数理解析研究所講究録
第 1595 巻 2008 年 6-13

6

A generalization of a problem of Fremlin
中部大学・工学部・理学教室

渕野昌 ( Saka\’e Fuchino)*

Department of Natural Science and Mathematics
School of Engineering, Chubu University
Kasugai, Japanfuchino@isc.chubu.ac.jp

February 29, 2008

1

Fremlin-Miller Covering Principle

The following result is stated in A. Miller [3] as an answer to a question by David
Fbremlin:

Theorem 1. (Theorem 3.7 in A. Miller [3]) The following holds in the genericextension obtained by adding at least
Cohen reals to a model of CH:$\aleph_{3}$(1.1)

For any family
a subfamily$\mathcal{F}$of Borel sets Utth$\mathcal{F}’\subseteq \mathcal{F}$with$|\mathcal{F}’|\leq\aleph_{1}$$|\mathcal{F}|=\aleph_{2}$suchsuch that$\cap \mathcal{F}’=\emptyset$$that\cap \mathcal{F}=\emptyset$, there is.

Note that by moving to complements of elements of , the assertion (1.1) can bealso conceived as a covering property resembling,Lindel\"of property of topologicalspaces. Thus we shall call here the property (1.1) the IFlremlin-Miller Covering, let us denote with FMCP
Principle. More generally, for cardinals
thefollowing parametrized Fremlin-Miller Covering Principle:$\mathcal{F}$$\kappa\geq\lambda$

FMCP

For any family of Borel sets with.
there issuch$(\kappa, \lambda)$:$\mathcal{F}$$\mathcal{F}’\in[\mathcal{F}]<\lambda$$(\kappa, \lambda)$$|\mathcal{F}|<\kappa$

such$that\cap \mathcal{F}=\emptyset$$that\cap \mathcal{F}’=\emptyset$’Supported by Grant-in-Aid for Scientific Research (C) No. 19540152 of the Ministry of Education, Culture, Sports, Science and Ibchnology Japan.

7
Lemma 2. ([3]).
FMCP(1) FMCP(2) FMCP(3) FMCP

For cardinals(0)$FMCP(\kappa, \lambda)$$\kappa\geq\kappa’\geq\lambda’\geq\lambda,$

implies$(\kappa’, \lambda’)$holds for any cardinal .
does not hold.
does not hold.
is one ofor then FMCP$(\kappa, \kappa)$$\kappa$$(c^{+}, c)$$(\aleph_{2}, \aleph_{1})$

If

(4)

$a,$

$\kappa$$b,$$b^{*}$$\ldots$$(\kappa^{+}, \kappa)$

does not hold.

Proof. (0), (1): Trivial by definition.
(2): Let $A$ be a maximal almost disjoint frnily
, let
each

$\subseteq[\omega]^{\aleph_{0}}$of cardinality

$c$

. For

$a\in \mathcal{A}$$X_{a}=$

Then
but

{

$X_{a}\neq\emptyset$(3): Letfor any$I\subseteq\omega_{1}$is almost disjoint $homa$ }.
$a \in \mathcal{A}_{\bm{t}}d\bigcap_{a\in A}X_{a}=\emptyset$$A’\subset\sim A$$\mathcal{A}$.

f_{\alpha}\leq {}^{t}f\leq*g_{\alpha}\}$are Borel setsby the maximality of

be a Hausdorff gap. For each$\langle\langle f_{\alpha}\rangle_{\alpha<w_{1}}, \langle g_{\beta}\rangle_{\beta<\omega_{1}}\rangle$$X_{\alpha}=\{f\in ww :$X_{\alpha}’ s$$x$for all$X_{a}\in Borel(\mathcal{P}(\omega))$$\bigcap_{a\in A},$

Then

:

$x\in P(\omega)$$\alpha<\omega_{1}$

, let

.

$\bm{t}d\bigcap_{\alpha<w_{1}}X_{\alpha}=\emptyset$

but$\bigcap_{\alpha\in I}X_{\alpha}\neq\emptyset$

for any countable.

(4): Similarly to (2) and (3).

(Lemma 2)$O$

By Lemma 2,
is the first non-trivial instanoe ofand FMCP
the principle FMCP.
is a generalIt is easy to show that the following principle for cardinaJs
ization of the corresponding parametrized Fremlin-Miller Covering Principle:$\aleph_{2}<\kappa\leq c$$(\kappa, \aleph_{2})$$(\kappa, \lambda)$$\kappa\leq\lambda$, and$X_{0}\in[X]<\lambda$ such that

: For any projective relation
GFMCP, there isunbounded in.
in$(\kappa, \lambda)$$\langle \mathbb{R}, R\rangle$

if $X$ isis unbounded$X\in[\mathbb{R}]<\kappa$$R\subseteq \mathbb{R}^{2}$$X_{0}$$\langle \mathbb{R}, R\rangle$

Here we say $X$ is unbounded in$\langle \mathbb{R}, R\rangle$

if$\forall r\in \mathbb{R}\exists x\in X\urcorner(xRr)$holds.
Proposition 3. GFMCP$(\kappa, \lambda)$

implies FMCP$(\kappa, \lambda)$

for any cardinals

holds and suppose that
Proof. Assume that GFMCP
sequence of Borel subsets of for somesuch that, let be a Borel code ofand letFor, let
For any$(\kappa, \lambda)$$\mathbb{R}$$\alpha<\delta$$x\in \mathbb{R}$$c_{\alpha}$$\delta<\kappa$$X_{\alpha}$$\langle X_{\alpha} :$\bigcap_{\alpha<\delta}X_{\alpha}=\emptyset$$X^{*}=\{c_{\alpha} :$\kappa\geq\lambda$.

\alpha<\delta\rangle$.

\alpha<\delta\}$

.

is a

8

(1.2)

$B_{x}=\{\begin{array}{ll}the Borel set coded by x, if x is a Borel code\emptyset, otherwise.\end{array}$Let$R\subseteq \mathbb{R}^{2}$

be defined by
is a non empty $subsetofB_{x}$$xRy\Leftrightarrow B_{y}$

for

$x,$

(1.3)

$y\in \mathbb{R}$$X$. The relation

is easily seen to be$R$is unbounded in$\Pi_{1}^{1}$$\langle \mathbb{R}, R\rangle\Leftrightarrow\cap\{B_{x} :. Clearly, we havex\in X\}=\emptyset$

for any. In particular, $X$ “ above is unbounded in. By GFMCP,““
there isof cardinality
such that $X$ is already unbounded in.
Thus, again by (1.3),
for. $0$ (Proposition 3)$X\subseteq \mathbb{R}$$(\kappa, \lambda)$$\langle \mathbb{R}, R\rangle$$<\lambda$$X^{**}\subseteq X^{*}$$\langle \mathbb{R}, R\rangle$$I=\{\alpha<\delta :$\bigcap_{\alpha\in I}X_{\alpha}=\emptyset$

c_{\alpha}\in X^{**}\}$The proof of Theorem 1 in [3] can be recast to show the following consistency
result on GFMCP:$(c, \aleph_{2})$Theorem 4. Let
such that(1.4)$\kappa<\mu$

be regular cardinals. Suppose thatfor all$\mathbb{P}_{\{\alpha\}}\cong \mathbb{P}_{\{0\}}$$\mathbb{P}=\prod_{\alpha<\mu}^{f1n}\mathbb{P}_{\alpha}$(16)$|\mathbb{P}_{\{0\}}|\leq\kappa=\kappa^{\aleph_{0}},$$|\vdash P$

“

GFMCP

$\alpha<\mu$

are posets;

satisfies the $c.c.c.$ ;

(1.5)

Then

$\alpha<\mu$$\mathbb{P}_{\langle\alpha\}},$$\kappa^{+}<\mu$$(\mu, \kappa^{+})$.

.

We shaJl give the details of the proof of Theorem 4 in the next section.
The formulation of GFMCP
has a certain resemblance to that ofof J. Brendle and S. Fuchino [1]. This feeling is also supported by the fact thatthey both hold in Cohen models. The following proposition shows however thatthese principles are rather independent to each other:$HP(\aleph_{2})$$(\kappa, \aleph_{2})$PropositIon 5.

(2)

(1)

$c\geq\aleph_{3}\wedge GFMCP(c, \aleph_{2})\wedge\urcorner HP(\aleph_{2})$$\neg GFMCP(\aleph_{8}, \aleph_{2})\wedge HP(N_{2})$is consistent.

is consistent.

Proof. (1): The arguments used in the proof of Theorem 4 are also vaJid for thegeneric extension with (measure theoretic) side-by-side product of random forcing.
It is known that
does not hold in a random extension (see [1]).
(2): In a model of $HP(\aleph_{2})\wedge c=\aleph_{2}$ we haveby Lemma
2, (2).
(Proposition 5)$HP(\aleph_{2})$$\neg GFMCP(\aleph_{3}, \aleph_{2})$$\square$Problem 1. Is $\urcorner GFMCP(c, \aleph_{2})\wedge HP(\aleph_{2})$ consistent under$c\geq\aleph_{3}$?

9Proof of the consistency result

2

In this section we prove Theorem 4.
be regular cardinals andLet, we denote
For$\kappa<\mu$$\alpha<\mu$$\mathbb{P}_{\{\alpha\}},$

satisfy (1.4), (1.5) and (1.6).

$X\subseteq\mu$(21)$\mathbb{P}_{X}=\prod_{\alpha\in X}^{fin}\mathbb{P}_{\alpha}$.

. We assume that finite support product is introduced just as in [1].
Thus.
for all
In particular, we haveinduces an automorphism of and this induces in turn
A bijection :an automorphism on P-names. We shall denote both of these automorphisms by$\mathbb{P}=\mathbb{P}_{\mu}$$X\subseteq Y\subseteq\mu$$\mathbb{P}_{X}\leqq \mathbb{P}_{Y}\leqq \mathbb{P}$$f$$\tilde{f}$$\mathbb{P}$$\muarrow\mu$.
All of the following Lemmas 6, 7 and 8 are folklore:

Lemma 6. Suppose that(in the sense of ) such that

$X\subseteq\mu$$V^{P}$(2.2)$X\backslash \cup\{supp(\dot{x}_{\xi}) :and$\dot{x}_{\xi},$$\xi<\delta$$supp(\dot{x}_{\xi})\subseteq X$\xi<\delta\}i8$

are -names of elements of$\mathbb{P}$for all$\xi<\delta$$\mathcal{H}(\aleph_{1})$. Ifuncountable,

then we have

(2.3)$|\vdash r\langle \mathcal{H}(\aleph_{1})^{V[\delta\cap P_{X}]}, \{\dot{x}_{\xi} ; \xi<\delta\}, \ldots, \in\rangle\prec\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :“Proof. Suppose that-formula and -names
criterion, it is enough to show that$p|\vdash P$$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$\dot{a}_{1},\ldots,\dot{a}_{n}$$\mathbb{P}_{X}$$\mathcal{L}_{ZF}$$\varphi$$p|\vdash p\langle \mathcal{H}(\aleph_{1}), \{\dot{x}\epsilon :\xi<\delta\}, \ldots, \in\rangle$

for a\xi<\delta\}, \ldots, \in\rangle\models\exists x\varphi(x,\dot{a}_{1}, \ldots, a_{n})$of elements of$\mathcal{H}(\aleph_{1})$.

. By the Tarski-Vaught\xi<\delta\}, \ldots, \in\rangle\models\varphi(\dot{c},\dot{a}_{1}, \ldots,\dot{a}_{n})$.
for some -name of an element ofBy (1.5), we may assume without loss of generality that$\mathbb{P}_{X}$(2.4)$\mathcal{H}(\aleph_{1})$$\dot{c}$$supp(\dot{a}_{1}),\ldots,$ $supp(\dot{a}_{n})$are all countable.

By (2.2), we may assume that $supp(p)\subseteq X$ . Let(2.5)$X’=\cup\{supp(\dot{x}_{\xi}) :\xi<\delta\}\cup\cup\{supp(\dot{a}_{1}) :.

i\in n+1\backslash 1\}\cup supp(p)$is still
By the assumptions above, we have $X’\subseteq X$ . By (2.2) and (2.4),
uncountable. By Maximal PrInciple, there is a -name of an element of$X\backslash X’$$\dot{b}$$\mathcal{H}(\aleph_{1})$$\mathbb{P}$such that$p|\vdash r\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :\xi<\delta\}, \ldots, \in\rangle\models\varphi(\dot{b},\dot{a}_{1}, \ldots,\dot{a}_{n})$.

10By (1.5), we can find such with countableLet :be a bijection such that$\dot{b}$$f$.

$\muarrow\mu$$f\square X’=id_{X’}$Let$supp(\dot{b})$$\dot{c}=\tilde{f}(\dot{b})$and $f”supp(\dot{b})\subseteq X$ .

. Then is a P-name and$\dot{c}$$p|\vdash r\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\zeta} :\xi<\delta\}, \ldots, \in\rangle\models\varphi(\dot{c},\dot{a}_{1}, \ldots,\dot{a}_{n})$.
$O$(Lemma 6)

Lemma 7. $Supp_{08}e$ thatis infinite andis countable.
Let
be -names of elements of(in the sense of ) such that.
for alland-sentenceIf
for somethen we have.
Thus we have
$X\subseteq\mu,$$\dot{x}_{\xi},$$\xi<\delta$$\mathbb{P}$$supp(\dot{x}_{\xi})\subseteq X$$p|\vdash r$“$X_{0}\subseteq\mu\backslash X$$V^{P}$$\mathcal{H}(\aleph_{1})$$\xi<\delta$$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$p|\vdash r_{X\cup X_{0}}$$\varphi$$\mu\backslash X$“\xi<\delta\}, \ldots, \in\rangle\models\varphi$$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$|\vdash r\langle \mathcal{H}(\aleph_{1})^{V[G\cap(X\cup X_{0})]}, \{\dot{x}_{\xi} :’$p\in \mathbb{P}_{X}$$\mathcal{L}_{ZF}$\xi<\delta\}, \ldots, \in\rangle\models\varphi’$\xi<\delta\}, \ldots, \in\rangle\equiv\langle \mathcal{H}(\aleph_{1})^{V[q}, \{\dot{x}_{\xi} :Proof. It is enough to show the following (2.6) for all
by induction on :$\mathcal{L}_{ZF}$$\psi$-formula\xi<\delta\}, \ldots, \in\rangle$.

$\psi=\psi(x_{1}, \ldots, x_{n})$$\psi$$(2.6)_{\psi}$

For any P-names$\dot{a}_{1},\ldots$

,

$an$

of elements of(2.6a)$supp(\dot{a}_{i})\subseteq X\cup X_{0}$(2.6b)$X_{0}\backslash \cup$if$q\in \mathbb{P}_{X\cup X_{0}}${ $supp$ a :$i$$\mathcal{H}(\aleph_{1})$such thatfor $i\in n+1\backslash 1$ and$i\in n+1\backslash 1$} is infinite,

and $q\leq pp$, then$q|\vdash p\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :\xi<\delta\}, \ldots, \in\rangle\models\psi(\dot{a}_{1}, \ldots,\dot{a}_{n})$if and only if$q|\vdash p_{X\cup X_{0}}$$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :\xi<\delta\}, \ldots, \in\rangle\models\psi(\dot{a}_{1}, \ldots,\dot{a}_{n})$The crucial step in the induction proof of (2.6)$\exists x\eta(x,x_{1}, \ldots,x_{n})$Suppose that
(2.6b),$\psi$

is when$\psi(x_{1}, \ldots, x_{n})$.
$\dot{a}_{1},\ldots,$$q\in \mathbb{P}_{X\cup X_{0}},$$a_{n}$$q\leq rp$

are P-names of elements ofThen there is a P-name$\dot{a}$\xi<\delta\}, \ldots, \in\rangle\models\psi(\dot{a}_{1}, \ldots,\dot{a}_{n})$of an element of$q|\vdash r\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$\mathcal{H}(\aleph_{1})$.

such that\xi<\delta\}, \ldots, \in\rangle\models\eta(\dot{a},\dot{a}_{1}, \ldots,\dot{a}_{n})$is of the form

satisfying (2.6a) and$\mathcal{H}(\aleph_{1})$and$q|\vdash p^{(}\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :

.

.

11
By (1.5), we may assume thatsuch that(2.7)$supp(\dot{a})$is countable. Let$f$: $\muarrow\mu$ be a bijection$f(X’=id_{X’}$

where $X’=X\cup\cup\{supp(\dot{a}_{i}) : i\in n+1\backslash 1\}\cup supp(q)$ ;(2.8)$f”(supp(r)Usupp(\dot{a}))\subseteq XUX_{0}$(2.9)$X_{0}\backslash (\cup\{supp(\dot{a}_{i}) :and

i\in n+1\backslash 1\}\cup supp(\dot{a}))$is infinite.

Then by induction’s hypothesis, we have$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$q|\vdash p_{X\cup X_{0}}$\xi<\delta\}, \ldots, \in\rangle\models\eta(f(\dot{a}),\dot{a}_{1}, \ldots, a_{n})$.

It follows that$\langle \mathcal{H}(\aleph_{1}), \{\dot{x}_{\xi} :$q|\vdash p_{X\cup X_{0}}$\xi<\delta\}, \ldots, \in\rangle\models\psi(\dot{a}_{1}, \ldots,\dot{a}_{n})$.

The “only if” direction of this induction step can be shown similarly and more
$0$ (Lemma 7)
easily.
IfG isa (V,the set$\{\dot{x}^{G} :$\mathbb{Q}$)-generic set fora poset\dot{x}\in V^{\mathbb{Q}}\cap M\}$$\mathbb{Q}$andM isaset, $wedenotewithM[G]$.

is a poset andLemma 8. Suppose that
regular . If is a (V, )-generic set then we have$\mathbb{P}\in M\prec \mathcal{H}(\theta)$$\mathbb{Q}$$\theta$(2.10)$G$$\mathbb{Q}$$M[G]\prec \mathcal{H}(\theta)[G]$.

Proof. Note that
Tarski-Vaught criterion.
Suppose that$\mathcal{H}(\theta)[G]=\mathcal{H}(\theta)^{V[G]}$(2.11)

for. We check again the forcing version of“ $\mathcal{H}(\theta)\models\exists x\varphi(x,\dot{a}_{1}, \ldots,\dot{a}_{n})$and -names$a_{n}\in M$ . $(2.11)$ is equivalent to$\mathcal{L}_{ZF}$$\dot{a}_{1},\ldots,$$p|\vdash Q$

for sufficiently large-formula$\mathbb{Q}$$\varphi$$\dot{a}_{1},\ldots,\dot{a}_{n}$$\mathcal{H}(\theta)\models p|\vdash Q\exists x\varphi(x,\dot{a}_{1}, \ldots,\dot{a}_{n})$Then by elementarity we have
$M\models p|\vdash\Phi\exists x\varphi(x,\dot{a}_{1}, \ldots,\dot{a}_{n})$.

of elements of $M$ . We may assume that.

12
It follows that there is some $\dot{a}\in V^{P}\cap M$ such that
By elementarIty of $M$ this is equivalent to“, a ).
(
turn, is equivalent to$M\models p|\vdash \mathbb{Q}$“$\mathcal{H}(\theta)\models p|\vdash \mathbb{Q}$$p|\vdash \mathbb{Q}$$\mathcal{H}(\theta)\models\varphi$$\varphi$(

Proof of Theorem 4: Suppose that
and$\kappa,$$\mu,$, a ).
(, a ) This, in(Lemma 8)$\dot{a},\dot{a}_{1},$$\ldots$$\ldots$$n$$n$$\square$$n$$\mathbb{P}_{\{\alpha\}},$$\varphi$$\dot{a},\dot{a}_{1},$$\dot{a},\dot{a}_{1},$$\ldots$“$\alpha<\mu,$are as in Theorem 4,$\mathbb{P}$$p\in \mathbb{P}$$\{\dot{x}_{\alpha} :\alpha<\delta\}$$R=\{\langle x, y\rangle :

where
Let

$\delta\leq\kappa,$

is unbounded in“$p|\vdash P$(2.12)$X\subseteq\lambda$

with respect to\mathcal{H}(\aleph_{1})\models\varphi(x, y,\dot{a})\}$

is a-formula andbe such that$\mathcal{L}_{ZF}$$\varphi$$\mathcal{H}(\aleph_{1})$$\dot{a}$is a P-name of an element of$X\supseteq\cup\{supp(\dot{x}_{\alpha}) :\alpha<\delta\}\cup supp(p)\cup supp(\dot{a})$is uncountable.
andLet
be a (V, )-generic filter with $p\in G$ and let
regular cardinal. By Lemma 7, we have$|X|<\kappa$$X\backslash \{supp(\dot{x}_{\alpha}) :$G$(2.13)

Let(2.14)$M\prec \mathcal{H}(\theta)$$\mathbb{P},$. Then“ $\{\dot{x}_{\alpha}^{G} : \alpha<\delta\}$

is unbounded in$\theta$

be a sufficiently large$\mathcal{H}(\aleph_{1})$with respect to $R’$ .

be such that$\{\dot{x}_{\alpha} :(2.15)$[M]^{\aleph_{0}}\subseteq M$(2.16)$|M|\leq\kappa$\alpha<\delta\}\in M$

;

; and

.

The last two conditions are possible since(2.17).

\alpha<\delta\}$$\mathbb{P}_{X}$$\mathcal{H}(\theta)[G]\models|\vdash p_{v}$$\mathcal{H}(\aleph_{1})$$\kappa^{\aleph_{0}}=\kappa$. By Lemma 8, we have

$M[G]\prec \mathcal{H}(\theta)[G]$and hence(2.18)$M[G]\models|\vdash p_{\omega}$“ $\{\dot{x}_{\alpha}^{G} : \alpha<\delta\}$

is unbounded in$\mathcal{H}(\aleph_{1})$with respect to $R’$ .

forY as below and thus we crnot
Note thatis an element ofM but not
apply the elementary submodel argument to the latter poset.
Let $Y=\delta\cap M$ . Since 1 $Y|\leq\kappa$ by (2.16), it is enough to show the following
claim:$\mathbb{P}_{w}$Claim 8.1.

$\mathbb{P}_{\mu\backslash Y}$$\mathcal{H}(\theta)[G]\models|\vdash P_{\mu\backslash X}$” $\{\dot{x}_{\alpha}^{G} : \alpha\in Y\}$is unbounded in$\mathcal{H}(\aleph_{1})$with respect to $R’$ .

. Suppose thatIn the folowing we work always inand ab is a
$Z=supp(\dot{x})\cup
supp(p)$$X_{0}\in
M$ be a. Let
. Let-name of an element of
be a bijectIon suchcountable subset of disjoint $homY\cup Z$ . :that$\mathcal{H}(\theta)[G]$$\vdash$$q\in \mathbb{P}_{\mu\backslash X}$$\mathcal{H}(\aleph_{1})$$\mathbb{P}_{\mu\backslash X}$

$\mu$

$f$

$\mu\backslash Xarrow\mu\backslash X$

13(2.19)$f”Z\subseteq Y\cup X_{0}$and $f\lceil Y=id_{Y}$ .

. By (1.5) and (2.15), we may
is aNote that-name of an element of.
assume that $f(\dot{x})\in M$ . Also note thatBy (2.18), there are
and $\alpha^{*}\in\delta\cap M(=Y)$ such that$\tilde{f}(\dot{x})$$\mathcal{H}(\aleph_{1})$$\mathbb{P}_{X_{0}}$$\mathbb{P}_{X_{0}}\cong \mathbb{P}_{w}$$\tilde{r}\leq P_{X_{0}}f(q)$(2.20)$M[G]\models\tilde{r}|\vdash P_{X_{0}}$“ $\neg(\dot{x}_{\alpha}^{G}. R\tilde{f}(\dot{x}))$“By (2.17), it follows that
By Lemma 6, it follows that$\tilde{r}|\vdash P_{X_{0}}$(2.21)$\tilde{r}|\vdash r_{\mu\backslash X}$

Letobtain$r=\tilde{f}^{-1}(\tilde{r})$(2.22)

Since$q$(2.23). Then$r|\vdash r_{\mu\backslash X}$

and$\dot{x}$“ $\neg(\dot{x}_{\alpha}^{G}. R\tilde{f}(\dot{x}))$$q$$r\leq p_{\backslash }$“ $\neg(\dot{x}_{\alpha}^{G}. R\dot{x})$.

$\neg(\dot{x}_{\alpha}^{G}.

Rf(\dot{x}))$.

.
. By mapping the parameters in (2.21) by$\tilde{f}^{-1}$

, we

.

were arbitrary, it follows that“$|\vdash P_{\mu\backslash X}$$\{\dot{x}_{\alpha}^{G} :\alpha\in Y\}$is unbounded in$\mathcal{H}(\aleph_{1})$with respect to $R’$ .
(Claim 8.1)
$0$ (Theorem 4)$\dashv$

References[1] J. Brendle and S. Fuchino, Coloring ordinals by reals, Fundamenta Mathematicae, 196, No.2 (2007), 151-195.

covers and large thin sets of reals, Proceedings of London Mathematical Society (3) 53 (1986), 518-538.

[2] J. Jasinski and D.H. Reiin,$G_{\delta}$[3] A. Miller, Infinite Combinatorics and Definability, Annals of Pure and Applied Logic, 41 (1989), 179-203.