数理解析研究所講究録
第 1619 巻 2008 年 83-90
83
The Cardinal Invariants of certain Ideals related to
the Strong Measure Zero Ideal
大須賀昇 (Noboru Osuga)
大阪府立大学理学系研究科
Graduate school of Science,
Osaka Prefecture University
1
Motivation and Notation
In 1919, the new class of Lebesgue measure zero sets was introduced by
Borel[l]. This class is called strong measure zero sets today. The family of
all strong measure zero sets become a-ideal and is called the strong measure
zero ideal. In 2002, the results were given by Yorioka[2] about the cofinality
of the strong measure zero ideal. In the proof, he introduced the ideal
for each strictly increasing function
is a subset of
on . The ideal
Lebesgure measure zero sets, so it relates to the structure of the real line.
We have been interested in how the cardinal invariants of the ideal
behave.
We deal with the consistency problems about the relationship between the
cardinal invariants of the ideals
. Mainly, we treat of the minimum and
supremum of cardinal invariants of the ideals
for all . In this paper, we
add inequalities for the minimum of the additivity and the supremum of the
cofinality as new results to the past results.
So, we explain some notation which we use in this paper. Our notation is
quite standard. And we refer the reader to [3] and [4] for undefined notation.
For sets X and $Y$ , we denote by $xY$ the set of all functions from $X$ to $Y$ .
We denote by $<t02$ the set of all finite partial function from to 2. We write
”
and
infinitely many” and “for all but finitely
to mean that
many” respectively. For a family
of subsets of , we define the following
cardinals.
$\mathcal{I}_{f}$
$f$
$\omega$
$\mathcal{I}_{f}$
$\mathcal{I}_{f}$
$\mathcal{I}_{f}$
$g$
$\mathcal{I}_{g}$
$\omega$
$’\exists^{\infty}$
$\forall^{\infty}$
”
${}^{t}for$
$\mathcal{X}$
$\mathcal{A}$
add
cov
non
$( \mathcal{A})=\min\{|\mathcal{F}|$
:
$\mathcal{F}\subset \mathcal{A}$
$( \mathcal{A})=\min\{|\mathcal{F}|$
:
$\mathcal{F}\subset \mathcal{A}$
:
$Y\subset \mathcal{X}$
$( \mathcal{A})=\min$
$\{$
$|Y|$
and
and
and
$\cup \mathcal{F}\not\in \mathcal{A}\}$
,
$\cup \mathcal{F}=\mathcal{X}\}$
$Y\not\in \mathcal{A}\}$
,
,
and
84
cof $( \mathcal{A})=\min$
$\{$
$|\mathcal{F}|$
:
and
$\mathcal{F}\subset \mathcal{A}$
$\forall A\in \mathcal{A}\exists B\in \mathcal{F}(A\subset B)\}$
.
It is easy to check that
implies non
non
and cov
cov . If is a proper a-ideal on , that is, is a -ideal and contains
all singletons of
and does not contain , it holds that
add
cov
cof
and add
non
cof . We often use the notation
for a closed formura if formula is consistent. And CH, GCH and
CON
MA stand for the continuum hypothesis, the general continuum hypothesis
and the Martin’s axiom respectively.
We will work on the topological spaces; the Baire space
or the Cantor
space
instead of the real line . We call an element of any of these spaces
a real. We denote by
and
the ideal of meager subsets, the ideal
of Lebesgue measure zero subsets and the ideal of the strong measure zero
subsets of the real line respectively. Each cardinal (the additivity, covering
number, uniformity or cofinality) defined by
or
is constant in any
of the above topological spaces.
The ideals
are introduced by T. Yorioka to study the cofinality of
the strong measure zero ideal. The following definitions are not original
definitions which Yorioka introduced, but these are the same ideals as Yorioka
defined.
$\mathcal{A}\subset \mathcal{B}$
$(\mathcal{B})$
$\mathcal{I}$
$(\mathcal{A})\leq$
$\mathcal{X}$
$\mathcal{I}$
$(\mathcal{I})\leq$
$\mathcal{I}$
$\sigma$
$\mathcal{X}$
$\mathcal{X}$
$(\mathcal{I})\leq$
$(\mathcal{I})$
$(\mathcal{I})\leq$
$(\mathcal{A})\geq$
$(\mathcal{B})$
$\omega_{1}\leq$
$(\mathcal{I})\leq$
$(\mathcal{I})$
$(\varphi)$
$\varphi$
$\varphi$
$\omega\omega$
$\omega 2$
$\mathbb{R}$
$\mathcal{M},$
$S\mathcal{N}$
$\mathcal{N}$
$\mathcal{M},$
$\mathcal{N}$
$S\mathcal{N}$
$\mathcal{I}_{f}$
Definition 1.1 (T. Yorioka [2]) Let
by
the relation $<<$ on
”
$f\ll g$
$\mathcal{A}nd$
iff
$f\in\omega\omega$
be strictly increasing.
$\forall k<\omega\forall^{\infty}n<\omega(f(n^{k})\leq g(n))$
we denote by
Define
$\omega\omega$
$\mathcal{I}_{f}$
.
the family
$\mathcal{I}_{f}=\{X\subset\omega 2:\exists\sigma\in\omega(<\omega 2)(\langle|\sigma(n)|:n<\omega\rangle\gg f$
and
$\forall x\in X\exists^{\infty}n<\omega(\sigma(n)\subset x))\}$
.
To make the ideal
a -ideal for each strictly increasing function
Yorioka introduced the order $‘\ll’$ .
$\mathcal{I}_{f}$
$\sigma$
Fact 1.2 (T. Yorioka [2]) Let
-ideal.
$\sigma$
$f\in\omega\omega$
be strictly increasing. Then
$\mathcal{I}_{f}$
$f$
,
is a
$\square$
85
It is the fact that $f\leq*f’$ implies
is a subideal of
. By this fact,
we have that $f\leq*f’$ implies cov
cov
and non
non
. It
means that $\min$ { cov
:
and is strictly increasing}
cov
and $\sup$ { non
:
and is strictly increasing}
where
non
is the identity function from to
About the additivity and cofinality
of the ideals , we have the following fact.
$\mathcal{I}_{f}/$
$(\mathcal{I}_{f})\leq$
$f\in\omega\omega$
$(\mathcal{I}_{f})$
$(\mathcal{I}_{f})$
$\mathcal{I}_{f}$
$(\mathcal{I}_{f’})$
$(\mathcal{I}_{f})\geq$
$f$
$=$
$f$
$f\in\omega\omega$
$id_{\omega}$
$=$
$\omega$
$\omega$
$(\mathcal{I}_{f’})$
$(\mathcal{I}_{id_{\omega}})$
$(\mathcal{I}_{id_{\omega}})$
,
$\mathcal{I}_{f}$
Fact 1.3 (S. Kamo) Let $f,$
be strictly increasing.
$(f(n+1)-f(n)\leq f’(n+1)-f’(n))$ holds, then add
cof
cof
hold.
$f’\in\omega\omega$
If
$\forall^{\infty}n<\omega$
$(\mathcal{I}_{f})\geq$
$(\mathcal{I}_{f})\leq$
add
$(\mathcal{I}_{f’})$
and
$\square$
$(\mathcal{I}_{f’})$
The supremum of the additivitv of
and the minimum of the cofinality
of
are detarmined by the above fact. These are add
and cof
respectively. So, we define the following cardinal invariants related to the
ideals . We describe the consistency results of these invariants.
$\mathcal{I}_{f}$
$(\mathcal{I}_{id_{w}})$
$\mathcal{I}_{f}$
$(\mathcal{I}_{id_{w}})$
$\mathcal{I}_{f}$
{ add
supcov $= \sup$ { cov
minnon $= \min$ { non
supcof $= \sup$ {cof
minadd
$= \min$
$(\mathcal{I}_{f})$
$(\mathcal{I}_{f})$
$(\mathcal{I}_{f})$
$(\mathcal{I}_{f})$
2
:
:
:
:
$f\in\omega\omega$
$f\in\omega\omega$
$f\subset\omega\omega$
$f\in\omega\omega$
and
and
and
and
$f$
$f$
$f$
$f$
is
is
is
is
strictly
strictly
strictly
strictly
increasing},
increasing},
increasing},
increasing}.
Summary of ZFC results
is the subideal of the ideal
It can be easily proved that the null ideal
cov
. So, we have that cov
and
for all strictly function
, it can be easily
. Also, for each strictly function
non
non
are isogonal. Therefore it
and the meager ideal
proved that the ideal
. About the additivity
cov
and non
non
holds that cov
and cofinality of , the following theorem is proved in 2006.
$\mathcal{N}$
$\mathcal{I}_{f}$
$f\in\omega\omega$
$(\mathcal{I}_{f})\leq$
$(\mathcal{N})$
$(\mathcal{I}_{f})\geq$
$f\in\omega\omega$
$(\mathcal{N})$
$\mathcal{M}$
$\mathcal{I}_{f}$
$(\mathcal{I}_{f})\leq$
$(\mathcal{M})$
$(\mathcal{I}_{f})\geq$
$(\mathcal{M})$
$\mathcal{I}_{f}$
Theorem 2.1 (S. Kamo [5]) add
$(\mathcal{I}_{f})\leq b$
and cof
$(\mathcal{I}_{f})\geq 0$
.
$\square$
is the
It is the known fact that the additivity of the meager ideal
minimum of the unbounding number and the uniformity of the strong mea, M. Kada
. About the cofinality of the meager ideal
sure zero ideal
$\mathcal{M}$
$S\mathcal{N}$
$\mathcal{M}$
86
is the maximum of
showed a fact that the cofinality of the meager ideal
that is introduced
the dominating number and the cardinal invariant
by M. Kada [6]. And we have the following lemma about the minimum of
. Because the strong measure zero ideal corresponds with
uniformity of
.
for all
the intersection of the ideals
$\mathcal{M}$
$0_{ubd}$
$\mathcal{I}_{f}$
$f\in\omega\omega$
$\mathcal{I}_{f}$
Lemma 2.2 minnon
$=$
non
$(S\mathcal{N})$
and supcov
$=\mathfrak{d}_{ubd}$
.
$\square$
hold by the
cof
and supcof
add
It remarks that minadd
theorem 2.1 and lemma 2.2. By Dr. Brendle, the following results are proved.
$(\mathcal{M})$
$\leq$
Theorem 2.3 (J. Brendle, 2008) add
$\geq$
$(\mathcal{N})\leq$
$(\mathcal{M})$
minadd and supcof
$\leq$
cof
$(\mathcal{N})$
hold,
These results have been newly added. So we have the new questions
) holds.
minadd (or supcof cof
whether it is consistent that add
$<$
$(\mathcal{N})<$
$(\mathcal{N})$
(Question 3.9)
We have the twenty cardinal invariants (the invariants in the Cicho\’{n}’s
and
and the continuum
diagram, the invariants related to the ideals
c . The following diagram (Figure 1) summarizes the relationships between
these cardinal invariants which is provable in ZFC. The arrows in the diagram
point toward larger invariant.
$\mathcal{I}_{f}$
$\omega_{1}$
$)$
cov
$(\mathcal{N})arrow$
$\ovalbox{\tt\small REJECT}$
$\omega_{1}arrow$
add
$(\mathcal{N})arrow$
cov
add
$(i_{f})arrow$
supcov
$arrow non(\mathcal{M})arrow$
$b$
$(\mathcal{I}_{f})$
minadd
add
$arrow$
cof
supcof
$(\mathcal{M})$
cof
$cof(\mathcal{I}_{f})$
$0$
$(\mathcal{M})arrow cov(\mathcal{M})arrow$
$arrow$
minnon
$arrow$
non
$(\mathcal{N})arrow c$
$\ovalbox{\tt\small REJECT}$
$(\mathcal{I}_{f})arrow non(\mathcal{N})$
Figure 1: Cicho\’{n}’s diagram and the cardinal invariants related to the ideals
$\mathcal{I}_{f}$
87
Moreover, we introduce the rclationship between the cardinal invariants
and the cardinal invariants of the strong measure
related to the ideals
$\mathcal{I}_{f}$
for
zero ideal
. The strong measure zero ideal is included the ideals
. So, we have the following results about the supremum of the
all
covering numbers of . By the lemma 2.2, the minimum of the uniformity
.
is identical to the uniformity of the strong measure zero ideal
of
$S\mathcal{N}$
$\mathcal{I}_{f}$
$f\in\omega\omega$
$\mathcal{I}_{f}$
$S\mathcal{N}$
$\mathcal{I}_{f}$
Lemma 2.4 supcov
cov
$\leq$
$(S\mathcal{N})$
.
$\square$
And we have the following results for the additivity.
Lemma 2.5 minadd
$\leq$
add
$(S\mathcal{N})$
.
$\square$
We can expect the dual of the lemma above, that is, the supremum of the
. But it is possible
is an upper bound of the cofinality of
cofinality of
is larger than the continuum. We introduce a
that the cofinality of
.
number that is beyond the cofinality of the strong measure zero ideal
$S\mathcal{N}$
$\mathcal{I}_{f}$
$S\mathcal{N}$
$S\mathcal{N}$
Lemma 2.6 cof
3
$(S\mathcal{N})\leq 2^{v}$
.
$\square$
Summary of Consistency results
In this section, we introduce some consistency results. At the first, we introduce the consistency results between the cardinal invariants related to the
and the cardinal invariants in the Cicho\’{n}’s diagram. It is known
ideals
for all strictly increasresult that the Martin’s axiom implies add
. This is proved by using the forcing notion which is
ing function
introduced by T. Yorioka [2]. Therefore it is consistent that minadd
for all strictly
holds. And we have proved the consistency that cof
-stage countable
. This is proved by using a
increasing function
support iteration of forcing notions with the Sacks property [7]. Therefore it
$\mathcal{I}_{f}$
$(\mathcal{I}_{f})=c$
$f\in\omega\omega$
$>\omega_{1}$
$(\mathcal{I}_{f})<c$
$f\in\omega\omega$
$\omega_{2}$
is consistent that supcof $<c$ .
We proved the following lemma.
-stage
be the
Lemma 3.1 (CH) Let
Hechler forcing notion. Then it holds that
”,
add
$D_{\cup 2}$
$\omega_{2}$
finite support
$|\vdash D_{\omega_{2}}$
$(\mathcal{M})=\omega_{2}$
iteration
of
’ $\forall f\in\omega\omega(cov(\mathcal{I}_{f})=\omega_{1})$
the
and
$\square$
88
And we proved the dual of the lemma above.
Lemma 3.2 $(MA+c=\omega_{2})$ Let
be the -stage
of the Hechler forcing notion. Then it holds that
and cof
”.
$D_{\omega\iota}$
$\omega_{2}$
finite
“
$|\vdash D_{\omega}2$
support iteration
$\forall f\in\omega\omega$
(non
$(\mathcal{I}_{f})=$
$(\mathcal{M})=\omega_{1}$
$\omega_{2})$
$\square$
By these results, we have the following consistency results.
Corollary 3.3 CON $( \sup\subset ov<$ non
Corollary 3.4 CON (minnon
$(\mathcal{M}))$
$>co\vee(\mathcal{M})$
and CON (minadd
) and CON (supcof
$<$
add
$(\mathcal{M})$
$>\subset of(\mathcal{M})$
).
$\square$
).
$\square$
Also we studied about the consistency problems between the cardinal
invariants of
for each function
and the minimum or supremum
of the cardinal invariants of
for all
. We obtained the following
results for the covering number and uniformity.
$f_{0}\in\omega\omega$
$\mathcal{I}_{fo}$
$f\in\omega\omega$
$\mathcal{I}_{f}$
Theorem 3.5 (CH) For all strictly increasmg functions
and a forcing notion
a strictly increasing function
cov
.
countable chain condition such that
cov
there exist
which satisfies
$g\in\omega\omega$
$f\in\omega\omega$
$|\vdash \mathbb{P}$
$(\mathcal{I}_{f})>$
$\mathbb{P}$
$\square$
$(\mathcal{I}_{g})$
Theorem 3.6 $(MA+c=\omega_{2})$ For all strictly increasing functions
and a forcing notion
there exist a strictly increasing function
non
which satisfies countable chain condition such that
non
$g\in\omega\omega$
$f\in\omega\omega$
$|\vdash \mathbb{Q}$
$\mathbb{Q}$
$(\mathcal{I}_{f})<$
$(\mathcal{I}_{g})\square$
By these theorem, we can obtain the following corollary immediately.
Corollary 3.7 CON (
$\exists f$
(supcov
$>$
cov
$(\mathcal{I}_{f})$
)) and CON (
$\exists f$
(minnon
$<$
non
$(\mathcal{I}_{f}))$
$\square$
About the covering number and uniformity, we obtain some results. But
and the
we have no consistency results between the invariants of each
about the additivity(or
minimum (or supremum) of the invariants of all
$\mathcal{I}_{f}$
$\mathcal{I}_{f}$
cofinality).
).
89
Question 3.8 Is it consistent that there is a strictly increasing function $f\in$
such that minadd
add
? And is it consistent that there is a strictly
increasing function
such that supcof cof
?
$\omega\omega$
$<$
$(\mathcal{I}_{f})$
$f\in\omega\omega$
$>$
Question 3.9 Is it consistent that add
holds ?
$(\mathcal{N})<$
$(\mathcal{I})$
minadd (or supcof
$<cof(\mathcal{N})$
)
Next, we introduce the consistency results between the strong measure
zero ideal and the ideals . We have the three inequalties, that is, minadd
add
and supcov cov
and cof
. (The minimum of the
uniformity of the ideals
is equal to the uniformit, of the strong measure
)
zero ideal
$\mathcal{I}_{f}$
$(S\mathcal{N})$
$\leq$
$(S\mathcal{N})\leq 2^{0}$
$(S\mathcal{N})$
$\leq$
$\mathcal{I}_{f}$
$y$
$S\mathcal{N}.$
As the results related to the additivity and covering number, the following
results is known.
Fact 3.10 (Bartoszy\’{n}ski [3]) (CH) Let
be the
-stage
support iteration of the eventually equal forcing notion. Then
and add
.
$EE_{\omega_{2}}$
$\omega_{2}$
$\omega untable$
$|\vdash EE_{\omega_{2}}$
$=\omega_{1}$
cof
$(\mathcal{M})$
$(S\mathcal{N})=\omega_{2}$
By minadd
immediately.
$\leq$
supcov
$\leq$
cof
Corollary 3.11 CON(minadd
$(\mathcal{M})$
$<$
,
$\square$
the following corollary can be obtained
add
$(S\mathcal{N})$
) and CON(supcov
$<$
cov
$(S\mathcal{N})$
).
$\square$
About the cofinality of the strong measure zero ideal
fact is known.
Fact 3.12 (T. Yorioka [2]) CH implies cof
dominating number for the functions in
.
$S\mathcal{N}$
$(S\mathcal{N})=\mathfrak{d}_{\omega_{1}}$
,
, the following
where
$0_{\omega_{1}}$
is the
$\square$
$\omega_{i}\omega_{1}$
, GCH implies that the cofinality of the strong measure
By
zero ideal
is equal to .
Also, the cofinality of the strong measure zero ideal
is equal to the
continuum in the model satisfying the Borel conjecture. And it is consistent
that the Borel conjecture holds and the dominating number is equal to
$\omega_{2}\leq 0_{\omega 1}\leq 2^{\omega_{1}}$
$S\mathcal{N}$
$2^{v}$
$S\mathcal{N}$
$\mathfrak{d}$
90
the continuum. (By using the -stage countable support iteration of the
Mathias forcing notion, we can obtain a model in which the Borel conjecture
holds and the dominating number is equal to the continuum [8]. So it is
.
consistent that cof
$\omega_{2}$
$\mathfrak{d}$
$)$
$(S\mathcal{N})<2^{0}$
References
[1] E. Borel, “Sur la classification des ensembles de measure nulle,” Bulletin
de la Societe Mathematique de France, vol. 47, pp. 97-125, 1919.
[2] T. Yorioka, “The cofinality of the strong measure zero ideal,” Joumal
Symbolic Logic, vol. 67, no. 4, pp. 1373-1384, 2002.
of
[3] T. Bartoszy\’{n}ski and H. Judah, Set theory: on the structure of the real
line. 289 Linden Street Wellesley, Massachusetts 02181 USA: A. K. Peters,
Ltd., 1995.
[4] K. Kunen, Set Theory. North Holland, 1980.
[5 N. Osuga and S. Kamo, “The cardinal coefficients of the ideal
of Mathematical Logic, vol. 47, pp. 653-671, 2008.
$|$
”
$\mathcal{I}_{f},$
Journal
[6] M. Kada, Consistency results conceming shrinkability for positive sets
reals. $PhD$ thesis, Osaka Prefecture University, 1997.
[7] N. Osuga, “The covering number and the uniformity of the ideal
Mathematical logic Quarterly, vol. 52, no. 4, pp. 351-358, 2006.
of
”
$\mathcal{I}_{f},$
[8] J. E. Baumgartner, “Iterated forcing,” in Surveys $m$ Set Theory, In London Mathematical Society Lecture Note Series 87, pp. 1-59, Cambridge
University Press, 1983.