数理解析研究所講究録 1202 巻 2001 年 7-32

7Cardinal invariants of the continuum–Asurvey
J\"org Brendle *The Graduate School of Science and Technology
Kobe University
Rokko-dai1-1, Nada-ku
Kobe 657-8501, Japan
神戸市灘区六甲台 1–1

神戸大学大学院自然科学研究科
brendle(Dpascal . seg. kobe-u. ac.jp

March 16, 2001

Abstract
These are expanded notes of aseries of two lectures given at the meeting on
at Ky\={o}to University
axiomatic set theory
in November 2000. The lectures were intended to survey the state of theart of the theory of cardinal invariants of the continuum, and focused on theinterplay between iterated forcing theory and cardinal invariants, as well as on
important open problems. To round off the present written account of this survey,
we also include sections on $ZFC$-inequalities between cardinal invariants, and on
applications outside of set theory. However, due to the sheer size of the area, proofs
had to be mostly left out.
While being more comprehensive than the original talks, the personal flavor of
the latter is preserved in the notes. Some of the material included was presented
in talks at other conferences.
$(\overline{\mathit{5}1\backslash }\#\beta*l^{11}\mp^{k}\Re\Phi \mathrm{f}\mathrm{f}1\Re \mathrm{f}\mathrm{f}1$$(’\grave{\mathrm{A}}\mathrm{E}\mathfrak{X}\not\in^{\mathrm{x}^{-\mathrm{A}}}\square \mathrm{r}\mathrm{B}\mathrm{r})\equiv$$*\pi\overline{\mathrm{P}}F)$

upported by Grant-in-Aid for Scientific Research (C)(2)12640124, Japan Society for the Pron

8
1What are cardinal invariants?
We plan to look at certain basic features of the real line from the point of view of
combinatorial set theory. For our purposes, it is convenient to work with the Cantor
space 2’ or the Baire spaceinstead of itself. Here we put, as usual,$\mathbb{R}$$\omega^{\omega}$2$\omega$2’$\omega^{\omega}$$\mathbb{R}$$=${0, 1}$=$$\mathrm{N}=\{0,1,2,3, \ldots\}=\mathrm{t}\mathrm{h}\mathrm{e}$the set of functions fromthe set of functions from

$=$

$=$

$\omega$$\omega$natural numbers
to 2
to .
$\omega$

Both 2’ andcan be turned into topological spaces in anatural way: 2andcarry the discrete topology, andandare equipped with the product topology.
This means that basic open sets in, say, 2’ are of the form$\omega^{\omega}$$\omega$$2^{\omega}$$\omega^{\omega}$$[\sigma]:=\{f\in 2^{\omega};\sigma\subseteq f\}$where
, that is, is afinite sequence of 0’s and Is. Whereas open sets have aneasy description in any of these spaces (countable unions of open intervals in case of ,countable unions of sets of the formin case of 2’ and ), in 2’ (and ) closed sets
are nicely characterized as well, namely as sets of branches$\sigma\in 2^{<\omega}$$\sigma$$\mathbb{R}$$[\sigma]$$[T]:=${$\omega^{\omega}$$f\in 2^{\omega};f[n$ $\in T$

for all$n$$\omega^{(v}$}through trees, and this is one of the main reasons for using the Cantor spaceor the Baire space in combinatorial set theory.
Incidentally, there is an alternative description of this topology, as follows. Given, define
0if $f=g$$d(f,g)=\{$
if $f\neq g$ .
is easily seen to be ametric which turns 2’ into aPolish (complete metric)
space. The topology induced by the metric is identical to the product topology outlinedin the previous paragraph.
While ,
andare different objects from the topological point of view (e.g.,
is compact, is -compact and not compact, whereas
is not even a-compact),
they are still close enough to each other so that all notions we are interested in carryover from one space to the other. In fact, they are “homeomorphic modulo acountableset”. For example, the mapping $F$ sending
to the real $F(f)\in[0,1]$ whose binary
expansion is$T\subseteq 2^{<\omega}$$f,g\in 2^{\omega}$$\frac{-1}{2^{\min\{njf(n)\neq g(n\Pi}}$$d$$=\mathrm{s}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}$$2^{\omega}$$2^{\omega}$$\omega^{\omega}$$\mathbb{R}$$\mathbb{R}$$\sigma$$\omega^{\omega}$$f\in 2^{\omega}$$F(f)=0.f(0)f(1)f(2)f(3)\ldots$

is continuous and onto $[0, 1]$ , as well as injective on a -countable subsetof ,with the inverseof the; restriction $F[C$ being continuous too. Similarly, it’swell-known that the Baire space is homeomorphic to the irrational numbers.
Recall that asubset of atopological space $X$ is called nowhere dense if its closure
has empty interior.
is meager (or: of first category) if it’s aunion of countabl$\mathrm{c}\mathrm{o}$$C$$2^{\omega}$$\mathrm{m}\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{i}_{\mathrm{L}}\mathrm{n}\mathrm{g}$$A$$B\subseteq X$$\mathrm{y}$9. (Fordenote the a-ideal of meager subsets ofmany nowhere dense sets. Let
our purposes, it doesn’t matter on which space we consider . The point is the maps
and which we briefly discussed above send meager sets toalmost identifying ,is non-trivial.
meager sets and vice-versa.) By the Baire category theorem,with ameasure in anatural way: 2
As with topology, we can equip 2’ andcarries the measure giving
carries the measure giving both {0} and {1} measureboth get the product measure. Since this is;and 2’ as well as
measureanalogous to the usual construction of Lebesgue measure on the reals , we may referto this measure as “Lebesgue measure” as well. Let $N$ stand for the a-ideal of null
. (Again, the exact nature of the space is not relevant for subsequent
subsets ofconsiderations.)
These ideals provide us with our first examples of cardinal invariants of the continand (so that, in particular, they become$uum$ , that is, cardinals sitting between, holds) and retrivial if the continuum hypothesis $CH$ , that is, the statement
flecting part of the combinatorial structure of the real line. Here we put, as usual,$\omega^{\omega}$$\mathcal{M}$$\mathcal{M}$$\mathbb{R}$$\omega^{\omega}$$2^{\omega}$$\mathcal{M}$$\omega^{\omega}$$\frac{1}{2};\omega$$\omega^{\omega}$$\{n\}$$\frac{1}{2^{n+1}}$$\mathbb{R}$$\omega^{\omega}$$\aleph_{1}$$\mathrm{c}$

$c$

$=$

$\aleph_{1}$$=$$\mathrm{c}$$=\aleph_{1}$the first uncountable cardinalcardinality of the continuum.
$|\mathbb{R}|=\mathrm{t}\mathrm{h}\mathrm{e}$Namely let us defineadd(I)$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{I})$non(I)$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{I})${{$= \min$ {$= \min$ {$= \min$$|\mathcal{F}|;\mathcal{F}\subseteq \mathrm{I}$$\min$$|\mathcal{F}|;\mathcal{F}\subseteq \mathrm{I}$

$=$

and

$\cup \mathcal{F}\not\in \mathrm{I}$and$\cup \mathcal{F}$

and
and

$|X|;X\subseteq\omega^{\omega}$$|\mathcal{F}|;\mathcal{F}\subseteq \mathrm{I}$}$=\omega^{\omega}$$X\not\in \mathrm{I}$$\forall A\in \mathrm{I}$}$=\mathrm{t}\mathrm{h}\mathrm{e}$}$=$additivity of I$=\mathrm{t}\mathrm{h}\mathrm{e}$covering number of I

the uniformity of I$\exists B\in \mathcal{F}(A\subseteq B)$}

cofinality of I$=th\mathrm{e}$as well as
or $N$ . Note that one has add(I)where Iis either
be awitness for. To see, e.g., the last inequality, letaddand put $X=\{x_{A};A\in \mathcal{F}\}$ .
areal(I), choose for each
and $N$ are a-ideals is rephrased
is straightforward. Furthermore, the fact that bothand $N$, respectively) in this language. Since both
(add(N)
as add(M), resp.) is immediate ashave abasis consisting of Borel sets,, that is, if
is called abasis if it satisfies the defining clause ofwell. Herecontaining . Finally notice that add (I) is always aregular
there isfor allcardinal.
is given by: $f\leq^{*}g$ iff $f(n)\leq g(n)$ holds foron
The eventual dominance order
all but finitely many . In this case we say eventually dominates . (This relation istransitive and reflexive, but not antisymmetric. It can be turned into apartial order bylooking at equivalence classes of functions modulo the relation almost equal. However,is unbounded if it’s not bounded in
we have no use for doing this.) Let us say
such that $f(n)>g(n)$ forthere isthis order or, equivalently, if given any
infinitely many . The unbounding number is the size of the smallest unbounded family.
.
An easy diagonal argument shows that countable families are bounded so that$g(k)=\max\{f_{n}(k);n\leq
k\}$.
$n\in\omega$, define,(More explicitly, given functions
dominating if it is cofinal in the eventual dominancefor all $n.$ ) Call
Then$\leq \mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{I})$$\mathcal{M}$$\leq \mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{I})$$\mathcal{F}\subseteq \mathrm{I}$$(\mathrm{I})\leq \mathrm{n}\mathrm{o}\mathrm{n}(\mathrm{I})$$\leq \mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{X})$$X\not\in \mathrm{I}$$x_{A}\in\omega^{\omega}\backslash A$$A\in \mathcal{F}$$\mathcal{M}$$\mathcal{M}$$\geq\aleph_{1}$$\geq\aleph_{1}$$\mathrm{c}\mathrm{o}\mathrm{f}(\mathcal{M})$$\leq \mathrm{c}\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{I})$$\leq \mathrm{c}$$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{I})$$\mathcal{F}\subseteq \mathrm{I}$$A$$B\in \mathcal{F}$$A\in \mathrm{I}$$\leq*$$\omega^{\omega}$

$f$

$g$

$n$

$\mathcal{F}$$g\in\omega^{\omega}$$\subseteq\omega^{\omega}$$f\in \mathcal{F}$$\mathrm{b}$$n$$\aleph_{1}\leq \mathrm{b}$$f_{n}\in\omega^{\omega}$$f_{n}\leq^{*}g$$\mathcal{F}\subseteq\omega^{\omega}$

10order, that is, if, given any, there iswith $g\leq^{*}f$ . The dominating number
is the size of the least dominating family. Note that for the definition of it would
not matter if we considered domination everywhere, i.e., if we replaced$f\leq g$ iff $f(n)\leq g(n)$ for all. (However, such achange would affect ;for indeed,
the corresponding cardinal would be , and thus uninteresting.) Clearly. Alittle while’s thought lets us convince ourselves that is regular and less than $cf(V)$ , thecofinality of 0.
The relationship between these numbers is best illustrated by the following diagram,called’s diagram, where cardinals grow as one moves up
to the right.
$g\in\omega^{\omega}$$f\in \mathcal{F}$

$v$

$v$

$\leq*\mathrm{b}\mathrm{y}\leq \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$n$$\mathrm{b}$$\aleph_{0}$$v$$\leq \mathrm{c}$$\mathrm{b}$$Cicho\acute{n}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{c}\mathrm{o}\mathrm{v}(N)$$-$ non$(\mathcal{M})-\mathrm{c}\mathrm{o}\mathrm{f}(\mathcal{M})$$-\mathrm{c}\mathrm{o}\mathrm{f}(N)$$-$$\mathrm{c}$$|\begin{array}{lll}| |\mathrm{b} - v| |\end{array}|$$\aleph_{1}-$add(M)–

add$(\mathcal{M})$$-\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$-$ non(N)The inequalities between cardinals exhibited in this diagram (all theorems of $ZFC$ ,ZermelO-Fraenkel set theory with choice) were mostly proved in theand(see. The deepest and most important result is perhaps$70’ \mathrm{s}$$80’ \mathrm{s}$$[\mathrm{B}\mathrm{a}\mathrm{J}])$

Theorem 1.1 (Bartoszyiiski-Raisonnier-Stern Theorem, classical version
add.

$[\mathrm{B}\mathrm{a}\mathrm{J}]$)$(N)\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{M})$

None of the inequalities above is reversible (see Section 4). However, there are two
results each of which relates three of the cardinals.

Theorem 1.2 (Miller-Truss Theorem$\max${ , non$v$$(\mathcal{M})$}.

$[\mathrm{B}\mathrm{a}\mathrm{J}]$) add(M)$= \min\{\mathrm{b}, \mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})\}$

and$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{A}0$$=$

Except for this restriction, it has been shownthat all assignments of the valuesand H2 to the above cardinals which do not contradict the diagram are consistent
with $ZFC$ . (However, if the continuum is larger than , anumber of deep questions
remain open, see below, Sections 4and 5.)
There are quite afew more interesting and important cardinal invariants which weshall introduce as need arises. At this point, let us rather ask: Why are cardinal invariants
Why do we study them?
(1) They are interesting objects of study in their own right. For they describe theunderlying combinatorial structure of the real line.
$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\aleph_{1}$$\aleph_{2}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}^{7}$(2) There are lots of connections to other areas of mathematics. In particular, togeneral topology. But also to algebra (mainly group theory) and real analysis...
We shall illustrate this in Section 3.
(3) They provide us with “test problems” for forcing theory, in particular in light ofthe search for new iteration techniques. We shall come back to this in Section 4.
For amore detailed account of everything we discussed so far see eitheror [B12]$[\mathrm{B}\mathrm{a}\mathrm{J}]$11
2Proving inequalities in ZFC
In the preceding section we saw some easy examples of how to prove inequalities between
cardinal invariants in $ZFC$ . But how does one proceed in general? Usually, such proofs
are first done in an ad hoc fashion, but then adeeper analysis may yield either sharper
results or asimplified argument or the very essence of what’s going on. There are two
important, and not unrelated, ways to look at such arguments, namely(1) working over models for fragments of $ZFC$ ,(2) reformulating the argument in the language of s0-called Galois-Tukey connections.

We shall take acloser look at both.
FIRST METHOD. models OF FRAGMENTS OF $ZFC$ . The
Stern Theorem 1.1 is sometimes stated as$\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}-\mathrm{R}\mathrm{a}\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{i}\mathrm{e}\mathrm{r}-$) Assume-Raisonnier-Stern Theorem, model version
Theorem 2.1 (
$ZFC$
is
there
that
suppose.
Also$M$ is a modelof a large enough finite fragment ofnull set A which contains all Borel null sets coded in M. Then there is a meager set $B$
which contains all Borel meager sets coded in $M$ .
$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$a$. For assume,
Let us first check this indeed implies add(N)
$M$
of
size
$T$
amodelfind
thenof many meager sets. We canand choose afamily
as well as satisfying alarge enough finite
all members offorcontaining Borel codes, the union of the Borel null sets coded in $M$ is nullfragment of $ZFC$ . Since
so that there is satisfying the assumption of the above theorem. Therefore, we obtain, asameager set containing all Borel meager sets coded in $M$ . Afortiori,
required.
In principle, the proof of 2.1 goes like a“model-free” proof of 1.1. There is one slight, one
simplification, however. The point is to prove astatement like add(jV)of size less thanhas to proceed in the following, rather obvious, manner: fix
in such awayadd(N),
with(, still of size less thanadd(A ) associate afamily
is meager
is null (i.e. contained in aBorel null set ) entails
that the fact that$B$ ). Now, in this as well as in most interesting cases(i.e. contained in aBorel meager set
in such away that any(see below for some counterexamples), is constructed frommodel $M$ containing will also contain . Therefore the approach via models allows us
not to care a priori about the concrete nature of , and just take any Borel null set
which contains all Borel null sets coded in $M$ . Then use to construct the appropriate
Borel meager set $B$ . Only when doing the last step of the proof do we associate withfrom $M$ such(or from $M$ , for that matter) anull set
each meager set $X$ from. .(To appreciate what we mean by “not caring apriori” weentailsthat
recommend the reader to have a look at the proof of Theorem 2.12 below.)
There are, however, more important reasons for formulating statements like 2.1:$\kappa$$\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{j}\mathrm{V}\mathrm{f})$$<\mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{A}\mathrm{I})$$\kappa$$\kappa$$\mathcal{F}$$\kappa$$<\mathrm{a}\mathrm{d}\mathrm{d}(N)$$A$$\cup \mathcal{F}\subseteq B$$B$$\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{A}\mathrm{I})$$\mathcal{F}\subseteq \mathcal{M}$$;\subseteq N$$\mathcal{F}$$A$$\cup \mathcal{G}$$\cup \mathcal{F}$$\mathcal{F}$$\mathcal{G}$$\mathcal{G}$$\mathcal{F}$$A$$\mathcal{G}$$A$$\mathrm{Y}$$\mathcal{F}$$\mathrm{Y}\subseteq A$$X\subseteq B$(1) They are readily adapted to adescriptive set-theoretic context. For example,
2.1 in particular entails that if there is anull set containing all Borel null sets

12
coded in $L[x]$ , then there is ameager set containing all Borel meager sets coded
in $L[x]$ . Now, the former statement is well-known to be equivalent to (lightface), while the latter is the same as the (lightface)-Baireproperty. Therefore, we immediately get the (effective) descriptive set-theoretic
version of the Bartoszsynski-Raisonnier-Stern Theorem, namely the statement-measurability implies-Baire property.
$\Sigma_{2}^{1}(x)-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}\mathrm{i}1\mathrm{i}\mathrm{t}\mathrm{y}$$\Sigma_{2}^{1}(x)$$\Sigma_{2}^{1}(x)$$\Sigma_{2}^{1}(x)$(2) They are particularly useful when dealing with cardinal invariants which can bereformulated in forcing language (Martin axiom language).

Let me expand somewhat on the second point. Given a. , denote by
the least
such that Martin’s axiom
fails for , that is, there are many dense sets in suchthat no filter meets all of them. For aclass $P$ of.’s, let.
(So), i.e., the least cardinal such that the standard
denotes
Martin axiom
fails.
$\mathrm{P}$$\mathrm{m}(\mathrm{P})$$\mathrm{p}.0$$MA_{\kappa}$$\kappa$$\mathrm{P}$$\mathrm{p}.0$$\mathrm{m}(\{\mathrm{P}\})=\mathrm{m}(\mathrm{P}).$$\mathrm{m}(ccc)$$\mathrm{m}$$\mathrm{P}$$\kappa$$\mathrm{m}(P)=\min\{\mathrm{m}(\mathrm{P});\mathrm{P} \in P\}$$\kappa$$MA_{\kappa}$

Fact 2.2(ii)(i)$\mathrm{m}(\mathrm{C})=\mathrm{m}(\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e})=\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{m}(\mathrm{B})=\mathrm{c}\mathrm{o}\mathrm{v}(N)$.

.

Proof, (i) The first equality is an immediate consequence of the fact that every countable
forcing notion is equivalent to Cohen forcing.
Next we argue that
. For indeed if
is afamily of Borel
sets of size less than
, then there is amodel $M$ containing the Borel codes of themembers of and still of size less than
. Hence there is afilter which meets all
dense sets of of $M$ , that is is -generic over $M$ . From we can decode areal
which is Cohen-generic over $M$ . By Solovay’s characterization of being Cohen-generic(that is, areal is Cohen-generic over $M$ iff it avoids all Borel meager sets coded in $M$ ),we see that, so thatwas not acovering family.
To show, we simply
our way back in the argument of thepreceding paragraph.
(ii) is analogous.
$\mathrm{m}(\mathbb{C})\leq \mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathcal{F}$$\subseteq \mathcal{M}$$\mathrm{m}(\mathbb{C})$$\mathcal{F}$$F$$\mathrm{m}(\mathbb{C})$$F$$\mathbb{C}$$c\not\in\cup \mathcal{F}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{A}\mathrm{f})$$F$$\mathbb{C}$$c$$\mathcal{F}$$\leq \mathrm{m}(\mathbb{C})$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\square$Now, what is the use of this? Suppose for example, we want to prove the inequality
in Cichon’s diagram. Of course, this can be done directly rather easily,
but it becomes even more trivial in the “model languag\"e. One simply needs to recall
that a real Cohen over $M$ is unbounded over $M$ (in the sense that is not (eventually)
dominated by any member from At). Now, if
is of size less than, we
can find a model $M$ , still of size less thanand containing , and then areal
over $M$ . Since is unbounded over $M$ , we see immediately that 7was not adominating family. Hence.
In asimilar fashion, using that areal random over $M$ canonically defines ameager
set containing, one can prove. And
isanalogous!
The additivity numbers can also be formulated in forcing language.
$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{t}$$\leq v$$c$$c$$\mathcal{F}\subseteq\omega^{\omega}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{A}4)$$c\mathrm{C}\mathrm{o}\dot{\mathrm{h}}\mathrm{e}‘ \mathrm{n}$$\mathcal{F}$$c$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{A}\mathrm{l})$$\leq v$$r$

$M\cap 2^{\omega}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{N})$$\leq \mathrm{n}\mathrm{o}\mathrm{n}(\mathrm{A}\mathrm{l})$$j$Proposition 2.3 (see$[\mathrm{B}\mathrm{a}\mathrm{J}]$and [B4})$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\leq \mathrm{n}\mathrm{o}\mathrm{n}(N)$13where denotes Hechler forcing (the standard -centered forcing(i) add(M)for adding a dominating real).
$\mathrm{D}$$=\mathrm{m}(\mathrm{D})$$\sigma$where Adenotes amoeba forcing (the standard forcing(ii) add(M)for adding a measure one set of random reals), and LQC is localization forcing
and $|\phi(n)|\leq n$ for(the standard forcing for adding a function with domain
in the ground model, $f(n)\in\phi(n)$ holds for all but
all such that for all
finitely many $n.$ )$=\mathrm{m}(\mathrm{A})=\mathrm{m}(\mathrm{L}\mathbb{O}\mathbb{C})$$\phi$$\omega$$f\in\omega^{\omega}$

$n$

with
Here, we say aforcing notion is -centered if it can be written as. Similarly,
has alower boundbeing centered, that is, each finite
eachbeing linked, i.e. any two
with eachis -linked if it is of the formis -linked as well. A
forcingany
a-centeredare compatible. Clearly
members ofabove are -linked(but not a-centered).
and.
stand for the
denote the forcing for adding many Cohen reals; and letLeta-centered,isalgebra).
Thenadjoining many random reals (the product measureis a-linked (but not -centered). So we see
and$\mathrm{P}$$\mathrm{P}$$\sigma$$F\subseteq P_{n}$$P_{n}$$\mathrm{P}$$\mathrm{P}$$\sigma$$= \bigcup_{n}P_{n}$$p\in \mathrm{P}$$P_{n}$$= \bigcup_{n}P_{n}$$\sigma$$P_{n}$$\mathrm{L}\mathbb{O}\mathbb{C}$$\sigma$$\mathrm{B}_{\kappa}$$\mathrm{p}.0$$\kappa$$\mathbb{C}_{\kappa}$$\mathbb{C}_{c}$$\kappa$$\sigma$$\mathrm{B}_{c}$Observation 2.4(ii)( -centered).

(i)$\mathrm{m}(\mathbb{C})\geq \mathrm{m}(\mathbb{C}_{c})\geq \mathrm{m}$$\sigma$( -linked).

$\mathrm{m}(\mathrm{B})\geq \mathrm{m}(\mathrm{B})\geq \mathrm{m}$$\sigma$

In fact one can prove
Theorem 2.5 [B4]$\mathbb{C}_{c}$completely embeds into$\mathrm{L}\mathbb{O}\mathbb{C}$.

In view of 2.3 this entails
Corollary 2.6 [B4]$\mathrm{m}(\mathbb{C}_{c})\geq \mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{M})$SECOND METHOD. Galois-Tukey CONNECTIONS. Most cardinal invariants of the.
, and non is dual tocontinuum come in dual pairs. For example add is dual to
Similarly, and are dual. Now, it has been realized early on that dual inequalities
between cardinal invariants have similar proofs. This is reflected rather badly by the“model language” above, and some other approach is needed to economize on proofs.
Consider$\mathrm{c}\mathrm{o}\mathrm{f}$$\mathrm{c}\mathrm{o}\mathrm{v}$$v$$\mathrm{b}$Theorem 2.7 (Bartoszyn’ski-Raisonnier-Stern Theorem, Galois-Tukey version, see$arrow N$ and $A-*B_{A}$ : A:[Ba] or [PR] There are “definable”functions.
is contained in , $X$ is contained insuch that whenever$X\vdasharrow \mathrm{Y}_{X}$

$)$

$A$

$\mathrm{Y}_{X}$$[\mathrm{B}\mathrm{a}\mathrm{J}]$$arrow \mathcal{M}$$\mathcal{M}$$B_{A}$, as
Let us quickly argue this implies Theorem 2.1 above, and thus add(AT). For the former, let $M$ and be as in 2.1,
well as the dual inequality
and check that $B=B_{A}$ is as required. Indeed, let $X$ be ameager set coded in $M$ . By$arrow N$ ,. Hencealso belongs to $M$ . Therefore
definability of the mapping$N$. Thenofbeabasislatter,let, which is what we wanted to show. For theis easily seen to be abasis of , as required.
$\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{M})$$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{A}\mathrm{T})$$\mathcal{M}$$A$$\geq \mathrm{c}\mathrm{o}\mathrm{f}(\mathcal{M})$$\mathrm{Y}_{X}\subseteq A$$\mathrm{Y}_{X}$$\mathcal{F}$$X\subseteq B$$\mathcal{G}=\{B_{A};A\in \mathcal{F}\}$$\mathcal{M}$,14
Acomment concerning the word “definable” in the statement of the theorem is inorder. Of course, from the argument in the preceding paragraph it is clear that it sufficesthat the process leading from $X$ to(from, respectively) is constructive
to
$X$
$ZFC$
enough so that every-model containing( , resp.) will also contain,resp.). However, one can usually do much better and get functions which are Borel oreven (in most cases) continuous. (Let me briefly dwell on what would be meant by“Borel” or “continuous” in the above theorem. It is relatively easy to see there is aBorel
master meager set in the plane, that is, ameager set such that all verticalsections
are meager and for all meager sets $X$ there is areal such that.
Similarly there is aBorel master null set $D$ . Now, “continuity” simply means that there
are continuous maps :and :such that whenever,then. Thenis simply, and$D_{y}-*C_{\psi(y)}$is.
See [Ba] or [PR] for details.)
The language with pairs of functions, as exemplified by Theorem 2.7, is originally
due to Kremlin, has been put into amore general framework by Vojtas [Vo], and reformulated by Blass [B12]. The optimal results, using continuous functions, were obtained
by Pawlikowski and edaw [PR] (see also [Ba]).
Unfortunately, things are not always as easy as in 2.7, and sometimes one needs either
several functions or functions defined on products (several models, resp.) to arrive at areformulation of some inequality between cardinal invariants. We give some examples.
$A$$\mathrm{Y}_{X}$$B_{A}$

$A$

$C$

$C$

$\omega^{\omega}\cross\omega^{\omega}$$C_{x}$$X\subseteq C_{x}$$x$$\phi$$\omega^{\omega}arrow\omega^{\omega}$$X-\rangle$$C_{x}\subseteq C_{\psi(y)}$$\psi$$\omega^{\omega}arrow\omega^{\omega}$$C_{x}-\rangle$$\mathrm{Y}_{X}$$(\mathrm{A}$$\mathrm{Y}_{X}$$D_{\phi(x)}\subseteq D_{y}$$A\vdash\Rightarrow B_{A}$$D_{\phi(x)}$$\mathrm{R}$Theorem 2.8 (Truss’ Theorem, see also [Ba] or [PR]) There are functions /,:and $(f, B)|arrow yf)B$ :such that for all , , , if
$B\leq^{*}x$$yf$, thenand ).
$[\mathrm{B}\mathrm{a}\mathrm{J}]$$A_{f,x}$$($$2^{\omega}\cross\omega^{\omega}arrow \mathcal{M}$$2^{\omega}\cross \mathcal{M}$$arrow\omega^{\omega}$

$x$

$B$

$f$

$x)\vdasharrow$$f\not\in B$$B\subseteq A_{f,x}$

In terms of the “model language” this means that given $ZFC$-models
as well as reals $f\in M_{1}$ Cohen overand dominating over,contains all
Borel meager sets coded in. In the language of cardinal invariants this then entails add(Af), one inequality in the Miller-Truss Theorem 1.2 quoted
above. However, we used two models, and it is known one is not enough. For Pawlikowski [Pa] has shown that if is -generic(Hechler-generic) over $M$ , then the unionof the Borel meager sets coded in $M$ is not meager in $M[d]$ , yet $M[d]$ contains bothaCohen real and adominating real over $M(d$ is dominating, and modisCohen). We leave it to the reader to infer from 2.8 that for the dual inequality, namely$\leq\max${ , non(M)}, one model is indeed enough.
Let
abbreviate there are infinitely many . Similarly,
stands for for all but
finitely many .
$M_{0}\subseteq M_{1}$$M_{0}$$x$$M_{1}$$A_{f,x}$$M_{0}$$\geq\min\{\mathrm{b}, \mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})\}$$d$$\mathrm{D}$$d$$2\in 2^{\omega}$$v$$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{M})$$\exists^{\infty}n$$\forall^{\infty}n$

$n$

$n$

Theorem 2.9 (-Miller characterization of non
also [Ba] or [PR] There are functions,::
Cx . andsuch that whenever$\exists^{\infty}n(z_{A,y}(n)=v(n))$ , then.

$(\mathcal{M})$$\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$$A\vdash\Rightarrow x_{A}$$)$$\mathcal{M}$$arrow\omega^{\omega}$and$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})[\mathrm{B}\mathrm{a}\mathrm{J}]$$(A, y)\vdash\Rightarrow z_{A,y}$:$\mathcal{M}$$\omega$$(y, v)\vdasharrow w_{y,v}$$\omega^{\omega}\cross\omega^{\omega}arrow\omega^{\omega}$, see$\cross\omega^{\omega}arrow$$\exists^{\infty}n(x_{A}(n)\leq \mathrm{y}\{\mathrm{n}))$and$w_{y,v}\not\in A$

Now this shows that given $ZFC$-models
as well as reals $y\in M_{1}$ unboundedover
and infinitely often equal over
, thenis Cohen over
. (Check details!)
In this case it is not known whether one model is sufficient or not:$M_{0}\subseteq M_{1}$$M_{0}$$v$$M_{1}$$w_{y,z}$$M_{0}$

15
Problem 2.10 (half-a-Cohen-real problem) Assume there is a real $ EtJ’which isinfinitely often equal over M, that is, given any yEw’ M, there are infinitely many
. Does there exist a Cohen real over ysuch that$n$$\mathrm{r}^{\ovalbox{\tt\small REJECT}}1$$\mathrm{t}$$\ovalbox{\tt\small REJECT} \mathrm{r}(\mathrm{r}\mathrm{r})\ovalbox{\tt\small REJECT}$$y(\mathrm{n})$It is easy to see aCohen real is infinitely often equal in this sense, and by the above,from two infinitely often equal reals, one over the other we can reconstruct aCohen real.
Therefore an infinitely often equal real is sometimes labeled “half aCohen real”, and2.10 asks whether half aCohen real is the same as one Cohen real. In terms of cardinalinvariants this gives us
Theorem 2.11 ($\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$= \min${-Miller characterization of non(M)By}$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}$)$\in \mathcal{F}\forall^{\infty}n(x(n)\neq y(n))$$|\mathcal{F}|;\mathcal{F}\subseteq\omega^{\omega};\forall x\in\omega^{\omega}$, one
Again we leave it to the reader to verify that for the characterization of nonmodel is enough.
We finally discuss arecent result of the author, dealing with asituation which is even-constantly predicts aapredictor. Say
more complex. Call afunction :$i\inI$that $x(i)=\pi(x[i)$ .
such
,
is
there
if for almost all intervals I of length
real
constantly predicts . The-constantly predicts for some , say that
In caseis the smallest size of aset of predictors such that
constant prediction number
is constantly predicted by some $\pi\in\Pi$ . The concept of prediction, which
every
we shall encounter again in Section 3below, is originally due to Blass [Bll] who also
put it into amuch more general framework in [B12, Section 10]. The notion of constant, however, are due to Kamo (see [Kal] and [Ka2]),
prediction and the definition ofis due to Kada.
and the notation
may[Kal]. He also proved that,
Kamo observed thatbe larger than all cardinal invariants in Cichon’s diagram [Kal], and smaller than thedominating number 0[Ka2]. He asked whether it can even be smaller than the unbounding number . The following is the main step towards the solution of his problem.
$(\mathcal{M})$$\pi k$$2^{<\omega}arrow 2$

$\pi$

$k$

$x\in 2^{\omega}$

$\pi$

$k$

$k$

$x$

$x$

$\pi$

$\Pi$

$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$x\in 2^{\omega}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}\geq \mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$0_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathrm{c}\mathrm{o}\mathrm{v}(N)$$\mathrm{b}$Theorem 2.12 [B3] Fix $k\in\omega$ . Let $\ell=2^{k}-1$ . Assume there are ZFC-models
is dominating oversuch thatand reals /0, ,which is not -constantly predicted by any predictor from. Then there is.
$\ldots$$k$$x\in 2^{\omega}\cap M_{\ell}$$M_{i}$$f_{i}\in M_{\dot{l}+1}$$f_{\ell-1}\in\omega^{\omega}$$M_{0}\subset M_{1}\subset\ldots\subset M_{\ell}$$M_{0}$

are strictly increasing, $f_{i}(0)>0$ and $f_{\dot{1}}(n+1)>$
by the recursion $h_{i}(0)=f_{i}(0)$ and $h_{:}(n+1)=f_{\dot{l}}(h:(n))$ .

Proof. Assume without loss all$f_{i}$. Define
for all . Listfor all . Clearly
Without loss we may assume$\{s\in 2^{k}; s\neq 0\}$ (where 0denotes the sequence with constant value 0) as $\{s:;i<\ell\}$ .
$f_{i}(n)+k$$h_{i}\in\omega^{\omega}\cap M_{\dot{l}+1}$$\mathrm{r}\mathrm{a}\mathrm{n}(h_{i+1})\subseteq \mathrm{r}\mathrm{a}\mathrm{n}(h_{i})$Define$x\in 2^{\omega}$$i$$h_{:}\geq f_{i}$as follows:

0if{ $h_{0}(m)+j;m\in\omega$ and $j<k$ }$s:(j)$
if is of the form $h_{i}(m)+j$ , $i<\ell-1$ and $j<k$ ,$x(n)=\{$
and
if is of the form
$n\not\in$$n$$h_{i}(m)\not\in \mathrm{r}\mathrm{a}\mathrm{n}(h_{i+1})$$s_{\ell-1}(j)$$n$$\mathit{1}_{\mathit{1}\mathit{4}-1}(m)+j,j<k$

$i$

16
We also define, for each$x_{t,0}$$=$$x_{t,:}(n)$$=$$t\in 2^{<\omega}$

and each(this means$t^{\wedge}\mathrm{O}$$x_{t,0}$$i\leq\ell$, areal$x_{t,i}\in 2^{\omega}\cap M_{i}$is constantly 0past

$|t|$

:

) and

$\{\begin{array}{l}t(n)\mathrm{i}\mathrm{f}n\in|t|0\mathrm{i}\mathrm{f}n\not\in\{h_{0}(m)+j,.m\in\omega \mathrm{a}\mathrm{n}\mathrm{d}j<k\}\cup t|s_{\dot{\iota}’}(j)\mathrm{i}\mathrm{f}n\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}h_{\dot{\iota}’}(m)+ji’<i-1\mathrm{a}\mathrm{n}\mathrm{d}j<kh_{\dot{l}’}(m)\not\in \mathrm{r}\mathrm{a}\mathrm{n}(h_{\dot{l}+1},),\mathrm{a}\mathrm{n}\mathrm{d}n\not\in|t|s_{\dot{\iota}-1}(j)\mathrm{i}\mathrm{f}n\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}h_{i-1}(m)+j,j<k,\mathrm{a}\mathrm{n}\mathrm{d}n\not\in|t|\end{array}$for $i>0$ . So $x=x$ ( , . Moreover, thecan be thought of as approximations toinitial segment in the intermediate models Mi..
Fix apredictor. In , $i<\ell$ , defineby$\rangle$$x_{t,:}$$\ell$

$x$

with

$t$

$\pi\in M_{0}$$M_{\dot{l}}$go $(n)= \min\{m$ ;for all$g:\in\omega^{\omega}$:if there is $m’\geq n$ such that$\pi(x_{t,0}[m’ +j)\neq x_{t,0}(m’+j)$ for all $j<k$ , then $m>m’+k$} and$g_{i}(n)= \min\{m$ ;for all:if there is, $m’\geq n$ , such that$\pi(x_{t,:}[m’+j)\neq x_{t,\dot{*}}(m’+j)$ for all $j<k$ , then $m>m’+k$}$t\in 2^{n}$$t\in 2^{n}$$m’\in \mathrm{r}\mathrm{a}\mathrm{n}(h:-1)$

for $i>0$ . We digress briefly on why models are useful in this proof. In aminute wewill use thatand that . is dominating over
to infer that
. If we
had not chosen andin this fashion apriori, we would have had to choose
only
after getting , then construct theand , then choose , etc... This is rather
cumbersome, and this is what we meant at the beginning of this section by “not caring$g_{\dot{1}}$$\in M_{\dot{l}}$$f.\cdot$$f_{*}$$g_{i}\leq*f_{i}$$M_{\dot{l}}$$M_{\dot{1}}$$f_{0}$$g_{0}$$x_{t,1}$$f_{1}$$g_{1}$

apriori”.
Now, there issuch that for all $i<\ell$ and all $n\geq n_{0}$ we have $ft(n)>g_{i}(n+k)$ .
The following is clear from the way things were set up.
$n_{0}$

Claim 2.13 For all $i<\ell$ , all , $n’>n_{0}$ , all $t\in 2^{n+k}$ such that and are consecutive
members of ran(h:): if there is no $m’\in \mathrm{r}\mathrm{a}\mathrm{n}(h:-1)\cap[n+k, n’-k](m’\in[n+k, n’-k]$in case $i=0$ ) such that $\pi(x_{t,i}(m’+j)\neq x_{t,i}(m’+j)$ for all , then it’s not true that$\pi(x_{t,:}[n’+j)\neq x_{t,:}(n’+j)$for all .
$n$

$n$

$n’$

$j$

$j$

Proof. If

$n$

,

$n’$

are consecutive members of$g_{i}(n+k)<f_{\dot{l}}(n)$$\mathrm{r}\mathrm{a}\mathrm{n}(h_{i})$, the claim follows.

Put $s_{-1}=0$ (the sequence in$2^{k}$, we must have $n’=f_{i}(n)$ . SinceEl

with constant value 0).

Claim 2.14 For all , all , $n’>n_{0}$ , all as in Claim 2.13: if there is no $m’\in[n+$, $n’-k]$ such that $\pi(x_{t,i}[m’+j)\neq x_{t,i}(m’+j)$ for all , then for all $i’<i$ , it’s not truethatfor all .
$i$

$n$

$t$

$k$

$j$

$\pi(x_{t,:}\square n’s_{\dot{l}’}\wedge[j)\neq(x_{t,:}[n’s_{i’})\wedge(n’+j)$$j$

Proof. We make induction on :the case $i=0$ is clear from Claim 2.13.
$i$$iarrow i+1$ . $n$ and $n’$ are consecutive members of$\mathrm{r}\mathrm{a}\mathrm{n}(h_{i+1})$. So there is$n^{*}\geq n$suchthat and are consecutive members of ran(h:). Let $t^{*}:=x_{t,:+1}[n^{*}+k\in 2^{n^{*}+k}$ . Notethat
. So we may apply the induction hypothesis to get the conclusion
the claim for all $i’<i$ . The case $i’=i$ , however, follows from Claim 2.13 (for $i+1$ ).
$n^{*}$$n’$$x_{t\dot{*}}*,[n’=x_{t,:+1}\square n’$$\mathrm{o}\mathrm{f}-$17Applying Claim 2.14 to $i=\ell-1$ , we see that if , $n’>n_{0}$ are consecutive members ofand $t\in 2^{n+k}$ , then there is $m’\in[n+k, n’]$ such that $\pi(x_{t,\ell}[m’+j)\neq xt,\ell(m’+j)$for all . (Using that $x_{t},\ell[n’=x_{t,\ell-1}[n’$ , we see that if there is no $m’\in[n+k, n’-k]$with this property, then, by the claim, $\pi(xt,\ell-1[n^{\wedge}’ s\ell-1\lceil j)\neq(xt,\ell-1[n’s_{\ell-1})\wedge(n’+j)$ for) This completes the proof of the theorem.
all . However,$n$$\mathrm{r}\mathrm{a}\mathrm{n}(h_{\ell-1})$$j$$x_{t,\ell-1}[n’s_{\ell-1}\wedge\lceil k=x_{t,\ell}[n’+k.$$j$$\square$

Before proceeding we mention that this result is optimal.
Theorem 2.15 [B3] Fix
be -generic overLetina predictor :stage iteration of Laver forcing L.
. Given with $\ell<2^{k}-1$ , there is, and letwhich -constantly predicts .

$\ell\in\omega$$G_{\ell}$$\mathrm{L}_{\ell}$$\pi$$2^{<\omega}arrow 2$

$V$
$V$

. Denote bythe$\mathrm{L}_{\ell}$$\ell$$k$$x\in 2^{\omega}\cap V[G_{\ell}]$

$k$

$x$

in
It is thus clear that if we want to solve Kamo’s question by showing
many dominating reals corresponding to an ;-sequence of$ZFC$ , we need at leastmodels. In fact, the following holds.
$\mathrm{b}\leq 0_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathrm{a}$$\omega$and reals
Lemma 2.16 [B3] Assume there are $ZFC$ -models
areassume.
Alsosuch that $f_{i}\in M_{i+1}$ is dominating over, , ,.
and $f\in N_{1}$ is dominating over,
$ZFC$ -models containing.
which is not constantly predicted by any predictor from
Then there is$M_{0}\subset M_{1}\subset\ldots\subset M_{i}\subset\ldots$$f_{0}$$f_{i}$$N_{0}\subset N_{1}$$M_{i}$$\ldots\in\omega^{\omega}$$\ldots$$\langle M_{i};i\in\omega\rangle$$N_{0}$$\langle f_{i};i\in\omega\rangle$$M_{0}$$x\in 2^{\omega}\cap N_{1}$Thus we infer
Theorem 2.17 [B3]$\mathrm{b}\leq \mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$.

For indeed, if we had, andthen $M_{i}(i>0)$ , , Now,
Thus we reach acontradiction.

Proof.

, andof size, we could find first amodel
lemma.
previous
which satisfy the hypotheses of the$M_{0}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}<\mathrm{b}$$N_{1}$$f_{i}$$f$$\square$So we get the following “local” diagram for$- \max\{\mathrm{b}, \mathrm{c}\mathrm{o}\mathrm{v}(N)\}-$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{M})$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$.

$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}-$$\mathrm{c}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})|$$\mathrm{b}|$is in order. Shortly before we obtained
One more comment concerning andour result outlined above, Kamo (unpublished) proved that an stage iteration of Laverwhich is not constantly predicted by any predictor from theforcing adjoins
after adding u2 Laver reals with countableground model. This shows that
support over amodel for $CH$ . This was strong evidence, and also an incentive, for
our 2.16 and 2.17. For Zapletal [Z] has proved, assuming aproper class of measurable
Woodin cardinals, that the iterated Laver model is aminimal model for in the senseinthat whenever acardinal invariant with areasonably easy definition has value
indeed falls into Zapletal’s framework.
is provable. Now,
that model, then
However our result does not follow from Kamo’s and Zapletal’s work because the latte$\mathrm{b}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\omega$$x\in 2^{\omega}$$\mathfrak{d}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}=\aleph_{2}$$\mathrm{b}$$\aleph_{2}$$i$$\mathfrak{y}_{2}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}}$$\mathrm{b}$$\leq \mathrm{i}$$\mathrm{r}$18
uses alarge cardinal assumption while ours is in $ZFC$ alone. Moreover, it turns out our
proof of 2.17 is much simpler than Kamo’s argument referred to above.
As further reading which contains most of the material of this section with proofs werecommend [B12] and [Ba], two survey articles written for the Handbook on set theory
which is scheduled to appear on Sankt-Nimmerleins-Tag.

3An example from algebra
Much of the original motivation for studying cardinal invariants came from set-theoretic
topology. In fact, the two standard survey articles on cardinal invariants which havebeen published so far appeared in topology books. Even though somewhat outdated,
they are still good introductions to the subject, seeand [Va]. In the meantime,
cardinal invariants have found many more applications, in particular in topology andalgebra.
We shall give one example the choice of which is determined by our wish to illustrate
two quite distinct phenomena at the same time. The first of these is fairly common andcan be roughly described as follows.
$[\mathrm{v}\mathrm{D}]$If a certain cardinal invariant is small )$/large$
object of a certain kind $exists/does$ not exist.
$\mu_{1}$$(\mathrm{c})$, then a $topological/algebraic$

Of course, one ideally looks for statements (that is, characterizations of the (non)existence of certain objects in terms of certain cardinal invariants assuming certain values),
but this is often difficult, if not impossible, to achieve.
Let be an at most countable field. Assume is an uncountable-dimensional vector
space over , and :is asymmetric bilinear form (i.e. $(a, ) $=\Phi(b, a)$ for all
$b\in
E$
,
and 4is linear in both coordinates). $(E, \Phi)$ is called aGross space if for any
infinite-dimensional subspace $U\leq E$ , the orthogonal complement$(x, ) $=0$
for all $x\in U$ } has dimension less than $\dim(E)$ . In case one even has thatis atmost countable for all such , $(E, \Phi)$ is called astrong Gross space. (Of course, the two
notions agree in case
So aGross space is rather different from Hilbert
space.
Let us introduce the cardinal invariant which plays the main role in the study of Gross
spaces. Let $D$ be an infinite subset of . Afunction :
with
for
$D$
all $n\in D$ is called aslalom. Givenand aslalom with domain , we say that
localizes if $f(n)\in\varphi(n)$ holds for all but finitely many $n\in D$ . The linear evasion
number is the size of the least
such that no slalom localizes all members of. By
characterization of add(N), the size of the least $T$
such that no slalom with domain $D=\omega$ localizes all members of is equal to add(N).
Moreover, the standard proof of the Bartoszynski-Raisonnier-Stern Theorem 1.1 uses
this combinatorial characterization, and it turns out the same proof still workes withthe present concept of slalom (with variable $D$ ). Therefore
sits between the two
additivities [Bll], i.e. we get the following “local” diagram for .
$E$$\mathrm{K}$$\mathrm{K}$$\Phi$$E^{2}arrow \mathrm{K}$

$b$

$a$

$U^{[perp]}=\{x;$$y$$\dim(U^{[perp]})$$U$$\dim(E)=\mathrm{N}\mathrm{i}.)$$\omega$$\varphi$$f\in\omega^{\omega}$$|\varphi(n)|\leq n$$\varphi$$f$$\varphi$$\mathrm{c}_{\ell}$$\mathcal{F}$$Darrow[\omega]^{<\omega}$$\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}’ \mathrm{s}$$\mathcal{F}\subseteq\omega^{\omega}$$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\subseteq\omega^{\omega}$$\mathcal{F}$$e_{\ell}$$t\ell$19$\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{M})|$$l\ell$$\mathrm{a}\mathrm{d}\mathrm{d}(N)\aleph_{1}||$

We are ready to characterize existence of strong Gross spaces over countable fields.
Theorem 3.1 [B4] The following are equivalent.
(i) There is a strong Gross space over some countable field.

(ii) There is a strong Gross space over every countable field.
(iii)$e_{\ell}=\aleph_{1}$.

Sketch of proof. To give arough idea of the proof we need some “intermediate combinainfinite, calltorics” Given an at most countable field and
$n\in
D$
the
connection to the concept of
(Note
.
all
for
is -linear:
predictor if
if $f(n)=\pi_{n}(f[n)$“prediction” discussed in Section 2above.) Such a predicts
holds for all but finitely many $n\in D$ . If does not predict we say that evades .
such thatis the cardinality of the smallest
The -linear evasion number
holds for any countableevery predictor is evaded by some member of . Then
, this was the original definition of [Bll]. This is the. In fact, forfield
is also true for finite fields. Yet,reason for its name, “linear evasion number”.
[B1]. In fact, it is consistently evenmay be strictly larger thanfor finite fields,
larger than the dominating number 0.
is a -Luzin space if every preSay an countable-dimensional subspace$E$ . Now, for countable,dictor predicts at most countably many members ofisdirection
course,the reverse
is equivalent to the existence of a -Luzin space. Of
trivial and holds for finite fields as well. The forward direction, however, is the main
point of the proof of 3.1, see [B4]. As shown in [B1], it is consistently false for finite
fields. This should be seen as saying that there is no reasonable characterization of theexistence of strong Gross spaces over finite fields in terms of cardinal invariants.
Finally, for any at most countable field , the existence of a -Luzin space is equivalent to the existence of astrong Gross space over [B1]. Again, the reverse direction
is rather straightforward. In fact, asimilar argument works to derive the existence ofageneralized -Luzin space from aGross space. The forward direction, however, uses
in a . rucial way.
the combinatorics of$\pi_{n}$$\mathrm{K}^{n}arrow \mathrm{K}$$\pi=\langle\pi_{n};n\in D\rangle \mathrm{a}$$D\subseteq\omega$$\mathrm{K}$$\mathrm{K}$$f\in \mathrm{K}^{\omega}$

$\pi$

$f$

$f$

$\pi$

$\mathcal{F}$$\mathrm{K}$$\pi$$\subseteq \mathrm{K}^{\omega}$$e_{\mathrm{K}}$$\mathcal{F}$$e_{\mathrm{K}}=e_{\ell}$$\mathrm{K}=\mathbb{Q}$$\mathrm{K}[\mathrm{B}\mathrm{S}]$$\mathrm{t}\ell$$e_{\mathrm{K}}\geq \mathrm{t}\ell$$\mathrm{t}\ell$$e_{\mathrm{K}}$$E\leq \mathrm{K}^{\omega}$$\mathrm{K}$$\mathrm{K}$$e_{\mathrm{K}}=\aleph_{1}$$\pi$$\mathrm{K}$$\mathrm{K}$$\mathrm{K}$$\mathrm{K}$$\mathrm{K}$$\square$$\omega_{1}$$\mathrm{c}$Let us now turn to the second phenomenon which can be described roughly as follows

20
Cardinal invariants may be a useful tool to prove something which doesn’
depend on the values of cardinal invariants at all. E. g. they may be used to
split the proof of an “absolute” result into cases, according to the value ofcertain cardinal invariant.

$t$

$a$

As an illustration we address
Question 3.2 Do Gross spaces exist$q$

An old result of Gross says that they exist under the continuum hypothesis $CH$ (ofcourse, this also follows from 3.1 above for countable fields). On the negative side one
has
Theorem 3.3 (Shelah-Spinas [SSp]) It is consistent thatspaces over any finite field.

$c$$=\aleph_{2}$and there are no Gross

This is proved by acountable support iteration of proper forcing (see below, Section 4,
for more on forcing). With avery complicated new iteration technique involving “mixed
support”, Shelah got
Theorem 3.4 (Shelah [S2]) It is consistent that
Gross spaces over any (at most countable) field.

$c$$=\aleph_{3}$(or larger) and there are no

See below (Theorem 4.12) for amore explicit formulation of this result.
But what about?It should be remarked here that, apart from trivial exceptions, almost all combinatorial problems on the reals which are independent of $ZFC$ arealso independent of $ZFC+c$. However, this is not true in our situation.
$c$$=\aleph_{2}$$=\aleph_{2}$Theorem 3.5 [B4]$c$field.

$=\aleph_{2}$implies the existence of a Gross space over every countable

Sketch of proof. The proof splits into two cases according to the pattern sketched above.
Case 1.
. Then use Theorem 3.1: there are strong Gross spaces (in particular,
Gross spaces of dimension ) over every countable field.
Case 2. $t\ell=c$ . One argues in two steps that this implies that Martin’s axiom MA
holds for any Cohen algebra: first show that , the forcing for adding many Cohen
reals, completely embeds into [B4], the standard a-centered forcing for making large(this is like the proof of Theorem 2.5 mentioned above); then show that
[B4].
(Of course,
is obvious. What we need here, however, is. This isproved by showing that, given amodel $M$ of $ZFC$ of size less than , one can reconstruct
aP-generic filter over $M$ using aslalom localizing all members of $M$ , as well as aCohenreal and dominating reals (over A#). The latter is OK because,
in $ZFC.$ )Taken together, the two results imply that
. By
, MA holds for Cc.
By an old result of Baumgartner and Spinas [BSp], saying that MA for
implies
existence of Gross spaces of dimension over any at most countable field, we ’re done$\epsilon_{\ell}=\aleph_{1}$$\aleph_{1}$$\mathbb{C}_{c}$$c$$\mathrm{P}$$\mathrm{t}\ell$$e_{\ell}=\mathrm{m}(\mathrm{P})$$\mathrm{c}_{\ell}\geq \mathrm{m}(\mathrm{P})$$t\ell\leq \mathrm{m}(\mathrm{P})$$\mathrm{t}\ell$$e_{\ell}\leq \mathrm{b}$$e_{\ell}\leq \mathrm{m}(\mathbb{C}_{c})$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{M})$$e_{\ell}=c$$\mathbb{C}_{c}$$\mathrm{c}$$-\mathrm{t}\mathrm{h}\mathrm{e}$21
As mentioned already, unlike for the first phenomenon, there seem to be few otherexamples for the second phenomenon so far. More specifically, there seem to be few
non-trivial problems which are decided by
but independent of. See [Ko]for one example. Let us briefly discuss two cases which are good candidates though
independence fromhas not been shown yet in either case.
Let ,. We say that is almost included or almost contained in (and write
ifis finite. Given afilter base, callapseudointersection$A\subseteq*X$of if
for all. The pseudointersection number is the size of the least
filter basewhich has no pseudointersection. can be characterized in forcing
language (see Section 2).
$c$$=\aleph_{2}$$c$$\geq\aleph_{3}$$\mathrm{c}\geq\aleph_{3}$$A$$A\subseteq^{*}B)$$A$$B\in[\omega]^{\omega}$$B$$A\backslash B$$A\in[\omega]^{\omega}$$\mathcal{F}\subseteq[\omega]^{\omega}$$X\in \mathcal{F}$$\mathcal{F}$$\mathfrak{p}$$\mathcal{F}\subseteq[\omega]^{\omega}$$\mathfrak{p}$Theorem 3.6 (Bell’s Theorem, seeor [B12])$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\mathfrak{p}$$=\mathrm{m}$( -centered).
$\sigma$Let be acardinal. Call asequenceatower of height if it isdecreasing modulo almost inclusion and has no pseudointersection. The tower number
is the least such that there is atower of height .
is immediate from thedefinition. The Piotrowski-Szymaiiski Theorem
says that. In fact[La]. So we get the following “local diagram” for and .
$\mathcal{T}=\langle T_{a};\alpha<\kappa\rangle$$\kappa$$\mathrm{t}$$\kappa$$\kappa$$\kappa$$\leq \mathrm{t}$$\mathfrak{p}$$\mathrm{t}$$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\mathrm{t}\leq \mathrm{t}\ell$$\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{M})$$\mathrm{t}$$\mathfrak{p}$add(N)–$\mathrm{t}\ell$$\aleph_{1}|-$

Theorem 3.7 [B12] Assume$c$$=\aleph_{2}$$-$ add$(\mathcal{M})$$\mathfrak{p}\mathrm{t}||$. Then$=\mathrm{t}$$\mathfrak{p}$.

Sketch of proof. Split into two cases according to the value of D.
Case 1.
. Sincein $ZFC$ , we ’re done.
. There is awell-known combinatorial lemma to the effect that given
Case 2.
of size less than 0as well as acountable filter base such thatis infinite$T$for all, there is apseudointersectionandof which still has infinite
intersection with all members of . Now assume. Let
be afilter baseof size . Using the lemma one can recursively construct adecreasing sequenceoflengthbelow 7(that is, for all
there isalmost contained in $X$ ). By
$T$
[Il
,
is not atower. Ergo has apseudointersection, and we ’re done.
$v$$=\aleph_{1}$$v$$=\mathrm{c}$$\aleph_{1}\leq \mathfrak{p}$$\leq \mathrm{t}\leq v$$\mathcal{G}\subseteq[\omega]^{\omega}$$X\cap \mathrm{Y}$$\mathcal{F}$$X\in \mathcal{F}$$A$$\mathrm{Y}\in \mathcal{G}$$\mathrm{t}=\aleph_{2}$$\mathcal{G}$$\mathcal{F}$$\subseteq[\omega]^{\omega}$$\aleph_{1}$$\mathcal{T}$$X\in \mathcal{F}$$\omega_{1}$$\mathrm{t}=\aleph_{2}$$\mathrm{Y}\in Ci$$\mathcal{T}$Note the above proof works more generally under the assumption. The
consistency ofis still open (see Problem 5.1).
We turn to the second example. Recall that an ultrafilter on is called aP-point
if any countable
has apseudointersection belonging to .
is a -point if given
any partitionof into finite sets, there is aselector belonging to , i.e.
$A\cap X_{n}$ has at most one element for all. Shelah proved it is consistent there are no$P$ -points
and Miller showed it is consistent there are no -points(seefor bothproofs). In both models
holds. Howeve$\mathfrak{p}$$=\aleph_{1}$$<\mathrm{t}$$\mathfrak{p}$$\mathcal{U}$$\mathcal{F}\subseteq \mathcal{U}$$\mathcal{U}$$\langle X_{n};n\in\omega\rangle$$\omega$$\mathcal{U}$

$Q$

$A$

$\omega$$\mathcal{U}$

$n$

$Q$

$\mathrm{c}$$=\aleph_{2}$$[\mathrm{B}\mathrm{a}\mathrm{J}]$22
Theorem 3.8 [BaJ] Assume c
Q-point.

$\ovalbox{\tt\small REJECT}$$\mathrm{R}_{2}$.

Then either there is a $P$ -point or there is$a$

Sketch of proof. Again we split into two cases according to the value of D.
, and we can use Canjar’s Theoremsay. Then
Case 1. )can being that under the latter assumption every filter base of size less thanextended to aQ-point.
saying that under this assumpCase 2. $=c$ . Then we use Ketonen’s Theorem
tion every filter base of size less than 0can be extended to a -point.
$=\aleph_{1}$$v$$=\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$[\mathrm{B}\mathrm{a}\mathrm{J}]$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$v$$[\mathrm{B}\mathrm{a}\mathrm{J}]$$P$$\square$Again, the simultaneous consistency of the non-existence of -points and Q-points
is still open (see Problem 5.8).
$P$

4Cardinal invariants and iterated forcing
Forcing, invented by Cohen to prove the independence of $CH$ , was soon transformed
into ageneral and powerful technique to obtain aplethora of independence results. In
particular, Solovay and Tennenbaum developed iterated forcing to show the consistencyof Martin’s axiom MA and, afortiori, Suslin’s hypothesis.
In our context, using this technique, quite anumber of statements of the form
have been proved to be consistent where and are cardinal invariants. For example
add(AT) is consistently less than add(M) and, in fact, any of the inequalities between
cardinal invariants shown in the diagrams in the preceding sections may be strict except
for the restriction imposed by the Miller-Truss Theorem 1.2 and the still open p-versusproblem 5.1 (seefor most of the consistency proofs). However, though many
consistency results have been proved concerning combinatorial properties of the reals,
this has been done almost exclusively with two very specific iteration techniques which
describe only asmall fraction of what an iteration is in general, namely$i<\mathrm{i}$$\mathrm{i}$$\dot{)}$$\mathrm{t}$$[\mathrm{B}\mathrm{a}\mathrm{J}]$(1) finite support iteration of $ccc$ forcing, the technique originally developed by Solovay
and Tennenbaum,(2) countable support iteration of proper forcing, created by Shelah [SI].

Both approaches are well-understood by now, the latter being much more complicatedas well as more powerful.
In fact, whenever two cardinals have been shown to be consistently different under, this could be proved as well using acountable support iteration of proper forcing
over amodel of $CH$ though the latter proof may be more complicatedof lengththan the original one. There is ageneral reason for this, see Zapletal’s work [Z] which
we mentioned at the end of Section 2. To enjoy countable support iteration in its fullbaroque splendor see [RS].
On the other hand, one of the nice features of finite support iterations is that consistency results usually dualize in afairly straightforward manner. For example, as\geq cof(\^A) is dual to add(N)but likewise
mentioned in Section 2,$\mathrm{c}$$=\aleph_{2}$$\omega_{2}$$\mathrm{c}\mathrm{o}\mathrm{f}(\mathrm{A}\mathrm{f})$$\leq \mathrm{a}\mathrm{d}\mathrm{d}(\mathrm{M})$23. The latter proof pr0is dual to$CH$ while theceeds by along finite support iteration of Hechler forcing over amodel offormer uses ashort finite support iteration of the same forcing over amodel of MA.
Both methods have rather obvious flaws. An iteration of the first kind (of length
at least the value of the continuum in the extension) must force Martin’s axiom MA$=c$ .
), and thus in particular,
for any Cohen algebra (that is,, yet, and thus
, still forces
shorter iteration, say of lengthofanumberare. So there
The second technique necessarily forcesnon(M)
natural problems for which none of the techniques can work.
Of course, there are alternative constructions. One may use alarge measure algebra(adding many random reals) or force with alarge (countable support) product. The
latter approach has been used quite successfully to obtain models where many cardinalinvariants assume different values, see e.g. [GS]. Yet, this works only for arather smallclass of invariants, namely those which can be increased by forcing with compact infinitely
often branching trees, afortiori only for cardinals which are consistently larger than D.
Sometimes one may also use an$CON(\mathrm{a}\mathrm{d}\mathrm{d}(N) <\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{M}))$$CON(\mathrm{c}\mathrm{o}\mathrm{f}(N) >\mathrm{c}\mathrm{o}\mathrm{f}(\mathcal{M}))$$(\mathrm{A}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{m}(\mathbb{C}_{c})=\mathrm{c}$$\kappa$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{m}(\mathbb{C}_{\kappa})\geq\kappa$$\geq\aleph_{1}$$\mathrm{c}$$\leq\kappa.)$$\geq \mathrm{i}\mathrm{S}$$\leq\aleph 2$(3) iterated forcing construction with mixed support
but, unlike the first two methods, this one is not very well understood yet. It consists
merely of anumber of consistency results, due mainly to Shelah, in part in joint work(see for example [FSS], [DS1], and [S2]), and scattered over the literature. Below, weshall give an example illustrating both how this method works and why it is needed.
However let us first mention arelatively recent technique, namely(4) non-well-founded iteration (of reasonably definable $ccc$ forcing)developed by Shelah to solve along-standing open problem on cardinal invariants byshowing

Theorem 4.1 (Shelah [S4]) It is consistent that$a$$<v$ .

is called an almost
Here, denotes the almost disjointness number. Afamily$A$$A$ is finite.
disjoint family ( . family) if the intersection of any two members ofis amad family (maximal almost disjoint family) if it is almost disjoint and cannot be. family, is the size of the smallest mad family, is
extended to astrictly larger
well-known to be larger or equal than , and it has been known for awhile there were noother lower bounds, being consistently smaller than any cardinal which is not provably
below . However, concerning upper bounds, not much had been known, and Shelah’s
result, and its variations for other cardinals, can be construed as saying essentially there
are none except . So we get the following “local diagram” for .
$A\subseteq[\omega]^{\omega}$

$n$

$a$

$\mathrm{a}.\mathrm{d}$

$a$

$a$

$\mathrm{a}.\mathrm{d}$$\mathrm{b}$$a$$\mathrm{b}$$a$$\mathrm{c}$$\mathrm{c}|$

$a$

$|$

$\mathrm{b}$24More specifically, Shelah proved
Theorem 4.2 (Shelah [S4])(i) Assume is measurable, and $<\mu<\lambda$ are regular. Then there is a
forcing that $\mu=\mathrm{b}=V$ and $=c$.
$\kappa$$\kappa$$ccc\mathrm{p}.0$.

$=\mathrm{A}$$a$(ii) The same conclusion is true, assuming only the consistency of $ZFC$ , for any triple
andof uncountable regular cardinals ic $<\mu<\lambda$ . For example,
is consistent (assuming the consistency of $ZFC$ ).
$\mathrm{b}=v$$a=\mathrm{c}$$=\aleph_{2}$$=\aleph_{3}$Sketch of proof First add Hechler reals in afinite support iteration of length . Callthis forcing P. It adds a -scale(that is, it forces $=v$ $=\mu$ ). Now consider
whereis an ultrafilter on the measurable cardinal (in case (i)). This is still a ccc forcing.
completely embeds into
so that we may think ofas atw0-step iteration. An
analysis of P’/&-names shows that the scale added by is preserved, while every.
family of the intermediate extension is not maximal in the extension via P’/&. (Thelatter uses thatin the intermediate model).
One now tries to iterate this procedure of taking ultrapowers for Amany steps. The
main problem occurs in limits of of countable cofinality, and this is where the new
method, non-well-founded iteration comes in.
In case (ii), taking ultrapowers is replaced by repeatedly inserting new coordinatesin the forcing and then using an-names argument.
In fact, the proof shows more than just the consistency of $>v$ . By astraightforward
modification one gets e.g.
. Furthermore, case (i) also works for otherinvariants like the ultrafilter number and the independence number . See [S4] fordetails.
ts
$\mu$

$\mu$

$\mathrm{b}$$\mathrm{P}^{\kappa}/\mathcal{U}$$\mu$$\mathcal{U}$$\mathrm{P}$$\kappa$$\mathrm{P}^{\kappa}/\mathcal{U}$$\mathrm{P}^{\kappa}/\mathcal{U}$$\mathrm{P}$$a$$\mathrm{a}.\mathrm{d}$$>\kappa$$\mathrm{P}$$\mathrm{i}\mathrm{s}\mathrm{o}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{m}-\mathrm{o}\mathrm{f}$$a$$CON(a >\mathrm{c}\mathrm{o}\mathrm{f}(N))$$i$$\mathrm{u}$Using the same method Shelah also got

Theorem 4.3 (Shelah [S4]) ais consistently singular (of uncountable cofinality).

As promised we now turn to mixed support iterations. Recall the combinatorial
principle 4says there is asequenceis cofinal,is alimit)
such that for all uncountable
there is with. If we require only thatbe countable we get the weaker principle. Of course, this can bereformulated as saying there is afamilysuch that for all
there
$l$
is $B\in T$ contained in . It is well-known that$+CH$ while
is equivalent to$CH+\neg l$ (Jensen) and $l$ $+\neg CH$ (Shelah) are both consistent. Furthermore,a$CH$trivial consequence of bothand 4while
is consistent. We
want to address questions of the following pattern.
$(” \mathrm{c}\mathrm{l}\mathrm{u}\mathrm{b}")$$\langle$$A\subseteq\omega_{1}$$A_{a};A_{a}\subseteq\alpha$$A_{a}\subseteq\alpha$$\alpha<\omega_{1}$$A_{a}\subseteq A$$\alpha$$\uparrow(" \mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{k}")$$\mathcal{F}$$A\in[\omega_{1}]^{\omega_{1}}$$\subseteq[\omega_{1}]^{\omega}$$A$$\phi$$\uparrow \mathrm{i}\mathrm{s}$$\uparrow+\neg CH+\neg l[\mathrm{D}\mathrm{S}1]$

Question 4.4 Let be a cardinal invariant. Is $l$ $+i>\aleph_{1}$ consistent$i$

To this end first note
Fact 4.5

(i)

(ii)

implies$e_{\ell}\geq\aleph_{2}$$\mathrm{m}(\mathbb{C}_{\omega_{1}})\geq\aleph_{2}$$\neg\uparrow$

implies, a fortiori$\neg\uparrow$$\neg l$, a fortiori$\neg l$. Similarly for$\mathrm{B}_{\omega_{1}}$.

. Similarly for add(N), , and even .
$\mathrm{t}$$\mathfrak{p}$25. Let $M$ be a model of a large
is awitness for stick. So
Proof, (i) Assume, there is aand containing . By
enough fragment of $ZFC$ , still of size
-generic filter over $M$ . Prom this filter, we can reconstruct an uncountable subset ofis awitness forwhich contains no countable set of $M$ , contradicting
Use the same argument for the measure algebra.
. Or see, see the proof of 3.5. Then use add(N), ,(ii) For , use
Theorem 2.6 for add(jV), and use Observation 2.4 and Bell’s Theorem 3.6 for . For
add(A), there is an alternative argument: huss [T] proved
implies the failure of .
$|\mathcal{F}|$$\mathcal{F}$$=\aleph_{1}$$\mathrm{m}(\mathbb{C}_{\omega_{1}})\geq\aleph_{2}$$\mathcal{F}$$\aleph_{1}$$\mathbb{C}_{\omega_{1}}$$\uparrow.$$M\cap[\omega_{1}]^{\omega}$$\omega_{1}$$\mathfrak{p}$$t\ell\leq \mathrm{m}(\mathbb{C}_{c})$$\mathrm{t}\leq\epsilon_{\ell}$$e_{\ell}$$\mathfrak{p}$$\min\{\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M}), \mathrm{c}\mathrm{o}\mathrm{v}(N)\}\geq\aleph_{2}$$\square$$\uparrow$

In the other direction we have
Theorem 4.6 (Fuchino, Shelah and Soukup [FSS]) Assume $CH$ . Let$=c$and $l$ .
there is a cardinal-preserving extension satisfying$\kappa$$=\kappa^{\omega}$. Then$=\kappa$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$This is proved by adding many Cohen reals with amixed support product. Why
are other constructions not possible? Let us note first that finite support products anditerations add generics for large Cohen algebras, and are thus excluded by the argument
above (Fact 4.5). Acountable support product of Cohen forcing collapses the continuum.
The case of countable support iterations is less clear, that is, we do not know whether
they can preserve 4, but in any case, as mentioned above, they yield only models with$\mathrm{c}$$\leq\aleph_{2}$.

Now, how does “mixed support” work? As mentioned above, we subsume under thisname several constructions which do not fit well into asingle framework (yet??), but
all work roughly as follows: conditions have two components, we use finite support on
one and countable support on the other. The order is a-closed on the countable pieces. is proper. Moreover, if we assume
so that apressing-down argument shows the$CH$ and work either with aproduct (as e.g. in [FSS]) or restrict the countable piece in
the iteration to something from the ground model (as e.g. in [S2] or [B2]) or consider. So cardinals are(as e.g. in [DS2]), we will have theonly iterations of length
preserved. Finally an elaboration of the pressing-down argument showing properness
usually gives combinatorial principles like 4As an illustration consider$\mathrm{p}.0$$\aleph_{2}-\mathrm{c}\mathrm{c}$$\omega_{2}$Theorem 4.7 [B4] Assume $CH$ .
$=c$extension satisfying$=\kappa$$\mathrm{c}\mathrm{o}\mathrm{v}(\cdot \mathrm{M})$Let
and $l$ .

$\kappa=\kappa^{\omega}$.

Then there is a cardinal-preserving

be non-empty. Givenbe acardinal and let
Sketch of proof Let
and $n<s(\alpha)$ } and: $Carrow\omega$ , let $a_{s}=$ {( , )apartial function.
consists of all triples $(s, f, A)$ such that : $Carrow\omega$ is an at most.
The
for allis Borel with, andcountable partial function, :(which is equipped$x\in A$ and such thathas positive measure as asubset of the space
,
$(s,
f,
A)\leq(t,
g,
B)$
iff
with the product measure as usual). We let,, $s(\alpha)=t(\alpha)$ for all but finitely manyfor all$x\inA$. (In particular,
for all(that is $f[a_{t}=g$ ) and)
The following is
still has positive measure insideeasy to show
$C\subseteq\kappa$$\kappa>\aleph_{1}$$\alpha$$s$$b_{s}=$$n$$;\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(s)$$(\mathrm{d}\mathrm{o}\mathrm{m}(s)\mathrm{x} \omega)\backslash a_{s}$$\mathrm{p}.0$$s$$\mathrm{P}_{C}$$f$$a_{s}arrow 2$$f\subseteq x$$A\subseteq 2^{\mathrm{d}\mathrm{o}\mathrm{m}(s)\cross\omega}$$2^{b_{s}}$$A$$\mathrm{d}\mathrm{o}\mathrm{m}(s)\supseteq \mathrm{d}\mathrm{o}\mathrm{m}(t)$$\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(t)$$s(\alpha)\geq \mathrm{t}(\mathrm{a})$$\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(t)$$f\supseteq g$$\{x (\mathrm{d}\mathrm{o}\mathrm{m}(\mathrm{s})\mathrm{x} \omega);x\in A\}\subseteq B$$x(\mathrm{d}\mathrm{o}\mathrm{m}(\mathrm{s}) \cross\omega)\in B$$2^{b_{t}}.$26
Fact 4.8(ii)(i)

forces$\mathrm{P}_{\kappa}$is$\mathrm{P}_{C}$$\aleph_{2}-cc$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{A}\mathrm{f})$.

$=c$ $=\kappa$.

The main point for proving cardinals are preserved isLemma 4.9 (i) Let, and letThen there is $q=(s^{q}, f^{q}, A^{q})\leq p$ such that {decides } is predense below .
$p=(s^{p}, f^{p}, A^{p})\in \mathrm{P}_{C}$$\dot{\gamma}$

be a$\mathrm{P}_{C}$-name for an ordinaland$r\leq q;\mathrm{d}\mathrm{o}\mathrm{m}(s^{f})=\mathrm{d}\mathrm{o}\mathrm{m}(s^{q})$$\dot{\gamma}$(ii)$\mathrm{P}_{C}$is proper.

(i) Assume not, and construct recursively(a sequence of countable partial functions

Proof,
and$\langle p_{\zeta};\zeta<\omega_{1}\rangle$$\langle s_{\zeta};\zeta<\omega_{1}\rangle$(i)$p_{\zeta}\leq p$(ii)$\zeta\leq\xi$

(iii)

,

implies$s_{\zeta}\subseteq s_{\xi}$and we have:for all
for all but finitely manyfor all$\mathrm{d}\mathrm{o}\mathrm{m}(s\zeta)=\mathrm{d}\mathrm{o}\mathrm{m}(s^{\mathrm{P}\zeta})$$s\zeta(\alpha)=s^{p_{\zeta}}(\alpha)$$s\zeta(\alpha)=s^{p_{\zeta}}(\alpha)$(v)(a sequence of conditions)) such that

$Carrow\omega$$s_{\zeta}\supseteq s^{p}$$s\zeta(\alpha)\leq s^{p_{\zeta}}(\alpha)$

(iv)

$r$

$q$

$\zeta\leq\xi$

implies

decides$p_{\zeta}$(vi) all$p_{\zeta}$$\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(s_{\zeta})$$\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(s_{\zeta})$$\alpha\in \mathrm{d}\mathrm{o}\mathrm{m}(s_{(})\backslash \bigcup_{\xi<\zeta}\mathrm{d}\mathrm{o}\mathrm{m}(s_{\xi})$$f^{p_{\zeta}}\square a_{s_{\zeta}}=f^{\mathrm{P}\zeta}[a_{s_{\zeta}}$$\dot{\gamma}$

are pairwise incompatible

This can be done easily: assume we are at step of the construction. Putand. Then let
where
and
. So
. By assumption, does not satisfy the conclusionof the lemma. Therefore there isdeciding and incompatible with all ,.
Now letand check clauses (i) to (vi) are satisfied.
Astandard pressing-down argument (i.e. an application of Fodor’s Lemma) now
gives us astationary
and apartial function :withof size at
mostsuch thatfor any ((by (ii) and (iii)) and such thatforfrom (by (iv)). The standard proof thatis ccc shows there arebothfrom such thatand are compatible, acontradiction.
(ii) is an easy consequence of (i).
Bl$\zeta$$t_{\zeta}= \bigcup_{\xi<\zeta}s_{\xi}$$q_{\zeta}=(t_{\zeta},g_{\zeta}, B_{\zeta})$$g_{\zeta}= \bigcup_{\xi<\zeta}(f^{p_{\zeta}}\square a_{s_{\zeta}})$$x[(\mathrm{d}\mathrm{o}\mathrm{m}(s^{p})\cross\omega)\in A^{p}\}$$B_{\zeta}=\{x\in 2^{\mathrm{d}\mathrm{o}\mathrm{m}(t_{\zeta})\cross\omega};g_{\zeta}\subseteq x$$q_{\zeta}\leq p$$p_{\zeta}\leq q_{\zeta}$$q_{\zeta}$$\dot{\gamma}$$p_{\xi}$$\xi<\zeta$$s_{\zeta}=t_{\zeta}\cup(s^{p_{\zeta}}\square \mathrm{d}\mathrm{o}\mathrm{m}(s^{p_{\zeta}})\backslash \mathrm{d}\mathrm{o}\mathrm{m}(t_{\zeta}))$$h$$S\subseteq\omega_{1}$$\in S$$s^{p_{\zeta}}\subseteq h$$\aleph_{1}$$S$$\zeta\leq\xi$$S$$Carrow\omega$$\mathrm{d}\mathrm{o}\mathrm{m}(\mathrm{f}\mathrm{t})$$f^{p_{\zeta}}\subseteq f^{p_{\xi}}$$\zeta\neq\xi$$\mathrm{B}_{\omega_{1}}$$p_{\xi}$$p_{\zeta}$

The following lemma completes the proof of Theorem 4.7.
Main Lemma 4.10(ii) If$\theta$holds in$V$(i) If $CH$ holds in the ground model,$then|\vdash_{\mathrm{P}_{\kappa}}l$.

$V$, then$\mathrm{I}\vdash_{\mathrm{P}_{\kappa}}\uparrow$.

27
The proof of this is similar to the proof of Lemma 4.9 above but technically more
involved and longer.
$\square$Next, by retrieving as much as possible from the flawed [DS2], one can prove

Theorem 4.11 (Dzamonja and Shelah [DS2]) Assume $CH$ . Then there is a cardinaland $l$ .
preserving extension satisfying add(M)$=\mathrm{c}$$=\aleph_{2}$Finally, let us give arough idea of what Shelah did to prove the consistency of the
non-existence of Gross spaces (Theorem 3.4 above). Let $S_{0}^{2}=\{\alpha<\omega_{2};cf(\alpha)=\omega\}$ .
such that for every
cofinal,means there is a sequence.
there is withof size$\langle$$A\subseteq\omega_{2}$$\alpha\in S_{0}^{2}\rangle$$A_{a};A_{a}\subseteq\alpha$$l_{S_{0}^{2}}$$\aleph_{2}$$A_{a}\subseteq A$$\alpha$Theorem 4.12 (Shelah [S2]) It is consistent that(or larger)(a)$c=\aleph_{3}$(b)$\mathrm{m}(\sigma-\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{d})=\aleph_{2}$(c) Given
single slalom$\mathcal{F}$(d)$\subseteq\omega^{\omega}$of size$\aleph_{2}$, there is$\mathcal{G}\subseteq \mathcal{F}$, still of size$\aleph_{2}$, which is localized by$a$$l_{S_{0}^{2}}$This is proved using acomplicated mixed support iteration. In fact, this may arguably
be the most sophisticated proof produced so far in iterated forcing theory, the definitionof the iteration alone taking roughly seven pages.
, so by 3.1 there are no Gross spaces
To derive 3.4 from 4.12, note (b) implies
still strong enough to get rid of Grossof dimension . (c) is aweakening of(by an argument analogous to the one in the proof of 3.1). In
spaces of dimension
entails
must hold in Shelah’s model. Finally,
fact, by the argument in 3.5,.
there are no Gross spaces of dimension$e_{\ell}\geq\aleph_{2}$$\mathrm{t}\ell>\aleph_{2}$$\aleph_{1}$$\aleph_{2}$$e_{\ell}=\aleph_{2}$$l_{S_{\mathrm{O}}^{2}}$$c=\aleph_{3}$The preceding discussion shows there is atw0-way interplay between iterated forcing
and cardinal invariants.

On the one hand, ate need forcing to prove consistency results on the possible
values of and possible inequalities between, cardinal invariants. On the other
hand, cardinal invariants provide us with test cases which might help us toeither develop nern iteration techniques or prove $ZFC$ -results saying this is
impossible.

5Open problems
We close our considerations with aselection of open problems on cardinal invariants, thisselection being guided mainly by personal interest, but also by choosing problems which
seem to require further development of new iteration techniques so that their solution
may shed new light on the interplay between forcing and cardinal invariants. For more

28
comprehensive problem lists, not only on cardinal invariants, see either [M2] (recent
update available on www) or [S3].
(1) PROBLEMS FOR FSI. Though the method is well-understood, there are still anumber of important problems which may be solvable by finite support iterations of ccc
forcing, for it is sometimes very difficult to find the right iterands.

Problem 5.1 (van Douwen’s problem [M2]) Is$\mathfrak{p}$$<\mathrm{t}consistent^{Q}$

See the end of Section 3for adiscussion of this.
Let ,. Say $X$ splits (or: reaps) if $X\cap A$ andare both infinite.
is asplitting family if every
is split by amember of . The splitting
number 5 is the size of the least splitting family. It is well-known thatand
$5\leq v$ , non(M), non(/V) [B12], and there are anumber ofconsistency results saying that
5 may be smaller
larger than any other cardinal invariant unless this is forbiddenby the aforementioned restrictions. So we get the following “local diagram” for 5.
$A$

$A$

$X\in[\omega]^{\omega}$$\mathcal{F}\subseteq[\omega]^{\omega}$$A\backslash X$$A\in[\omega]^{\omega}$$\mathcal{F}$$5\geq \mathrm{t}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$non(M)

$v$

$|$

5

$|$

–

$\min${ , non (N)} –non (N)
$V$

$|$

$\mathrm{t}$Furthermore, 5 must have uncountable cofinality.

Problem 5.2 (Vaughan’s problem [Va]) Can$\mathrm{s}$be $singular^{q}$(2) PROBLEMS FOR MIXED support. Theorem 4.11 suggests the following problem.

Problem 5.3 Is it consistent that add(A4)$>\aleph_{2}$yet 4holds.

For afailed attempt see [B2].
The following also seems to be agood candidate for “mixed support”.
Problem 5.4 (Judah [M2]) Let
without adding Cohen reals over$V$be any model of set theory. Can one force$\mathrm{b}$$>\aleph_{2}$$V^{q}$

Note that this is easy for some models: add Hechler reals
with afinite
support iteration over $W$ . Then add
random reals
overwith themeasure algebra. Let $V=W[r_{a};\alpha<\omega_{1}]$ and note there are no reals Cohen over in. So the problem asks this for arbitrary models, e.g. for .
$d_{\beta}(\kappa>\omega_{2})$$\kappa$$r_{a}$$\omega_{1}$$W[d_{\beta};\beta<\kappa]$$V$$V[d\beta;\beta<\kappa]$$L$(3) PROBLEMS AROUND . It is conceivable that some of the problems below can besolved by the method of [S4], namely, non-well-founded iteration. First note that since
Shelah only proved the consistency of $<a$ with(see 4.2), the following is
$a$

$v$

$v$

$\geq\aleph_{2}$$\mathrm{s}\mathrm{t}\mathrm{i}1$29
Problem 5.5 (Roitman’s problem [M2]) Assume$v$$=\aleph_{1}$. Does$a$$=\aleph_{1}^{q}$such that $X\cap A$ is at
Recall that aLuzin set is an uncountable set of reals
most countable for every meager set . The existence of aLuzin set obviously implies(the converse is false in general, see [JS]). The consistency ofnon (4)holds inis well-known (this is true, e.g., in the random real model), andin that model, there cannot be a
Shelah’s model [S4]. However, since non(M)Luzin set. So the following is yet unknown.
$X\subseteq\omega^{\omega}$

$A$

$a$

$=\aleph_{1}$$<\mathrm{n}\mathrm{o}\mathrm{n}(\mathrm{M})$$a>\mathrm{n}\mathrm{o}\mathrm{n}(\mathrm{M})$$\geq\aleph_{2}$Problem 5.6 (Fleissner’s problem [M2]) Assume there is a Luzin set. Does$a$$=\aleph_{1}$?

In view of Theorem 4.3 we may ask

Problem 5.7 Can$a$be singular of countable cofinality?

There are anumber of problems concerning relatives of
discussion at the end of [BSZ].

$a$which are still open, see the(4) MISCELLANEA.

Problem 5.8 (van Mill’s problem [M2]) Is it consistent there are neither $P$ -points nor$Q-points^{q}$.

See the end of Section 3for adiscussion of this. Since the continuum must be at leastR3 and cov(\^A) must be small, neither finite nor countable support iteration will work.
by asingle real, that is foris an unreaped family if it is not reaped. The reaping numberorsuch that either
there isall
is the size of the least unreaped family. Note that is dual to 5. So the ZFC-results
denote the size
. Also let,,
about 5 mentioned above dualize to,that is for allsetwhich is not reaped by acountableof the least.
orsuch that for all $X\in B$ , either
there iscountable, and we get the following “local diagram” for and .
Clearly$\mathcal{F}$$(=\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t})$$\subseteq[\omega]^{\omega}$$A\subseteq^{*}X$$A\in \mathcal{F}$$X\in[\omega]^{\omega}$$A\subseteq^{*}\omega\backslash X$$\mathrm{r}$$\mathrm{r}$$\mathrm{r}$$\geq \mathrm{b}$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{c}\mathrm{o}\mathrm{v}(N)$$\mathrm{r}_{\sigma}$$\mathit{1}\mathit{3}\subseteq[\omega]^{\omega}$$\mathcal{F}\subseteq[\omega]^{\omega}$$B\subseteq[\omega]^{\omega}$$A\subseteq^{*}X$$A\in \mathcal{F}$$\mathrm{r}$$\mathrm{r}$$\leq \mathrm{r}_{\sigma}$$A\subseteq^{*}\omega\backslash X$$\mathrm{r}_{\sigma}$$\mathrm{r}_{\sigma}$$|$$\mathrm{c}\mathrm{o}\mathrm{v}(N)$$- \max\{\mathrm{b},$$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{N})$$-$$\mathrm{r}$

$|$

$|$

$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$\mathrm{b}$As explained in [BJL], neither afinite nor acountable support iteration can be used
to solve the following
Problem 5.9 (Vojtas’ problem) Is it consistent that$\mathrm{r}$.

$<\mathrm{r}_{\sigma}^{q}$. In view of this, apositive answer
Both and may be singular. However,to the following problem would also yield apositive answer to 5.9 above$\mathrm{r}$$cf(\mathrm{r}_{\sigma})>\omega$$\mathrm{r}_{\sigma}$

30
Problem 5.10 (Miller [M2]) Is it consistent that$cf(\mathrm{r})$$=\omega$?

Say thatis an -splitting family if given any countable
thereiswhich splits all members of A.
is the size of the least -splitting family.
Clearly.
$\mathcal{F}\subseteq[\omega]^{\omega}$$\omega$$A\subseteq[\omega]^{\omega}$$X\in \mathcal{F}$$\epsilon_{\omega}$$\omega$$5\leq\epsilon_{\omega}$

Problem 5.11$(\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}\mathrm{r}\overline{\mathrm{a}}\mathrm{n}\mathrm{s}[\mathrm{S}\mathrm{t}])$Is it consistent that

$5<g_{\omega}\mathit{9}$Let us close with two problems on cardinal invariants in Cichori’s diagram.
Problem 5.12 (Miller [M1]) Is non(M)sistent ?

$=\aleph_{3}$and$\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$$=\aleph_{2}$(simultaneously) con-

In fact, in all known models which satisfy non(M), one has.
Such models are gotten either by acountable support iteration of proper forcing, or byadding many random reals, or by adding asingle Laver real to amodel of MA.
$>\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M})$Problem 5.13$(\mathrm{B}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{s}\mathrm{z}\mathrm{y}\acute{\mathrm{n}}\mathrm{s}\mathrm{k}\mathrm{i}-\mathrm{J}\mathrm{u}\mathrm{d}\mathrm{a}\mathrm{h}[\mathrm{M}2])$Is$cf(\mathrm{c}\mathrm{o}\mathrm{v}(\mathcal{M}))<\mathrm{a}\mathrm{d}\mathrm{d}(\mathcal{M})$$\mathrm{c}\mathrm{o}\mathrm{v}(\lambda 4)$$=\aleph_{1}$consistent?

We formulated almost all problems as consistency questions and, indeed, this is ourbias in most cases. However there may well be $ZFC$-results lurking behind, too.

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