppose that.
and
, and let
. Then $a\in x\cross x$ but
Hence $p^{*}(a)=\gamma\not\in x$ . Therefore is not closed under .
is a descending sequence in
Next we check (iv). Suppose thatbe a lower bound ofwhich has a lower bound. Let. IFYom thistois a restriction of
Then.
it is clear that$p$$p^{*}$$q$$x\subseteq d^{*}$$x\subseteq d^{*}$$x\subseteq d_{p}$

$x$

$p$

$x$

$x$

$q$

$x$

$x\subseteq d_{q}$$p^{*}$$p^{*}$$x$$p^{*}$$x\not\subset d_{p}$$a$$\beta\in x\backslash d_{q}$$a\in x\backslash d_{p}$$x\not\subset d_{q}$$a\not\in d_{p}\cross d_{p}$$:=\langle a,\beta\rangle$$a\not\in d_{q}\cross d_{q}$$p^{*}$$x$$\langle p_{n}1n\in w\rangle$$\langle p_{n}|n\in w\rangle$$p^{*}$$\mathbb{C}(S)$$p^{*}$$\bigcup_{\mathfrak{n}\in w}p_{n}$.

$( \bigcup_{n\in w}d_{p_{n}})\cross(\bigcup_{n\in w}d_{Pn})$$\square$$\bigcup_{n\in w}p_{n}\in \mathbb{C}(S)$Club shootings which we iterate will be T-complete for some stationary
to be T-complete:. Here we present a sufficient condition for$\mathbb{C}(S)$$T\subseteq \mathcal{P}_{w_{1}}\omega_{2}$Deflnition 5.5. For$S,T\subseteq \mathcal{P}_{w_{1}}w_{2}$let $\Phi(S,T)$ be the following principle:

58
There enist a regular cardinal
and an expansionof the$M$
structuresuch that if
is a countable elementary submodd$M\cap\omega_{2}\in
T$
then
.
of with$\theta>2^{w_{2}}$$\mathcal{M}$$\langle \mathcal{H}_{\theta}, \in\rangle$$S\cap \mathcal{P}(M)\subseteq M$$\mathcal{M}$

While we do not use, the standard argument shows that $\Phi(S,T)$ is equivalent
with the following:
If is a sufficiently large regular cardinal, and $M$ is a countableelementuy submodel ofwith $M\cap w_{2}\in T$ then$\theta$$S\cap$$\langle \mathcal{H}_{\theta}, \in,S,T\rangle$$\mathcal{P}(M)\subseteq M$.

Now we prove that $\Phi(S,T)$ is a sufficient condition forcomplete:

Lemma 5.6. Suppose that

$S,T\subseteq \mathcal{P}_{w_{1}}\omega_{2}$T-complete.

$\mathbb{C}(S)$to be T-and that $\Phi(S,T)$ holds. Then$\mathbb{C}(S)$isandbe witnesses of $\Phi(S,T)$ . Suppose that $M$ is acountableelementary submodel ofwith $M\cap\omega_{2}\in T$ and thatis an-generic $sequen\infty$ . By Lemma 3.3 it suffices to show thathas alower bound. Moreover it suffices for this to show thatis.
acondition in. ThenLetbeis afunction $bomd^{*}\cross d^{*}$ to. We show that if $x\in S$ andthen is not closed under .
$x\inS$. First note that
andSuppose thatfor eai $n\in w$,
becauseacountable set whii belongs toHence. Thus $x\in M$ by $\Phi(S,T)$ .
and soThen the set $D:=\{p\in \mathbb{C}(S)|x\subseteq d_{p}\}$ belongs to M. Moreover $D$ is dense
open inby Lemma 5.4 (1). Hence there exists $n\in w$ with $p_{n}\in D$ . Then
, and is not closed under. Therefore is not
because$c1_{08}ed$
which extends .
ako
under

Proof. Let$\mathcal{M}$$\theta$$\langle p_{n}|n\in w\rangle$$\mathcal{M}$$(M, \mathbb{C}(S))$$\langle p_{n}|n\in w\rangle$$p^{*}$$:= \bigcup_{n\in w}p_{n}$$\mathbb{C}(S)$$d^{*}$$d^{*}\in \mathcal{P}_{w_{1}}w_{2},$$\bigcup_{n\in w}h_{\mathfrak{n}}$$\bm{t}dp^{*}$$x\subseteq d^{*}$$w_{1}$$x$$p^{*}$$x\subseteq d’$$d_{p_{\hslash}}\subseteq M$$M\prec(\mathcal{H}_{\theta},$$d_{Pn}i_{8}$$\in\rangle$.

$d^{*}\subseteq M$$x\subseteq M$$\mathbb{C}(S)$$x$$x\subseteq d_{Pn}$

$:=$

$x$

$\square$$p_{n}$

Next we present a stationary
iterate wil be T-complete. For a$T^{\delta}$$p_{n}\in \mathbb{C}(S)$$p_{n}$$p^{*}$

the set of alsuch that club shootings which we-sequencelet$T\subseteq \mathcal{P}_{w_{1}}\omega_{2}$$\square _{w_{1}}$$x\in \mathcal{P}_{w\iota}w_{2}$$c=arrow(c_{\alpha}|\alpha\in Lim\omega_{2}\rangle$

such that(i)and $\sup x\not\in x$ ,(ii) $0.t.c_{\epsilon upx}>x\cap w_{1}$ .
$x\cap w_{1}\in\omega_{1}$fromis the property (ii) of its elements.
and
The main difference ofis stationary using Lemma 4.1:
It is easy to see that-sequence .
is $stationa\eta$ inLemma 5.7.
for every-sequence.
is a
Proof. Suppose thato.t.
contains
For each $i\in w_{1}$ let $E_{i};=$ {}. Note that
. Hence is a stationary subset of .
a club in for every
that
note
Here
also
$T^{\partial}$$S_{0}^{8}$$S_{1}^{\mathcal{E}}$$T^{\partial}$$T^{\mathcal{E}}$$\square _{w_{1}}$$\mathcal{P}_{w_{1}}\omega_{2}$$c=arrow\langle c_{\alpha}|a\in Limw_{2}\rangle$$\alpha\in E_{0}^{2}|$$\beta$$\beta\in E_{1}^{2}$$E_{i}$$c\sim$$\Pi_{\omega\iota}$$c_{\alpha}>i$$E_{i}\cap\beta$$E_{0}^{2}$$T^{\delta}= \{x\in \mathcal{P}_{w_{1}}w_{2}|x\cap\omega_{1}\in\omega_{1}\wedge\sup x\not\in x\wedge\sup x\in Eae\cap\omega_{1}\}$

Therefore$T^{\mathcal{E}}$is stationary by Lemma 4.1.

.
$\square$59
We want to show something like that if is a nonreflecting subset oforthen
is -complete. For this we slightly reduce
as follows:andWe call a sequencesystem if)a
is a surjection
fromto for each
. For a -sequence , a surjection system
and $k=0,1$ let$S$$S_{1}^{c}\sim$$T^{c}\sim$$\mathbb{C}(S)$$S_{1}^{c}arrow$$S_{0^{\vee}}^{c}$$su\dot{\eta}ection$$\tilde{\pi}=\langle\pi_{\alpha}|\alpha\in w_{2}$$a$$\omega_{1}$$S_{0^{\vee}}^{c}$$\alpha\in\omega_{2}$$\pi_{\alpha}$$c\sim$$\square _{w_{1}}$$\tilde{\pi}=\langle\pi_{\alpha}|a\in w_{2}\backslash w_{1}\rangle$$S_{k}^{\partial,\#}$.

$:=\{x\in S_{k}^{c}\sim|\forall\alpha\in x, x\cap\alpha=\pi_{\alpha}(x\cap w_{1})\}$

Note thatis nonstationary.
We claim the following.
Lemma 5.8. Suppose that} is a surjection system.
I$S_{k}^{\partial}\backslash S_{k}^{\delta,\#}$$c\sim=\langle c_{\alpha}|\alpha\in Limw_{2}\rangle$

is a$\square _{w_{1}}$-sequence and that$\alpha\in\omega_{2}\backslash w_{1}$$\tilde{\pi}=\langle\pi_{\alpha}$(J) Let $S$ be a subset ofwhich does not reflect to any rdinal in-complete.
Thenis$S_{0}^{c}arrow,\pi$$\mathbb{C}(S)$$0$$E_{0}^{2}\backslash w_{1}$.

$T^{c}arrow$be a subset ofwhich does not reflect to any ordinal in . Thenis -complete.
To prove Lemma 5.8 we need the following easy lemma:-sequence. Let be) is aLemma 5.9. Suppose that$M$
be a countable $elementa\eta$ submodela sufficiently large regular rdinal and
\
l
a
n
g
l
e
such that $a^{*}< \sup(M\cap w_{2})$ ,
.
in
let
be
ordinalMoreoveranof(2) Let$S$$S_{1}^{\delta,P}$$E_{1}^{2}$$T^{\delta}$$\mathbb{C}(S)$$c’=\langle c_{\alpha}|\alpha\in Lim\omega_{2}$$\theta$$\square _{w_{1}}$$ca$$\mathcal{H}_{\theta},$$\in,$$a^{*}\not\in M$$\alpha^{*}$$c\gamma$$E_{0}^{2}$

and $\sup(M\cap a^{*})=a^{*}$ . Then o.t.

Proof. Let$\beta^{*}$$:= \min(M\backslash \alpha^{*})$. Then$c_{\alpha}\cdot=M\cap\omega_{1}$$\beta^{*}\in M\cap w_{2}$.

, and $\sup(M\cap\beta^{*})=a^{*}<\beta’$ .

Moreover it easily follows $hom$ the elementarity of $M$ that $\rho*\in E_{1}^{2}$be the increasing enumeration of . We claim thatby the elementarity of $M$ . HenceFirst note that
$M$ . Thus

$w_{1}\rangle$$c_{\beta}\cdot$$c_{\beta}\cdot\in M$. Let$\langle\beta_{i}|i\in$$\sup(M\cap\beta^{*})=\beta_{M\cap w_{1}}$.

$\{\beta_{i}|i\in M\cap w_{1}\}\subseteq$.

$\sup(M\cap\beta^{*})\geq\sup\{\beta_{i}|i\in M\cap w_{1}\}=\beta_{M\cap w_{1}}$On the other hand assume that $\sup(M\cap\beta^{*})>\beta_{M\cap d_{1}}$ . Then we can take.
. Let be the least ordinal $<w_{1}$ such thatwith
. But $j\in M\cap w_{1}$ by the elementarity ofThen $j\geq M\cap w_{1}$ because$M$ . This is a contradiction. Therefore.
. Recall that $a^{*}= \sup(M\cap$
Now we have shown thatby the coherency of .
. Hence. Then.
Therefore o.t.

$\beta\in M\cap\beta^{*}$$\beta_{j}\geq\beta$$j$$\beta\geq\beta_{M\cap\omega_{1}}$$\beta\geq\beta_{M\cap w_{1}}$$\sup(M\cap\beta^{*})\leq\beta_{M\cap w_{1}}$$\sup(M\cap\beta^{*})=\beta_{M\cap w_{1}}$$c_{\alpha}\cdot=\{\beta_{i}|i\in M\cap w_{1}\}$$a^{*}=\beta_{M\cap\omega_{1}}$$\beta^{*})$$c\sim$$\square$$c_{\alpha}\cdot=M\cap w_{1}$Now we prove Lemma 5.8:

.licity of our notation let$S_{0},$$S_{1}$

and$T$

denote$s_{0}^{c,\pi},s_{1}^{\zeta_{\pi_{andT^{\delta}re\epsilon pective1y}}^{Lemma5.8.Forsimp}}P_{\sim}mof_{0}$(1) By Lemma 5.6 it suffices to show that $\Phi(S,T)$ holds. Let be a sufficiently
large regular cardinal, and let $M$ be a countable elementwy submodel of$\theta$$(\mathcal{H}_{\theta},$with $M\cap w_{2}\in T$ . Moreover suppose that $x\in S$ andthat $x\in M$ . Before starting note that $x\cap w_{1}\leq M\cap w_{1}\in w_{1}$ .
First we claim the following:,$S_{C}^{\vee},R\rangle$$x\subseteq M$$\in$. We show

60
Claim 1. $supx\in M$ .

Proof of Claim. On the contrary assume that $supx\not\in M$ . Then note that

: If $supx=\sup(M\cap w_{2})$ then $M\cap w_{1}<0.t.c_{\sup x}$ because. On the other hand, if up $x< \sup(M\cap w_{2})$ then $M\cap\omega_{1}=0.t.c_{\sup x}$$M\cap w_{1}\leq 0.t.c_{\sup x}$$M\cap w_{2}\in T$$s$by Lemma 5.9.
Note also that $x\cap w_{1}>$ o.t.
because $x\in S_{0}$ .
o.t.
. This contradicts that.
$c_{\sup x}$

Hence $M\cap w_{1}\leq$$x\subseteq M$$c_{\epsilon upx}<x\cap w_{1}$$\square (Claim)$

Next we claim the following:

Claim 2. $x\cap w_{1}<M\cap\omega_{1}$ .

Proof of Claim. Assume not. Then $x\cap w_{1}=M\cap w_{1}$ . First note that $M\cap a=$$\pi_{\alpha}(M\cap w_{1})$for each$\alpha\in M\cap\omega_{2}$by the elementarity of $M$ . Hence$M \cap\sup x=\bigcup_{\alpha\in x}\pi_{\alpha}(M\cap w_{1})=\bigcup_{\alpha\in x}\pi_{\alpha}(x\cap w_{1})=x$.

The last equality follows from $x\in S_{0}$ .
Here note thatis nonstationary by the assumption on .
Moreoverby Claim 1. Hence there exists a function$f \in Mhom[\sup x]<w$ to $supx$ such that every element of
is not$x=M\cap\supx$, and so is closed under by the elementarity
closed under But$\square (Claim)$
of $M$ . Because
this is a contradiction.
$S$$S \cap \mathcal{P}_{\omega_{1}}(\sup x)$$supx\in M\prec\langle \mathcal{H}_{\theta}, \in, S\rangle$$f$$S \cap \mathcal{P}_{\omega_{1}}(\sup x)$

.

$x$

$f$

$x \in S\cap \mathcal{P}_{w_{1}}(\sup x)$

Now $x=\cup\{\pi_{\alpha}(x\cap w_{1})|\alpha\in c_{\sup x}\}$ because $x\in S_{0}$ . Hence is definableinthe parameters
and $supx$ . But bothand $supx$$M$
by Claim 1 and 2, andbelong to. Therefore $x\in M$ .
$x$$x\cap w_{1}$$\langle \mathcal{H}_{\theta}, \in, carrow,\tilde{\pi}\rangle hom$$x\cap w_{1}$$M\prec\langle \mathcal{H}_{\theta}, \in,\overline{c},\tilde{\pi}\rangle$(2) We show that $\Phi(S,T)$ holds. Letshow that $x\in S$ .
First we claim the following:$\theta,$

$M$

and

$x$

be as in the proof of (1). We

Claim 3. $supx\in M$ .

Proof of Claim. First note that $supx<\sup(M\cap\omega_{2})$ : Otherwise $supx=$$\sup(M\cap w_{2})$, and$M\cap w_{1}<0.t.c_{\epsilon upx}=x\cap\omega_{1}$because $M\cap w_{2}\in T$ and $x\in S_{1}$ . This contradicts that.
$supx\not\inM$$M\capw_{1}=$
Now assume that
. Theno.t.
by Lemma 5.9.
$M\capw_{1}=x\cap\omega_{1}$$x\in
S_{1}$
Hence
because. Then the same argument as in the
proof of Claim 2 shows that $M \cap\sup x=x$ .
, and thus
Let be $\min(M\backslash \sup x)$ . Then
is nonstationary
by the assumption on . Because
there exists a function$f\in Mhom[\mathcal{B}^{*}]<\omega$ tosuch that every element of
is not closed
under . But $x=M \cap\sup x=M\cap\beta^{*}$ , and so is closed under by the$\square (Claim)$
elementarity of $M$ . This contradicts that $x\in S$ .
$x\subseteq M$$c_{\sup x}$$\beta’\in E_{1}^{2}$$\beta^{*}$$S$$\beta^{*}\in M\prec(\mathcal{H}_{\theta},$$S\cap \mathcal{P}_{\omega_{1}}\beta^{*}$$\in,$$\beta$$f$$S\rangle$$S\cap \mathcal{P}_{w_{1}}\beta^{*}$

$x$

$f$

61
Note that $x\cap\omega_{1}=0.t.c_{\sup x}\in M\cap\omega_{1}$ by Claim 3 and the elementarity of. The rest of the proof is similar as (1).
First $x=\cup\{\pi_{\alpha}(x\cap\omega_{1})|a\in c_{8Upx}\}$ , and thus is definable inbelongs. Moreover both $supx$ andfrom the parameters $supx$ and
$M$
$x\in
M$
.
. Therefore
to , and$M$$x$$\langle \mathcal{H}_{\theta}, \in, carrow,\tilde{\pi}\rangle$$x\cap w_{1}$$x\cap\omega_{1}$$\square$$M\prec\langle \mathcal{H}_{\theta}, \in, carrow,\tilde{\pi}\rangle$

Now we can prove Theorem 5.2 by combining lemmata above:
in $V$ . We make all nonrenonstationary by a countable support iteration

Proof of Theorem 5.2. Take a surjection system

flecting subsets ofof club shootings.
First note thatand

$S_{0}^{\delta,\#}$$S_{1^{\vee}}^{c,i}$$\tilde{\pi}$. -distributive forcing
are absolute in all
respectively. Noteandand denote
extensions of . Let
extensions.
inall suchalso thatfor every
Then, by Lemmata 3.4, 3.5, 3.8, 5.4, 5.8, by GCH and by the standard bookkeeping method, we can construct a countable support iteration
and$S_{k}^{\overline{c},\pi}$$V$$S_{0},$$T^{\delta}$$w_{2^{-}}c.c$$T$$S_{1}$$|\mathbb{C}(S)|=w_{2}$$S_{0}^{\sim},$$\omega$$T^{c}\sim$$S_{1}^{\sim}$$S\subseteq \mathcal{P}_{w_{1}}w_{2}$$\langle \mathbb{P}_{\xi},\mathbb{Q}_{\eta}|\xi\leq$$\omega_{3},\eta<w_{3}\rangle$(i)$\mathbb{P}_{\xi}$with the $f_{0}n_{oW}ing$ properties:

has the$\omega_{2}- c.c$(ii) If $\eta<w_{3}$ then$|\vdash\eta$. and is -distributive for each$\omega$“ $\dot{\mathbb{Q}}_{\eta}=\mathbb{C}(\dot{S})$$|\vdash\eta$Hence(iii) If“$|\vdash\eta$$\xi<\omega_{3}$-name$\dot{S}$such that either

does not reflect to any ordinal in$E_{0}^{2}$,“ $\dot{S}\subseteq S_{1}\wedge\dot{S}$does not reflect to any ordinal in$E_{1}^{2}$“.

$\dot{\mathbb{Q}}_{\eta}$

andis T-complete and better$\dot{S}$is a$\mathbb{P}_{\xi}$$\wedge|\dot{\mathbb{Q}}_{\eta}|\leq\omega_{2}’$.

-name such that either“ $\dot{S}\subseteq S_{0}\wedge\dot{S}$does not reflect to any ordinal in$E_{0}^{2}$‘“ $\dot{S}\subseteq S_{1}\wedge\dot{S}$does not reflect to any ordinal in$E_{1}^{2}$“$|\vdash\epsilon$or$|\vdash\epsilon$$\mathbb{P}_{\eta}$.

” $\dot{S}\subseteq S_{0}\wedge\dot{S}$or$|\vdash\eta$for some$\xi\leq w_{3}$then there exists$\eta\in w_{3}\backslash \xi$such that$|\vdash\eta$“ $\dot{\mathbb{Q}}_{\eta}=\mathbb{C}(\dot{S})$.

. and is -distributive. Let be a -generic filterThen
has theover $V$ . Then the standard argument shows that the folowing both hold in$w_{2^{-}}c.c$$\mathbb{P}_{\omega_{3}}$$\omega$$G$$\mathbb{P}_{w_{3}}$$V[G]$ :$\bullet$$\bullet$andIf
nonstationary.
$S\subseteq S_{0}$Iftionary.

$S\subseteq S_{1}$$S$

does not reflect to any ordinal in

and $S$ does not reflect to any ordinal in$E_{1}^{2}$$E_{0}^{2}\backslash w_{1}$

then

$S$

is

then $S$ is nonsta-

That is, $SR_{k}(S_{k})$ holds for both $k=0,1$ in $V[G]$ . But note that
holds for both $k=0,1$ in $V[G]$ .
nonstationary. Therefore
This completes the proof.

$S_{k}^{\delta}\backslash S_{k}$is$SR_{k}(S_{k}^{c})\sim$$\square$62

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