数理解析研究所講究録
第 1790 巻 2012 年 60-64

60

NOTES ON MIYAMOTO’S FORCING AXIOMSTOSHIMICHI USUBAABSTRACT. In this short paper, we prove that Miyamoto‘s forcing axiom FA’ ( -closed)
is equivalent to $MA^{+}$ ( -closed). We also study some variants of FA ( -closed).
$\sigma$$*$$\sigma$$\sigma$

1. INTRODUCTION
Throughout this note, will denote a sufficiently large regular cardinal. For.
andmeans that
submodels $M,$
Miyamoto ([1]) introduced the following forcing axiom FA ( -closed):$\theta$$N\prec H_{\theta},$$M\subseteq N$$M\prec_{\omega_{1}}N$$M\cap\omega_{1}=N\cap\omega_{1}$$*$$\sigma$

Definition 1.1. FA’( -closed) is the assertion that for every -closed poset ,, every-generic condition , and every
every countable
withsuch that::in , there exists a directed set
dense subsets$\mathbb{P}$$\sigma$$\sigma$$\mathbb{P}\in M$$M\prec H_{\theta}$$i<\omega_{1}\rangle$$\{D_{i}$$(M, \mathbb{P})$$p$$F\subseteq \mathbb{P}$$\mathbb{P}$

,
(1)
(2)
for every(3) $q\in F$ for some $q\leq p$ ,(4) $M\prec_{\omega_{1}}M(F)$ .
$|F|\leq\omega_{1}$$F\cap D_{i}\neq\emptyset$$i<\omega_{1}$,Where $M(F)=\{h(F) : h : \mathcal{P}(\mathbb{P})arrow H_{\theta}, h\in M\}\prec H_{\theta}$.
containing $M\cup\{F\}$ .
elementary submodel of$M(F)$

is the minimal$H_{\theta}$

He showed that FA’( -closed) implies $MA^{+}$ ( -closed), where:$\sigma$$\sigma$( -closed) is the assertion that for every -closed poset ,Definition 1.2.
every -name of a stationary subset of , and every dense subsets
forand the set
such thatin , there exists a filter
is stationary in .
$MA^{+}$$\mathbb{P}$$\mathbb{P}$$\sigma$$\sigma$$\{D_{i} :$\dot{S}$$\omega_{1}$$S=\{\alpha<\omega_{1} :$F\cap D_{i}\neq\emptyset$$F\subseteq \mathbb{P}$$\mathbb{P}$\exists p\in F(p|\vdash\alpha\in \mathscr{S}’)\}$

i<\omega_{1}\}$$i<\omega_{1}$$\omega_{1}$We show that the converse direction is also true, hence we have:

Theorem 1.3. FA ( -closed)$*$$\sigma$$\Leftrightarrow MA^{+}$( -closed).
$\sigma$On the other hand, we shall consider some variants of FA ( -closed), which arealso suggested by Miyamoto.
$*$$\sigma$

Definition 1.4. FA’( -closed) is the assertion that for every, and every dense subsets
every countable
withIF’ such that
there exists a directed set$\sigma$$\mathbb{P}\in M$$M\prec H_{\theta}$$F\subseteq$$\sigma$-closed poset ,in ,$\{D_{i}:i<\omega_{1}\rangle$$\mathbb{P}$$\mathbb{P}$

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(1)
,
(2)
,
for every$F$(3)
contains an-generic condition,$M\prec_{\omega}1M(F)$(4).
$|F|\leq\omega_{1}$$F\cap D_{i}\neq\emptyset$$i<\omega_{1}$$(M, \mathbb{P})$Definition 1.5.
( -closed) is the assertion that for every -closed poset ,every countable, and every dense subsets
with
in ,
there exists a directed set
such that:$FA^{0}$$\sigma$$\mathbb{P}$$\sigma$$M\prec H_{\theta}$$\mathbb{P}\in M$$\{D_{i}:i<\omega_{1}\}$$\mathbb{P}$$F\subseteq \mathbb{P}$

(1)
,
(2)
for every$M\prec_{\omega_{1}}M(F)$(3).
$|F|\leq\omega_{1}$$F\cap D_{i}\neq\emptyset$$i<\omega_{1}$,Obviously
FA’$FA’$
We prove that( -closed) andreflection principles.

( -closed).
( -closed) can be characterized by known$(\sigma-closed)\Rightarrow FA^{0}$$FA^{*}(\sigma-closed)\Rightarrow$$\sigma$$FA^{0}$$\sigma$$\sigma$

Definition 1.6 (Shelah [2]). For
, let
and
:
$S(a\subseteq x$ and. SSR (Semi-Stationary Reflection principle) is theassertion that for every cardinal
and every stationary, there issuch that.
andis stationary in$\hat{S}=\{x\in[\kappa]^{\omega}$$S\subseteq[\kappa]^{\omega}$$\kappa>\omega_{1}$$\exists a\in$$a\cap\omega_{1}=x\cap\omega_{1})\}$$S\subseteq[\kappa]^{\omega}$$\kappa>\omega_{1}$$|X|=\omega_{1}\subseteq X$$X\subseteq\kappa$$\hat{S}\cap[X]^{\omega}$$[X]^{\omega}$

Definition 1.7. IRP (Internally approachable Reflection Principle) is the assertion
that for every and every stationary, there is an internally approachable
continuous sequence:of countable submodels ofsuch that:$\theta$$S\subseteq[H_{\theta}]^{\omega}$$\langle M_{i}$$M_{i}\in S\}$is stationary in$i<\omega_{1}\}$$\omega_{1}$$\{i<\omega_{1}$$H_{\theta}$.

We will show the following equivalences:
Theorem 1.8.
(1) FA’( -closed)(2)( -closed)
SSR.
$\sigma$$FA^{0}$$\Leftrightarrow$

IRP.

$\Leftrightarrow$$\sigma$

2. PROOFS

Proof of Theorem 1.3. The direction FA( -closed) is known, so
we show only the converse direction. Suppose $MA^{+}$ ( -closed). Let be a -closed
poset. Let be the set of all countablesuch that
but there are an-generic conditionand dense subsets
in such that there
is no directed set( -closed). We
satisfying (1) $-(4)$ in the definition of
see that is non-stationary in, this suffices to prove the theorem.
Suppose otherwise. Let be a P-name such that $|\vdash\dot{T}=\{M\in S:p_{M}\in \mathscr{L}\}$ ”.
$*(\sigma-closed)\Rightarrow MA^{+}$$\sigma$$\mathbb{P}$$\sigma$$S$$M\prec H_{\theta}$$(M, \mathbb{P})$$\sigma$$\mathbb{P}\in M$$\{D_{i}^{M}:i<\omega_{1}\rangle$$p_{M}$$\mathbb{P}$$FA^{*}$$F\subseteq \mathbb{P}$$S$$\sigma$$[H_{\theta}]^{\omega}$$\dot{T}$Claim 2.1. There is$p\in \mathbb{P}$such that$p|\vdash\swarrow \mathscr{T}$is stationaw in$[H_{\theta}^{V}]^{\omega}$”.

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Proof. Suppose otherwise. Then there is a -name ; such that$|\vdash f$$\mathbb{P}$:$[H_{\theta}^{V}]^{<\omega}arrow H_{\theta}^{V}$

such that there is no
closed under ”. Take a sufficiently large another regular. Take a countablecardinal
such thatand $;\in N$ . Then-generic condition. Thusis also an
but is closed under”This is a contradiction.
$x\in\dot{T}$$f$$\chi>\theta$$N\cap H_{\theta}\in S$$N\prec H_{\chi}$$p_{M}|\vdash N\cap H_{\theta}\in\dot{T}$$(N, \mathbb{P})$$p_{M}$$f$$\square$”Pickis stationary insuch that
By replacing by thesuborderfor $M\in S$ , we may assume that
by
and
is
”. Let be a -name of a -closed poset which adds a bijection
stationary infrom. Take a-generic $G*H$ and work in $V[G*H]$ . Lettobe the interpretation of by $G*H$ . Since is -closed, remains stationary in. We know, thus we can find a clubinsuchthat $E=$. $\{i<\omega_{1} : i=M_{i}\cap\omega_{1}, M_{i}\in T\}$ is stationary in . Let be a name of
$E$ ,
and
of .
, and $M\in S$ , if
Retum to . Notice that for“, then is compatible with. For
, let $D_{i}=\{\{p,$
and
:
”
”
“
$(p, q)$ decides$q\}|\vdash M_{i}=M$
and ifthen
Q. For $i,j<\omega_{1}$ , let
and $p\leq p_{M})\}$ .
is dense in: if$\exists M\in S$ $(\langle p, q\rangle|\vdash i\in\dot{E}\wedge M_{i}=M")$ , then $p\in D_{j}^{M}$
is also dense}. Each$MA^{+}$Q. Byin( -closed), we can find a directed set
such that$|F’|=\omega_{1},$for each $i,j<\omega_{1}$ , and $E^{*}=\{i<\omega_{1}$ :$q\}\in F’$
is stationary in . Let $F$ be the projection of $F’$$F$
into P.
is a directed set in with $|F|=\omega_{1}$ . For $i\in E^{*}$ , there is $M_{i}\in S$ and$p|\vdash\dot{T}$$p\in \mathbb{P}$$\{q\in \mathbb{P}:q\leq p\}$$\dot{\mathbb{Q}}$$\mathbb{P}$$|\vdash\dot{T}$$p_{M}\wedge p$$p_{M}$$[H_{\theta}^{V}]^{\omega}$$\mathbb{P}$$\sigma$$T$$(V,\mathbb{P}*\dot{\mathbb{Q}})$$H_{\theta}^{V}$$\omega_{1}$$[H_{\theta}^{V}]^{\omega}$$T$$\mathbb{Q}$$T$$\sigma$$\langle M_{i}:i<\omega_{1}\}$$|H_{\theta}^{V}|=\omega_{1}$$[H_{\theta}^{V}]^{\omega}$$[H_{\theta}^{V}]^{\omega}$$\dot{E}$$\omega_{1}$$M_{i}$$M_{i}$$V$$\langle p,$$M=\dot{M}_{i}$$q\}\in \mathbb{P}*\dot{\mathbb{Q}},$$p$$i<\omega_{1}$$i<\omega_{1}$$p_{M}$$i\in\dot{E}$$\langle p,$$\langle p,$$q\}|\vdash i\in\dot{E}$$q\rangle.\in \mathbb{P}*\dot{\mathbb{Q}}$$\exists M\in S(\langle p,$$D_{ij}=\{\langle p,$ $q\rangle\in D_{i}$$\mathbb{P}*$$D_{i}$$q\rangle|\vdash i\in\dot{E}$$D_{ij}$$\mathbb{P}*$$F’\subseteq \mathbb{P}*\dot{\mathbb{Q}}$$\sigma$$D_{i}\cap F’\neq\emptyset,$$D_{ij}\cap F’\neq\emptyset$$(\langle p, q\}|\vdash i\in\dot{E}")\}$$\exists\langle p,$$\omega_{1}$$\mathbb{P}$“such thatand. Thus we may
and. Moreover, for $i \in\lim(E^{*})\cap E^{*}$ , it is easy to checkassume thatfor
, and for each $i\in E^{*}$ and, we havethat.
Now take a countablesuch thatand $\{M_{i}:i\in E^{*}\},$ $F\in N$ .
$i^{*}=N\cap\omega_{1}$, thus we haveLet
. Then. This is acontradiction.
$\langle p,$$q\rangle\in F’$$\langle p,$$p_{M_{i}}\in F$$\dot{M}_{i}=M_{i}$$q\rangle|\vdash i\in\dot{E}$$p\leq p_{M_{i}}$$i\in E^{*}$$j<\omega_{1}$$M_{i}= \bigcup_{j\in E^{\wedge}\cap i}M_{j}$$D_{j}^{M_{i}}\cap F\neq\emptyset$$N\cap\omega_{1}\in E^{*}$$N\prec H_{\theta}$$M_{i^{*}}\prec_{\omega_{1}}M_{i^{*}}(F)$$M_{i^{*}}\prec_{\omega_{1}}N$$\square$

Proof of (1) of Theorem 1.8. First we prove

IRP. Fix a stationbe a -closed poset adding a bijection from
. Fix
to
:
such thatis an internally approachable club$FA’(\sigma-closed)\Rightarrow$

. Let
ary
:
-namesinFix another large regular cardinalandwith: $i<\omega\}\in$$N$ and. Let $M=N\cap H_{\theta}$ and $i^{*}=M\cap\omega_{1}$ . Note that if
isan”. For-generic condition, then, let:$\exists M’(p|\vdash M’=\dot{M}_{i}" )\}$ .
is dense in . Applying FA’( -closed), we can find a$F$ contains
directed set
such that-generic condition,an$\existsp\in
F(p|\vdash\dot{M}_{i}=M_{i}")$. For each
anddefineas. Then$S\subseteq[H_{\theta}]^{\omega}$$\{M_{i}$$\mathbb{P}$$[H_{\theta}^{V}]^{\omega}$$\mathbb{P}$$\sigma$$i<\omega_{1}\rangle$$\omega_{1}$$|\vdash\langle\dot{M}_{i}$$H_{\theta}$$i<\omega_{1}\rangle$”$\chi>\theta$$N\prec H_{\chi}$$\theta,$$N\cap H_{\theta}\in S$$p|\vdash M=\dot{M}_{i^{*}}$$D_{i}$$N\prec_{\omega_{1}}N(F)$$\mathbb{P},$$\langle\dot{N}_{i}$$p\in \mathbb{P}$$(N, \mathbb{P})$$F\subseteq \mathbb{P}$$S,$$i<\omega_{1}$$\mathbb{P}$$\sigma$$F\cap D_{i}\neq\emptyset,$$i<\omega_{1}$$(N, \mathbb{P})$$M_{i}$$D_{i}=\{p\in \mathbb{P}$63
is an internally approachable continuous sequence and$N(F)$ . Moreover, since $F$ contains an-generic condition, we have $M_{i^{*}}=M$ .
Finally we see that $\{i<\omega_{1} : M_{i}\in S\}$ is stationary, but this follows from that
and.
Next we prove the converse direction IRP
FA’( -closed). Fix a -closed
poset P. Fix a well-ordering. Let
onbe the set of all countable $M\prec$such that: there are dense open subsetsof IP such that$F$
there is no directed set satisfying (1) $-(4)$ in the definition of FA’( -closed). We
claim that is non-stationary, which completes our proof. Suppose to contrary
that is stationary. By IRP, we can find an internally approachable continuous
sequence:with $E=\{i<\omega_{1} : N_{i}\in S\}$ stationary. Since it is internallyapproachable, we may assume that for each $i<j<\omega_{1},$: $k<\omega_{1}\}\in N_{j}$ ,where we are lettingfor. By induction on, we would definea descending sequencesuch thatis-generic andfor $k,j\leq i$ .
First, take the -least-generic condition . Since $N_{0}\in N_{1}$ , we have$p_{0}\in N_{1}$ .
Moreover, since, there is the -least condition
with. Repeating this procedure -times; Whenis limitand:was defined, then is the -least lower bound of the. Thenwe know that is-generic and. Thus we can take $p_{i+1}\in N_{i+1}$ as$\langle M_{i}$:$i<\omega_{1}\}$$\{M_{i} :i<\omega_{1}\}\in$$(N, \mathbb{P})$$M_{i^{*}}\cap\omega_{1}=M\cap\omega_{1}=N\cap\omega_{1}$$N\prec_{\omega_{1}}N(F)$$\Rightarrow$$\triangle$$\langle H_{\theta},$$\in,$$\sigma$$\sigma$$S$$H_{\theta}$$\{D_{i}^{M} :$\triangle\rangle$

i<\omega_{1}\}$$\sigma$$S$$S$$i<\omega_{1}\rangle$$\langle N_{i}$$\langle D_{k}^{N_{i}}$$D_{k}^{N_{i}}=\mathbb{P}$$i\not\in E$$i<\omega_{1}$$\langle p_{i}:i<\omega_{1}\}$$\triangle$$p_{i+1}\in D_{k}^{N_{j}}$$(N_{i}, \mathbb{P})$$p_{i}$$(N_{0}, \mathbb{P})$$p_{0}$$\{D_{k}^{N_{0}}:k<\omega_{1}\}\in N_{1}$$p_{1}\leq p_{0}$$\langle p_{j}$$\triangle$$p_{1}\in D_{0}^{N_{0}}$$i<\omega_{1}$$\omega_{1}$$j<i\rangle$$\triangle$$p_{j}’ s$$p_{i}$$p_{i}$$p_{i}\in N_{i+1}$$(N_{i}, \mathbb{P})$intended.
Now we have a descending sequencesuch thatis-generic$F=\{p_{i}:i<\omega_{1}\}$and for every $k<\omega_{1}$ , there is some with
. Let
.
$F$
is a directed set with size
. Take a countablesuch that$F,$ $\{N_{i}:i<\omega_{1}\}\in M$$M\cap\omega_{1}\inE$$i^{*}=M\cap\omega_{1}$
and
. Let
. Then, inparticular we have
. However $F$ contradicts.
$\{p_{i}:i<\omega_{1}\rangle$$i,$$(N_{i}, \mathbb{P})$$p_{i}$$p_{j}\in D_{k}^{N_{i}}$$j$$M\prec\{H_{\theta}, \in, \triangle\}$$\leq\omega_{1}$$N_{i^{*}}\prec_{\omega 1}M$

$N_{i^{*}}\prec_{\omega_{1}}N_{i^{*}}(F)$$N_{i^{*}}\in S$

Proof of (2) of Theorem 1.8. First we prove$\square$

SSR. Suppose
closed). Fix
and -closed poset which adds a bijection fromto . Fixa -name such thatis a surjection fromto ”. To show that SSR, takea stationary. We will findsuch thatandis stationary in.
$S\in M$ and $M\cap\kappa\in S$ . For
Take a countable
with
, let
, and for, let:
1
. Eachandare dense in . By
( -closed), we
can find a directed set
such that, and $F$ intersects
each. Let $X=\{\alpha<\kappa :$ ョ$p\in F\exists i<\omega_{1}(p|\vdash\dot{\pi}(i)=\alpha")\}$ . Thenand
$X\in M(F)$ , and by the choice of the, we have’s and theand$\kappa\geq\omega_{2}$$FA^{0}(\sigma-closed)\Rightarrow$$FA^{0}(\sigma-$$\mathbb{P}$$\sigma$$\omega_{1}$$\kappa$$|\vdash\dot{\pi}$$\mathbb{P}$$\kappa$$\omega_{1}$$|X|=\omega_{1}\subseteq X$$X\subseteq\kappa$$S\subseteq[\kappa]^{\omega}$$\hat{S}\cap[X]^{\omega}$$[X]^{\omega}$$M\prec H_{\theta}$$D_{i}=\{p\in \mathbb{P} :$\alpha\in M\cap\kappa$\exists j<\omega_{1}(p|\vdash\dot{\pi}(j)=i" )\}$

$i<\omega_{1}(p|\vdash\dot{\pi}(i)=\alpha" )\}$

$D_{i}$$E_{\alpha}=\{p\in \mathbb{P}$$FA^{0}$$\mathbb{P}$$E_{\alpha}$$|F|=\omega_{1},$$F\subseteq \mathbb{P}$$D_{i}$$i<\omega_{1}$$\mathbb{P},\dot{\pi},$$\sigma$$M\prec_{\omega_{1}}M(F)$$E_{\alpha}$$D_{i}$$E_{\alpha}’ s$$|X|=\omega_{1}\subseteq X$

64. Then $M(F)\cap X\in\hat{S}\cap[X]^{\omega}$ . By the elementarity of $M(F),\hat{S}\cap[X]^{\omega}$. This completes the proof of one direction.
is stationary in
For the converse, let be a -closed poset. Let be the set of all countable:such that
but there are dense subsets
in such that$F$$-(3)$( -closed). We
in the definition ofthere is no directed set satisfying (1), which completes the proof.
see that is non-stationary in
Suppose to contrary that is stationary. By SSR, we can findsuch
$N$$|X|=\omega_{1}\subseteq
X$
. Let
that
is stationary inandbe the Skolem$M\cap\kappa\subseteq X$$[X]^{\omega}$$\mathbb{P}$$S$$\sigma$$\mathbb{P}\in M$$M\prec H_{\theta}$$\langle D_{i}$$i<\omega_{1}\rangle$$\mathbb{P}$$FA^{0}$$S$$\sigma$$[H_{\theta}]^{\omega}$$S$$X\subseteq H_{\theta}$

バ$\cap[X]^{\omega}$$[X]^{\omega}$. Then we can take a clubhull of $X$ underinsuch$E=\{i<\omega_{1}$. Since:is stationary, we havethatfor
$M_{i}\cap X\in\hat{S}\}$
belongs to ;is stationary in . Notice that for each $i\in E,$, there is
. If
with
Since
are dense$M_{i}\inS$$M’\inS$, then it also witnesses that
. For eachsubsets in witnessing$M_{i}\inS$$i\in E$ , fix dense subsets. Now take a countable:
witnessing$E,$$\{D_{j}^{i}:i\in
E,j<\omega_{1}\rangle\in
N$ andsuch that {. Let$i^{*}=N\cap\omega_{1}$ .
. Since is -closed, we can find a directed set
Then$F\in N$ such that $|F|=\omega_{1}$ andfor $i\in E$ and. Then, since
$F\in
N$
,
.
we
and
have that
This is a contradiction. ロ$\langle H_{\theta},$$\in,$$\{M_{i}:i<\omega_{1}\rangle$$\Delta\rangle$$[N]^{\omega}$バ$M_{i}\prec H_{\theta}$$i<\omega_{1}$$\cap[X]^{\omega}$$M_{i}$$\omega_{1}$$M_{i}\cap X\in\wedge$$M’\prec H_{\theta}$$M’\prec_{\omega_{1}}M_{i}$$S$$\langle D_{j}:j<\omega_{1}\rangle$$\mathbb{P}$$\{D_{j}^{i}$$N\prec H_{\theta}$$j<\omega_{1})$$N\cap\omega_{1}\in E$$M_{i}:i<\omega_{1}\rangle,$$M_{i^{*}}\prec_{\omega_{1}}N$$\mathbb{P}$$\sigma$$F\cap D_{j}^{i}\neq\emptyset$$M_{i^{*}}\prec_{\omega_{1}}N$$j<\omega_{1}$$M_{i}\prec_{\omega_{1}}M_{i}(F)$REFERENCES[1] T. Miyamoto, A Forcing Axiom and Chang’s Conjecture, RIMS 2011.
[2] S. Shelah, Proper and improper forcing, Perspectives in Mathematical Logic. SpringerVerlag, Berlin, 1998.

INSTITUTE FOR ADVANCED RESEARCH, NAGOYA UNIVERSITY, FURO-CHO, CHIKUSA-KU,
NAGOYA, 464-8601, JAPANE-mail address: usubadmath. cm. is. nagoya-u. ac. jp