数理解析研究所講究録1095 巻 1999 年 16-21

16Game theoretic properties of the club filter on$P_{\kappa}\lambda$

MASAHIRO SHIOYA

Let$\kappa$

be a regular cardinal$>\omega,$$\lambda$a cardinal$\geq\kappa$

and Prikry [JP] called $F$ precipitous when the poset

and $F$ a filter on$F^{+}$$P_{\kappa}\lambda$. Jechordered by inclusion forcesthat the generic ultrapower is well-founded. Refining an ingenious idea of Foreman,

Magidor and Shelah [FMS], Goldring [G] and Shelah independently established thatthe club filter$C_{\kappa\lambda}$on$P_{\kappa}\lambda$

can be made precipitous by Levy collapsing a Woodin

cardinal to the successor of when$\lambda$

be precipitous e.g. when$\lambda$$\lambda$

is regular. In [MS] we showed thatis strong limit and of cofinality$<\kappa$$C_{\kappa\lambda}$

cannot. On the other hand

Galvin, Jech and Magidor [GJM] observed that precipitousness is also definable interms of games (see below). This led us to investigate game theoretic properties of$C_{\kappa\lambda}$

in this paper.

We refer to Kanamori [K] for basic material. We use
cardinal. By$S_{\kappa}^{\omega}$

and $\lim X$ for$X\subset\kappa$

we denote the set {and $\{\alpha<\kappa : \sup(X\cap\alpha)=\alpha>0\}$ respectively. For$\pi_{\alpha}$:$|\alpha|arrow\alpha$$T_{n}\in F^{+}$

wins iff$\mu$

to denote a regular$\alpha<\kappa$$\alpha<\lambda$$\bigcap_{n<\omega}S_{n}=\emptyset$$S_{n}\supset T_{n}\supset S_{n+1}$. Recall from [GJM] that

$S_{n}\in F^{+}$. We define the game$F$$\alpha=\omega$}

we fix a bijection. Suppose that two players I and II take in turnrespectively so that: cf$\mathcal{G}_{0}(F)$andby: I

is precipitous iff I has no winning

1991 Mathematics Subject Classification.
.
Part of this work was done during the author’s stay at Boston University as one of the Japanese
acknowledges Prof. Kanamori’s hospitality. He also
Overseas Research Fellows. He
thanks Mr. Yoshinobu for pointing out Proposition 3.
$03\mathrm{E}05,03\mathrm{E}55$$\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{u}_{\mathrm{y}}$Typeset by$A_{\mathcal{M}}S-\mathrm{u}\mathrm{E}\mathrm{X}$17strategy in$\mathcal{G}_{0}(F)$$| \bigcap_{n<\omega}S_{n}|\leq 1$. Also the game(resp.

$\mathcal{G}_{1}(F)$$\bigcap_{n<\omega}s_{n}\in F^{*}$

I has no winning strategy in$\mathcal{G}_{*}(F)$(resp.

$\mathcal{G}_{*}(F)$) is defined by: I wins iff). As observed by Jech [J], $p+$ is. A winning strategy for I in$\omega_{1}$-Baire iff$\mathcal{G}_{1}(c_{\kappa})\mathrm{h}\mathrm{o}\mathrm{m}[\mathrm{G}\mathrm{J}\mathrm{M}]$yields the following

Proposition 1.

$C_{\kappa\lambda}^{+}$

is not$\omega_{1}$-Baire.

Proof. We give a winning strategy for I in
$S_{\kappa}^{\omega}\}$$\mathcal{G}_{*}(C_{\kappa\lambda})$. We let$S_{0}=\{x\in P_{\kappa}\lambda$

be the first move of I, and for $x\in S_{0}$ fix an unbounded set

Next suppose that

we have$\gamma_{n}<\kappa$$T$$\{\alpha_{n}^{x} ::$x\cap\kappa\in$n<\omega\}\subset x\cap\kappa$.

is an n-th move of II. Then by the regressive function lemmasuch that$\{x\in T:\alpha_{n}^{x}=\gamma_{n}\}$is stationary, which we let be the next

move of I.
Let$\langle S_{n} :above. Thenn<\omega\rangle$

be a play of I following the above strategy with$\bigcap_{n<\omega}S_{n}\subset\{x\in \mathcal{P}_{\kappa}\lambda :$\langle\gamma_{n} :n<\omega\rangle$, as desired.

x\cap\kappa=\sup_{n<\omega}\gamma_{n}\}\in C_{\kappa\lambda}^{*}$

as$\square$

The following lemma is proved similarly to the original version in [GJM].

Lemma 1. Let
unbounded in$S\subset\{x\in \mathcal{P}_{\kappa}\lambda :$x\cap\kappa$

x\cap\kappa\in S_{\kappa}^{\omega}\}$for $x\in S.$ Then for somebe stationary and$\{\alpha_{n}^{x} :n<\omega\}$$m<\omega\{\gamma<\kappa:\{x\in S:\alpha_{m}^{x}=\gamma\}$isstationary} is unbounded.
Now we have a$P_{\kappa}\lambda$

generalization of the Galvin-Jech-Magidor result [GJM],

which shows that Goldring’s result is sharp from a game theoretic point of view:
Proposition 2. II has no winning strategy in

Proof. Let$\tau$

suc.

be a strategy for II. We construct a play in which II follows

I wins. Build inductively$t\in \mathcal{T}$$\mathcal{G}_{0}(C_{\kappa\lambda})$$\tau(t)=\{\gamma<\kappa :$\langle S_{t} :t\in \mathcal{T}\rangle$

t*\langle\gamma\rangle\in \mathcal{T}\}$with$\mathcal{T}$a subtree ofis unbounded and ($\kappa^{<\omega}$$S_{t|i}$:$\tau$

andsuch that for any$i<|t|\rangle$

is a partial

18

play of I with which II follows

whereset$S\subset S_{\kappa}^{\omega}$$\{\alpha_{n}^{x} :$t\in \mathcal{T}\cap\kappa^{n}$and$S_{\kappa}^{\omega}-s$n<\omega\}\subset x\cap\kappa$

. Let

$T$

$\tau$

as follows: First set

are both stationary, and for. Next suppose that$\langle S_{t} :be the n-th move of II following

Then by the above lemma take $m<\omega$ such that {stationary} is unbounded, which we let be$\{x\in T:\alpha_{m}^{x}=\gamma\}$an unbounded set$\mathcal{T}$with$\gamma_{n}\leq b(n)$. Fix$\gamma\in S_{\kappa}^{\omega}-S$$\{\gamma_{n} :$S_{\emptyset}=\{x\in P_{\kappa}\lambda :n<\omega\}\subset\gamma$

with$\mathrm{s}\mathrm{u}\mathrm{c}\mathcal{T}(t)$$\tau$$x\in S_{\emptyset}$$\gamma\in\lim \mathrm{s}\mathrm{u}\mathrm{c}\mathcal{I}(t)$$\langle S_{t|i} :i<n\rangle$.

: $\{x\in T:\alpha_{m}^{x}=\gamma\}$ is$\gamma\in \mathrm{s}\mathrm{u}\mathrm{c}_{\mathcal{T}}(t)$for any. Build inductively a branch

for any $n<\omega.$ Thenis defined. Fix

when I plays. For,

fix an unboundedt\in \mathcal{T}\cap\kappa^{n}\rangle$$\gamma<\kappa$x\cap\kappa\in S\}$$\bigcap_{n<\omega}Sb|n\subset\{x\in S_{\emptyset} :set$S_{t*\langle\gamma\rangle}=$$t\in T\cap\gamma^{<\omega}$$b:\omegaarrow\gamma$, andthroughx\cap\kappa=\gamma\}=\emptyset$, asdesired.
For the rest of the paper we give some cases where I has a winning strategy in thegame, and show optimality of Goldring’s result in a different way. Let us$\mathcal{G}_{1}(C_{\hslash}\lambda)$

begin by the following observation, which should be folklore likeof$F^{+}$

for a fine filter $F$ on$P_{\omega_{1}}\lambda$$\mathcal{G}_{1}(F)$-Baireness:Proposition 3. Let $F$ be a normal filter onin$\mathrm{n}\mathrm{o}\mathrm{n}-\omega 1$$P_{\omega_{1}}\lambda$. Then I has a winning strategy.

Proof. We let$P_{\omega_{1}}\lambda$

be the first move of I, and for$x\in P_{\omega_{1}}\lambda$

fix a list$\{\alpha_{n}^{x} :n<\omega\}$of . Next suppose that $T\in F^{+}$ is an n-th move of II. Then by the normality of $F$$x$

we haveLet$\gamma_{n}<\lambda$$\langle S_{n} :above. Thenwith

n<\omega\rangle$$\{x\in T:\alpha_{n}^{x}=\gamma_{n}\}\in F^{+}$, which we let be the next move of I.

be a play of I following the above strategy with$\bigcap_{n<\omega}S_{n}=\{\gamma_{n}\}$$\langle\gamma_{n} :n<\omega\rangle$

as, as desired.

Our first result, Proposition 4 follows$\mathrm{h}\mathrm{o}\mathrm{m}$Proposition 3 and our Theorem below.

We nevertheless present it here because of the following lemma, which refines a fact

19

from [BT] and seems of independent interest.

Lemma 2. Let$\lambda<\kappa^{+\omega}$$\lambda(\mathrm{c}\mathrm{f}\sup(x\cap\mu)=\omega)\}$. Then there is aon which the map

Proof. We go by induction onin$P_{\kappa}\kappa$$P_{\kappa}\gamma$:. Next suppose that$\pi_{\gamma}^{-1}x=x\cap\lambda\in S$$\langle\sup(x\cap\mu) :“$\pi_{\gamma}$$S\subset P_{\kappa}\lambda$

stationary inset$S\subset\{x\in P_{\kappa}\lambda:\kappa\leq\forall\mu\leq$$x \vdasharrow\langle\sup(x\cap\mu):\kappa\leq\mu\leq\lambda\rangle$. For$\lambda=\kappa$

just note thatis as above. Then for\kappa\leq\mu\leq\lambda\rangle$$P_{\kappa}\lambda^{+}$. Fix

Then by stationarity of$\gamma\in S_{\lambda^{+}}^{\omega}-\lambda S_{\gamma}=\{x\in$

we have, since $x\cap\lambda=y\cap\lambda\in S$ . We claim that$f$

$S_{\gamma}$:$(\lambda^{+})^{<\omega}arrow\lambda^{+}$

some$x\in S_{\gamma}$. Takewithis injective.

is stationary$S_{\kappa}^{\omega}$

A $\sup x=\gamma$ } is stationary, and for any $x,$

\kappa\leq\mu\leq\lambda\rangle=\langle\sup(y\cap\mu) :$(y\cap\lambda)=y$

desired.

$\lambda<\kappa^{+\omega}$$stati_{\mathit{0}}naw$“$x=\pi_{\gamma}$$\cup\{S_{\gamma} :$\gamma\in S_{\lambda}^{\omega}+-\lambda$$x\cap\kappa\in\kappa$$y\in S_{\gamma}$with$(x\cap\lambda)=$\gamma\in S_{\lambda^{+}}^{\omega}-\lambda\}$with$f$is“ $\gamma^{<\omega}\subset\gamma$

is closed under$f$.

, as$\square$

Now the rest of the proof goes as in Proposition 1:
Proposition 4. I has a winning straiegy in$\mathcal{G}_{1}(C_{\hslash\lambda})$when$\lambda<\kappa^{+\omega}$

Proof. We let the stationary set as in Lemma 2 be the first move$x\in S_{0}$and a cardinal

suppose that$T$$\kappa\leq\mu\leq\lambda$

Let$\{\alpha_{\mu,n}^{x} :$S_{0}$of I, and forn<\omega\}\subset x\cap\mu$. Nextis an n-th move of II. Then by finite applications of the regressivefunction lenma we have$\gamma_{\mu,n})\}$

fix an unbounded set.

$\langle\gamma_{\mu,n} :\kappa\leq\mu\leq\lambda\rangle$

such that$\{x\in T:\kappa\leq\forall\mu\leq\lambda(\alpha_{\mu,n}^{x}=$is stationary, which we let be the next move of I.
$\langle S_{n} :n<\omega\rangle$

as above. Thenbe a play of I following the above strategy with$\bigcap_{n<\omega}S_{n}\subset\{x\in So :$\langle\gamma_{\mu,n} :n<\omega\rangle$,\kappa\leq\forall\mu\leq\lambda(\sup(x\cap\mu)=\sup_{n<\omega}\gamma_{\mu,n})\}$

which is at most a singleton, as desired.
For our main result below, we are indebted to Shelah’s notion [S] of internally20

approachable submodels of$H_{\theta}$(see also [BM]), as well as Baumgartner’s style [B]of purely combinatorial proofs.
Theorem. I has a winning strategy in

Proof. Let
between$\kappa$$\kappa$$\{\mu_{i} :andi<\omega\}$$S_{0}= \{\bigcup_{n<\omega}x_{n}$A $x_{n} \cup\{\sup(x_{n}\cap\mu_{i}) : i<\omega\}\subset x_{n+1}$ A$\pi_{\alpha}^{-1})\wedge\forall i<\omega$($\mathrm{c}\mathrm{f}\sup(Xn\cap\mu i)=\omega_{1}\wedge x_{n}\cap\mu_{i}$is stationary inFix$f$:$P_{\kappa}\lambda$$f$$\kappa>\omega_{1}$and:$\forall n<\omega(x_{n}\in P_{\kappa}\lambda$$\forall\alpha\in x_{n}(x_{n}$A

is closed undercontains a club subset of$x_{n}\cap\kappa\in$$\pi_{\alpha}$:$\alpha\in x_{0,\xi}$. We build inductively

as follows: First set$x_{\xi}= \bigcup_{\zeta<\xi}x0,\zeta$}

and$\sup(x_{n}\cap\mu_{i})$$\langle x_{n} :n<\omega\rangle$$x_{0}= \bigcup_{\xi<\omega_{1}}X\mathit{0}_{\xi},$

as above with)$\cup\cup\{\pi_{\alpha}^{-1}(X_{0},\xi \mathrm{n}\alpha) :for a limit . Next set, where$x_{0,\xi}$is defined in-\alpha\in x_{0,\xi}\}\cup\{\sup(x_{\mathit{0}},\xi\cap\mu_{i}) :$x_{n+1}= \bigcup_{\xi<\omega_{1}}xn+1,\xi$$\xi$fined inductively by $x_{n+1,0}=x_{n}\cup f$ “

, where$x_{n}^{<\omega} \cup\{\sup(x_{n}\cap\mu_{i}) :$(x_{0,\xi}\cap$i<\omega\}$$x_{n+1,\xi}$i<\omega\},$andis de-$X_{n+1,\xi}+1=$$x_{n+1,\xi} \cup\sup(x_{n}+1,\xi \mathrm{n}\kappa)\cup\cup\{\pi_{\alpha}"(xn+1,\xi \mathrm{n}|\alpha|);\alpha\in xn+1,\xi\}\cup\cup\{\pi_{\alpha}^{-1}(xn+1,\xi\cap\alpha)$$\alpha\in x_{n+1,\xi}\}\cup\{\sup(x_{n+}1,\xi\cap\mu_{i}) :$\{\sup(x_{n,\xi}\cap\mu_{i}) :We letwith$S_{0}$\xi<\omega_{1}\}\subset x_{n}$i<\omega\}$and$x_{\xi}= \bigcup_{\zeta<\xi}xn+1,\zeta$:

for a limit . Then$\xi$

is club in $\sup(x_{n}\cap\mu_{i})$ for any $n,$ $i<\omega$ .

be the first move of I, and for $x\in S_{0}$ fix$x= \bigcup_{n<\omega}x_{n}$$)\}$$\bigcup_{n<\omega}x_{n}$ductively by $x_{0,0}=\{\mu_{i} : i<\omega\},$ $x_{0,\xi+1}=x_{0,\xi} \cup\sup(x_{0,\xi}\cap\kappa)\cup\cup\{\pi_{\alpha}$ “$|\alpha|)$.

$\lambda<\kappa^{+\omega_{1}}$.

$\lambda^{<\omega}arrow\lambda$

closed underwhenlist (not necessarily injectively) the set of regular cardinals. First we claim that$\lambda$$\mathcal{G}_{1}(C_{\hslash}\lambda)$. Next suppose that

$T$n<\omega\rangle$

as aboveis an n-th move of II. Then by $n+1$

applications of the regressive function lemma we have$\{x\in T:\forall i\leq n(\sup(x_{n}\cap\mu_{i})=\gamma n,i)\}$$\langle x_{n} :$\langle\gamma_{n,i} :i\leq n\rangle$

such thatis stationary, which we let be the next moveof I.
Let$\langle S_{n} :n<\omega\rangle$

be a play of I following the above strategy with$\langle\gamma_{n,i} :i\leq n<\omega\rangle$

21

as above. To see that$\kappa\leq\mu\leq\lambda$$| \bigcap_{n<\omega}s_{n}|\leq 1$, fix$x,$ $y \in\bigcap_{n<\omega}s_{n}$we show that $x\cap\mu=y\cap\mu$ .

First take $i<\omega$ with$\bigcup_{i\leq n<\omega}y_{n}\cap\kappa=y\cap\kappa$$\mu_{i}=\kappa$. Then$x \cap\kappa=\bigcup_{i\leq n<\omega}Xn\cap\kappa=\bigcup_{i\leq n<\omega}\gamma_{n},i=$, as desired.

Next suppose that $x\cap\mu=y\cap\mu$ with
$i\leq n<\omega$. Then$x_{n}\cap y_{n}\cap\mu^{+_{\mathrm{i}\mathrm{S}}}$

Thus $x\cap\gamma_{n,i}=y\cap\gamma_{n,i}$ , since“$\pi_{\alpha}$. By induction on cardinals$\mu<\lambda$. Take $i<\omega$ with$\mu_{i}=\mu^{+}$

unbounded in $\sup(x_{n}\cap\mu^{+})=\sup(y_{n}\cap\mu^{+})=\gamma_{n},i$ .
$x \cap\alpha=\cup n\leq j<\omega Xj\cap\alpha=\bigcup_{n\leq j<\omega}\pi_{\alpha}$“$(x_{j}\cap\mu)=$$(x\cap\mu)=\pi_{\alpha}$ “ $(y \cap\mu)=\bigcup_{n\leq j<\omega}\pi_{\alpha}$ “ $(y_{j} \cap\mu)=\bigcup_{n\leq j<\omega}yj\cap\alpha=y\cap\alpha$$\alpha\in x_{n}\cap y_{n}\cap(\mu^{+}-\mu)$$y\cap\mu^{+}$. Fix. Now we havefor any$x \cap\mu^{+}=\bigcup_{i\leq n<\omega}x\cap\gamma n,i=\bigcup_{i\leq n<\omega}y\cap\gamma n,i=$

as desired, since $\sup(x\cap\mu^{+})=\sup(y\cap\mu^{+})=\sup_{i\leq n<\omega}\gamma n,i$ .

The limit case is clear from the inductive hypothesis.

$\square$

REFERENCES
[B]
[BT]
[BM]
[FMS]
[GJM]
[G]

J. Baumgartner, On the size of the closed unbounded set8, Ann. Pure Appl. Logic 54(1991), 195-227.
J. Baumgartner and A. Taylor, Saturation properties of ideals in generic extensions. I,Rans. Amer. Math. Soc. 270 (1982), 557-574.
M. Burke and M. Magidor, Shelah’s pcf theory and its applications, Ann. Pure Appl.
Logic 50 (1990), 207-254.
M. Foreman, M. Magidor and S. Shelah, Martin’s Manimum, saturated ideals and nonregular ultrafilters. Part I, Ann. Math. 127 (1988), 1-47.
F. Galvin, T. Jech and M. Magidor, An ideal game, J. Symbolic Logic 43 (1978), 284-292.
N. Goldring, The entire NS ideal on
can be precipitous, J. Symbolic Logic 62 (1997),$P_{\gamma}\mu$1161-1172.
T. Jech, A game theoretic property of Boolean algebras, Logic Colloquium ’77, North
Holland, Amsterdan, 1978, pp. 135-144.
T. Jech and K. Prikry, Ideals over uncountable sets: Application of almost disjoint
functions and generic ultrapowers, Mem. Amer. Math. Soc. 214 (1979).
A. Kanamori, The Higher Infinite, Springer, Berlin, 1994.
Y. Matsubara and M. Shioya, Nowhere precipitousness of some ideals, J. Symbolic Logic
63 (1998), 1003-1006.
S. Shelah, Cardinal Arithmetic, Oxford University Press, Oxford, 1994.

[J]
[JP]

[K]
[MS]
[S]

IN ST IT UTE OF M AT HEM AT ICS, UNIV ERSIT YOFTSUKUBA, TSU UBA, 305-8571 J A PA$\kappa$-mail address:$E$. tsukuba. ac. jp$\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{o}\mathrm{y}\mathrm{a}\emptyset \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}$$\mathrm{N}$