数理解析研究所講究録
第 1895 巻 2014 年 16-25

16

A VARIANT PROOF OF Con$(\mathfrak{b}<\mathfrak{a})$J\"ORG BRENDLE AND ANDREW D. BROOKE-TAYLOR, which in
Abstract. We present a variation of the proof in [2] of Con
particular removes some of the obstacles to generalising the argument to cardinals$(\mathfrak{b}<\mathfrak{a})$$\kappa>\omega.$\S 1. Introduction. The generalisations of cardinal characteristicsof the continuum to cardinals greater than has generated sigparticular result that has so far resisted
nificant interest recently.
attempts at generalisation is the statement thatis consistent.
Blass, Hyttinen and Zhang [1, Section 5] briefly survey the different, highlighting the difficulapproaches known for proving
ties each presents for a generalisation.
given
We present here a variation on the proof ofin [2], which we hope will be more amenable to generalisation. In
particular, the proof in [2] relies on a rank argument, which of course
cannot be naively generalised to uncountable . We show here thatit may be replaced by a suitable formulation in terms of games,
which does generalise to higher . Indeed, with this observation, thequestion of forcing
for some suitable large cardinal seems
to boil down to interesting questions about the existence of suitablefilters on$\omega$$\kappa$$A$$\mathfrak{b}<\mathfrak{a}$$Con(\mathfrak{b}<\mathfrak{a})$$Con(\mathfrak{b}<\mathfrak{a})$$\kappa$$\kappa$$\mathfrak{b}_{\kappa}>\mathfrak{a}_{\kappa}$$\kappa$$\kappa.$\S 2. Preliminaries. Let be an infinite cardinal. family
is called almost disjoint if $|A\cap B|<\kappa$ for any two distinct
is a maximal almost disjoint familyofmembersand$A$$\kappa$$\mathcal{A}\subseteq$$[\kappa]^{\kappa}$

$A$

$B$

$\mathcal{A}.$$\mathcal{A}$(mad family, for short) if is almost disjoint and maximal with thissuchthere isproperty. This means that for every
is the leastthat $|A\cap C|=\kappa$ . The almost disjointness number$\mathcal{A}$$A\in \mathcal{A}$$C\in[\kappa]^{\kappa}$$\mathfrak{a}_{\kappa}$

Received by the editors December 19, 2013.
Partially supported by Grant-in-Aid for Scientific Research (C) 24540126,
Japan Society for the Promotion of Science.
Written while holding a JSPS Postdoctoral Fellowship for Foreign Researchersat Kobe University and supported by JSPS Grant-in-Aid no. 2301765.

17
J\"ORG BRENDLE AND ANDREW D. BROOKE-TAYLOR

size of a mad family on of size at least(equivalently, of size$>cf(\kappa))$ . In casewrite forNow assume is a regular cardinal. For functions, saythat eventually dominates ( $f\leq^{*}g$ in symbols) if
holds for all beyond some
. The unbounding numberisthe least size of an unbounded family in the order. Thatis, for all
there iswith $f(\alpha)>g(\alpha)$ for cofinally many
. Again we write instead ofLet
be a filter on . Mathias forcing
with consists ofconditions $(s, F)$ such that, and $\max(s)<\min(F)$ .
$(t,
G)\leq(s,F)$
is ordered byifand. It iswell-known and easy to see thatis a -centered forcing which
introduces a pseudointersection of the filter . This means thatfor all, wheredenotes almost inclusion:iff
is finite.
In [2], the notion of pseudocontinuity is used. This notion andthe corresponding basic lemma can be nicely phrased in terms ofcontinuity with respect to an appropriate topology.
$cf(\kappa)$$\kappa$$\kappa=\omega$$\mathfrak{a}$$\mathfrak{a}_{\omega}.$$f,g\in\kappa^{\kappa}$$\kappa$$f(\alpha)\leq g(\alpha)$

$f$

$g$

$\alpha_{0}<\kappa$$\alpha$$\mathfrak{b}_{\kappa}$$\mathcal{F}$$(\kappa^{\kappa}, \leq^{*})$$f\in \mathcal{F}$$g\in\kappa^{\kappa}$$\alpha’ s$$\mathfrak{b}$$\mathfrak{b}_{\omega}.$$\mathcal{F}$$\mathcal{F}$$\mathbb{M}(\mathcal{F})$$\omega$$F\in \mathcal{F}$$s\in[\omega]^{<\omega},$$s\subseteq t\subseteq s\cup F$$\mathbb{M}(\mathcal{F})$$\mathbb{M}(\mathcal{F})$$\sigma$$Z$$Z\subseteq^{*}F$$F\in \mathcal{F}$$G\subseteq F$$\mathcal{F}$$A\subseteq^{*}B$$\subseteq^{*}$$A\backslash B$

DEFINITION 1. The initial segment topology on is the topology
which has the (von Neumann) ordinals as open
. We denote
endowed with this topology by$\omega$$\mathcal{S}et_{\mathcal{S}}$$\omega$$\omega_{i}.$DEFINITION 2.
function to or is pseudocontinuous if it iscontinuous as a function toorrespectively.
$A$$\omega^{\omega}$$\omega$$\omega_{i}^{\omega}$$\omega_{i}$Thus, a pseudocontinuous function $F$ : $Xarrow\omega$ is one such that forevery, the set of in $X$ with image at most is open.
$n\in\omega$

$x$

$n$

LEMMA 3. Compact sets inandany pseudocontinuous image in orbounded.
$\omega_{i}$$\omega_{i}^{\omega}$$\omega$$\omega^{\omega}$are bounded. In particular,of a compact set must be

PROOF. The Lemma is clear for . Similarly, compact
are in fact bounded in the strict (not just) sense. Otherwise,
there would be some $m$ in such that $f(m)$ is unbounded in for$f\in K$ , and then the open sets $\mathcal{O}_{m,n}=\{f\in\omega_{i}^{\omega} f(m)\leq n\}$ for$n<\omega$ would form an open cover of $K$ with no finite
subcover.
As usual we may identify
withby way of the map taking
sets to their characteristic functions,:. We givethe corresponding topology, making a homeomorphism fromto the Cantor space$K\subset\omega_{i}^{\omega}$$\omega_{i}$$\leq^{*}$$\omega$$\omega$$\dashv$$2^{\omega}$$\mathcal{P}(\omega)$$\chi$$\chi$$2^{\omega}$$X\mapsto\chi_{X}$$\mathcal{P}(\omega)$$\mathcal{P}(\omega)$

18
A VARIANT PROOF OF Con $(b<a)$a
DEFINITION 4. For any cardinal , we call a filter
filter if it is generated by the union of fewer than many compact. We write
subsets offor$\mathcal{G}\subseteq \mathcal{P}(\omega)$$\lambda$$K_{\lambda^{-}}$$\lambda$$K_{N_{1}}.$$K_{\sigma}$$\mathcal{P}(\omega)$$K_{n-1}$ are (finitely many) compact subsets
LEMMA 5. Ifof, then the pointwise intersection$K_{0},$$\ldots,$$\mathcal{P}(\omega)$$\bigwedge_{i<n}K_{i}=\{\bigcap_{i<n}G_{i}|(G_{0}, \ldots, G_{n-1})\in\prod_{i<n}K_{i}\}$and the pointwise union$i<nK_{i}= \{\bigcup_{i<n}G_{i}|(G_{0}, \ldots, G_{n-1})\in\prod_{i<n}K_{i}\}$are compact. Furthermore, for any compact set$K\subseteq \mathcal{P}(\omega)$, theupward closure$\overline{K}=\{A\in \mathcal{P}(\kappa)|\exists B\in K(A\supseteq B)\}$is also compact.

is compact by the Tychonoff theoPROOF. The productgiven by $(G_{0}, \ldots, G_{n-1})\mapsto$
rem, and the functions
are clearly continuous, so
andand $_{i<n}K_{i}$ are compact. Finally, for compact.
is just$\prod_{i<n}K_{i}$$\mathcal{P}(\omega)^{n}arrow \mathcal{P}(\omega)$$(G_{0}, \ldots, G_{n-1})\mapsto\bigcup_{i<n}G_{i}$$\bigcap_{i<n}G_{i}$$\bigwedge_{i<n}K_{i}$$K\subseteq \mathcal{P}(\omega)\dashv$$\overline{K}$$K\vee \mathcal{P}(\omega)$\S 3. The proof. We work in a model of ZFC in which, and there is an unbounded,
is a regular cardinal satisfying$\lambda=c^{V}$$V$$2^{\lambda}=\lambda^{+}$:of strictly increasing functions-well-ordered sequence
from to . For example, any model of GCH will suffice as a groundmodel, and these properties will be preserved in intermediate stagesof our forcing iteration.
Let be an infinite maximal almost disjoint family in of subsetsof$<^{*}$$\langle f_{\alpha}$$\omega$$\alpha<\lambda\rangle$$\omega$$V$$\mathcal{A}$$\omega.$

THEOREM 6, There is a $ccc$ forcing:
is not mad and$\langle f_{\alpha}$$|\vdash_{\mathbb{P}(\mathcal{A})}\mathcal{A}$$\mathbb{P}(\mathcal{A})$$\alpha<\lambda\rangle$

such thatis still unbounded.

be the dual filter of , that is, the filter
PROOF. Let
generated by the sets whose complements are finite or in . Notethat this filter is proper: if for some $k<\omega$ there were $\{A_{i}|i<k\}\subset$, any other element of would havesuch that
infinite intersection with one of the , violating almost disjointness.
Note that the generic subset of introduced by Mathias forcing with$\mathcal{A}$$\mathcal{F}=\mathcal{F}(\mathcal{A})$$\mathcal{A}$$\mathcal{A}$$\mathcal{A}$$| \bigcap_{i<k}\omega\backslash A_{i}|<\omega$$A_{i}$$\omega$19
J\"ORG BRENDLE AND ANDREW D. BROOKE-TAYLOR, or any filter extending
be almost contained in$\mathcal{F}$$\mathcal{F}$, will end the madness of$\mathcal{A}$, as it will

for every
First we add many Cohen reals. It is well-known that the unboundedness of:is preserved in this intermediate extension. In caseis not mad anymore in this extension we aredone. Also, if
filter in the intermediate
is contained in a
extension, we may simply force withfor it is well-known, andeasy to see [2, 3.2], that Mathias forcing with a-filter does not
destroy the unboundedness of. So assume that is not:contained in any
We shall recursively construct a filter
such that furthermore$\omega\backslash A$$A\in \mathcal{A}.$$\lambda$$\alpha<\lambda\rangle$$\langle f_{\alpha}$$\mathcal{A}$$\mathcal{F}$$K_{\lambda}$$\mathcal{G}$$\mathbb{M}(\mathcal{G})$$K_{\lambda}$$\mathcal{F}$$\alpha<\lambda\rangle$$\langle f_{\alpha}$$K_{\lambda^{-}}fi1ter.$$\mathcal{G}\supseteq \mathcal{F}$$(*)$$|\vdash_{\mathbb{M}(\mathcal{G})}\langle f_{\alpha}$:$\alpha<\lambda\rangle$

is unbounded.

Along the construction we shall take are of every potential
name for a function in , either “killing it” or “sealing it off’
To be precise: let us refer to partial functions :as preterms, and let
be an enumeration of the setof all preterms. Note in particular that if
is a filter and isan-name for a function in , thengiven by$\mathbb{M}(\mathcal{G})-$$c$$\omega^{\omega}$$\tau$$\mathcal{T}=\{\tau_{\beta} :$[\omega]^{<\omega}\cross\omega--\star\omega$\beta<\lambda\}$$\mathcal{G}\supseteq \mathcal{F}$$\dot{g}$$\omega^{\omega}$$\mathbb{M}(\mathcal{G})$$\tau=\tau_{\dot{g}}$$\tau(s, m)=n$ iff $\exists G\in \mathcal{G}((s, G)|\vdash\dot{g}(m)=n)$

is a preterm, the preterm associated with . We shall constrain attention to names such that 1, since every function
from to in the generic extension has such a name; we call such
names total names.
We construct filters, starting fromfor, suchthatfor eachis generated by
and afilterfor each limit ordinal
and either(KILL): for all filters
is not associated with any-name, or
total(SEAL): there is ansuch that for all filtersandall-names , ifthenClearly any filter
will then satisfy $(*)$ .
; we wish to find an
So suppose
has been defined for someappropriate. Note thatfilteris generated by and afilter ; without loss of generality we may assume that contains
all cofinite subsets of . Let
be a family of fewer than many
compact subsets ofgenerating . By Lemma 5, we may assume$\dot{g}$$\dot{g}$$\omega$$|\vdash_{\mathbb{M}(\mathcal{G})}\dot{g}\in\omega^{\omega}$$\omega$$\mathcal{G}_{\beta}$$\beta<\lambda,$$\bullet$$\bullet$$0\leq\beta\leq\lambda$$\mathcal{G}_{0}=\mathcal{F}$$\mathcal{G}_{\beta+1}$$\mathcal{G}_{\beta}$$K_{\sigma}$$\mathcal{H}_{\beta},$$\delta\leq\lambda,$$\mathcal{G}_{\delta}=\bigcup_{\beta<\delta}\mathcal{G}_{\beta}$$\mathcal{H}\supseteq \mathcal{G}_{\beta+1},$$\tau_{\beta}$$\mathbb{M}(\mathcal{H})$$\alpha<\lambda$$\mathbb{M}(\mathcal{H})$$\mathcal{H}\supseteq \mathcal{G}_{\beta+1}$$|\vdash_{\mathbb{M}(\mathcal{H})}\dot{g}\not\geq^{*}\check{f}_{\alpha}.$$\dot{g}$$\tau_{\dot{g}}=\tau_{\beta}$$\mathcal{G}\supseteq \mathcal{G}_{\lambda}$$\beta<\lambda$$\mathcal{G}_{\beta}$$K_{\sigma}$$\mathcal{H}_{\beta}$$\mathcal{G}_{\beta}$$\mathcal{F}$$K_{\lambda}$$\mathcal{F}$$\mathcal{G}_{\beta}’$$\omega$$\lambda$$\mathcal{K}_{\beta}$$2^{\omega}$$\mathcal{G}_{\beta}’$20
A VARIANT PRO OF OF Con$(b<\mathfrak{a})$is closed under finite pointwise intersections, and that for$K$ is upwards-closed under, so that
Everything that has come so far can actually be considered tohave occurred in a partial extension model, between the originalmodel and the full extension with -many Cohens. More explicitly,
belong to this intermediate model.
all (codes of) elements of: that is,denote the strict end-extension relation onLet
and $\max(s)<\min(s’\backslash s)$ ; define
if and only if
accordingly.
andIn [2], a rank function was used. For our generalisation, we takea different approach using games, but use these games to much thesame end as the rank function is used in [2]. It should be notedthat our games are very closely related to the games independently
introduced by Guzm\’an, Hru\v{s}\’ak, and Martinez [3], also in the context.
of a proof of Con
Let

that
all

$\mathcal{K}_{\beta}$$\subseteq$$K\in \mathcal{K}_{\beta},$$\mathcal{G}_{\beta}’=\cup \mathcal{K}_{\beta}.$$\lambda$$\mathcal{K}_{\beta}$$[\omega]^{<\omega}$$\subset_{ee}$$s\subset s’$$s\subset_{ee}s’$$\subseteq_{ee},$$\supseteq_{ee}$$\supset_{ee}$$(\mathfrak{b}<\mathfrak{a})$$\tau=\tau_{\beta}.$, the nominalisation exercise is the
DEFINITION 7. Given
following game. There are two players, Sensei and Student. On turnAt odd stages $2d+1,$, Sensei chooses an $m\in\omega$ andand a compact set
Student plays a filter set
such
At even stages $2d+2$ , Sensei plays an elementofthat$\tau\in \mathcal{T}$$\tau$$t_{0}\in[\omega]^{<\omega}$$0$$K(d)\in \mathcal{K}_{\beta}.$$F(d)\in \mathcal{F}$$t_{d+1}$$\bullet$$t_{d+1}$

end-extends$[\omega]^{<\omega}$$t_{d}$$\bullet t_{d+1}\backslash t_{d}\subseteq F(d)$meets every member of $K(d)$ .
such that $(s, m)\in$ dom ,end extending
If there isSensei declares Student to have passed and the game ends. If thegame continues for infinitely many stages, then (clearly) Student has$\bullet$$t_{d+1}\backslash t_{d}$$s\subseteq t_{d+1}$$t_{0}$$(\tau)$failed.

is a filter, and by compactness of $K(d)$ , aNote that, since
satisfying the requirements always exists. Also notice that if Student
wins, he wins after finitely many steps. Hence the game is open and,by the classical Gale-Stewart Theorem, determined.
As in [2], we now distinguish two cases (in [2] they are Subcases),corresponding to options (KILL) and (SEAL) above.
such that Sensei has
and
3.1. Case . There area winning strategy in the nominalisation exercise with Oth move$(m, t_{0})$ : play will continue for infinitely many steps. In this case we$t_{d+1}$$\mathcal{G}_{\beta}$$a$$m\in\omega$$\tau$$t_{0}\in[\omega]^{<\omega}$21
J\"ORG BRENDLE AND ANDREW D. BROOKE-TAYLOR

shall choose
in such a way that (KILL) holds: will not correspond to a name for a functionin the generic extension. The
reader may wish to remember which case is which by the mnemonic“the that can be named is not the eternal
We shall actually work in the extension of such the intermediate
model by one further Cohen function
Consider the tree of all possible sequences of plays
for Sensei according to his strategy, corresponding to all possible
plays of Student. Note that is infinitely branching since extends
the Frechet filter. Use the Cohen function to choose a branchthrough , and denote the union of theof this branch by $G.$
There is no $(s, m)$ with $m$ from Sensei’s first move andsuch that $(s, m)\in$ dom. Indeed otherwise, thenominalisation
exercise would have ended once Sensei played sufficiently long tocover . Thus, for any filter
for any total
name . We may therefore simply takein order to satisfy(KILL). To check thatgenerates a filter, consider any
$G’$
, say
and
is in the compact set. For every
there is a successor nodein the tree that is Sensei’s response,
according to his strategy, to Student playing
and $K$ , and so inparticular thismeets the intersection of and every member
of $K$ . Thus, by Cohen genericity we have that $|G\cap F\cap G’|=\omega,$completing Case . (Note that $G’$ may not belong to the intermediate
model; this, however, is irrelevant for it is sufficient that $K$ does. By
genericity the Cohen real will produce infinitely many such thatis contained inand meets every $G”\in K$ , and this isclearly absolute and thus also holds for $G’.$ )3.2. Case . The negation of Case a: for every Oth move $(m, t_{0})$by Sensei, Student has a winning strategy in thenominalisation
exercise. In this case we wish to choose
in such a way that
(SEAL) holds.
Since Sensei chooses his moves from a countable set, there areclearly only countable many filter sets
, which appear
as $F(d)$ in some $2d+1st$ move of Student playing according to his
strategy.
Suppose that for all but less than many members of , there issuch that $A\cap G$ is finite. Then, adding less than many setsof the form
, toresults in a
filter containing
This contradicts our initial assumption. Hence, for many
$A\cap G$ is infinite for all. Let, be countably many$\mathcal{H}_{\beta}$$\tau$$\omegaarrow\omega$”$\tau$$\tau.$$c:\omegaarrow\omega.$$T$$(t_{0}, t_{1}, t_{2}, \ldots)$$T$$\mathcal{F}$

$c$

$T$

$t_{i}$$t_{0}\subseteq_{ee}s\subseteq G$$(\tau_{\beta})$$\tau_{\beta}$$t_{d}$$\mathcal{H}\ni G,$$s$$\tau\neq\tau_{\dot{g}}$$\mathbb{M}(\mathcal{H})$$\mathcal{H}_{\beta}=\{G\}$$\dot{g}$$F\in \mathcal{F}$$\{G\}\cup \mathcal{G}_{\beta}$$t_{d}\in T,$$K\in \mathcal{K}_{\beta}$$G’\in \mathcal{G}_{\beta}’$$T$$t_{d+1}$

$F$

$F$

$t_{d+1}$

$a$

$d$

$c$

$F$

$t_{d+1}\backslash t_{d}$$b$$\tau_{\beta}$$\mathcal{H}_{\beta}$$F_{\ell}\in \mathcal{F},$$\ell\in\omega$$A$$\lambda$$\mathcal{A}$$\lambda$$G\in \mathcal{G}_{\beta}’$$\omega\backslash A,$$A\in \mathcal{A}$$\mathcal{F}.$$K_{\lambda}$$\mathcal{G}_{\beta}’$$\lambda$$G\in \mathcal{G}_{\beta}’$$A_{j},$$j\in\omega$$A\in \mathcal{A},$22
A VARIANT PROOF OF Con$(b<\mathfrak{a})$is almost contained in :such ’s such that for each andthis is possible because is the dual filter of the mad family, we, and finite subsetof
For eachas follows.
define a function
$A$

$j$

$\ell,$$F_{\ell}$$A_{j}$$\mathcal{A}.$$\mathcal{F}$$G’\in \mathcal{G}_{\beta}’,$$k\in\omega,$$T$$j\in\omega$$[\omega]^{<\omega}$$f_{G’,k,j,T}:\omegaarrow\omega$for any partition$f_{G’,k,j,T}(m)= \min\{n|$$A_{j}= \bigcup_{i<k}B_{i}$there is $i<k$ s.t.

$\forall t\in T\exists s\supset_{ee}t(s\backslash t\subseteq B_{i}\cap G’\wedge\tau_{\beta}(s, m)\leq n)\}.$LEMMA 8. For every$f_{G’,k,j,T}$ iswell-defined.

$G’\in \mathcal{G}_{\beta}’,$$k,$ $j\in\omega$, and$T\in[[\omega]^{<\omega}]^{<\omega},$. Given a partition $\{B_{i}|i<k\}$ of , let
PROOF. Fix
suffices for $\{B_{i}|i<k\}$ mean the natural thing in the context of, namely, that there is $i<k$ such that forthe definition of. So
andevery $t\in T$ there iswithnow fix a partition $\{B_{i}|i<k\}$ of ; we shall show that there is athat suffices for it. Let $i<k$ be such that $|B_{i}\cap G’\cap G|=\omega$ for
has infinite intersection: such an must exist, since
every
with every member of the filter . Finally, fix $t\in T.$
Consider a play of the naming exercise in which Student follows
his strategy, Sensei’s Oth move is $(m, t_{0})$ with $t_{0}=t$ , and his later
moves always satisfy the additional requirement
is almost contained in all $F(d)$ played by Student according
Since
has infinite intersection with all
to his strategy and since
Sensei always has a valid such move.
So we have that eventually Sensei plays a such that$m\in\omega$$n$$A_{j}$”$f_{G’,k,j,T}$$s\supset_{ee}t$$s\backslash t\subseteq B_{i}\cap G’$$\tau_{\beta}(s, m)\leq n$$A_{j}$$n\in\omega$$i$$G\in \mathcal{G}_{\beta}’$$A_{j}$$\mathcal{G}_{\beta}’$$\tau_{\beta}$$t_{d+1}\backslash t_{d}\subseteq B_{i}\cap G’.$

$B_{i}$$B_{i}$$G\in \mathcal{G}_{\beta}’,$$t_{d}$$\exists n_{t}\in\omega\exists s\subseteq t_{d}(s\supset_{ee}t\wedge\tau_{\beta}(s, m)=n_{t})$.

. Taking
Of course, by the construction of the game,such anfor each $t\in T$ and setting $n= \max_{t\in T}(n_{t})$ , we have thatsuffices for $\{B_{i}|i<k\}.$Now, with still fixed but allowing the partition $\{B_{i}|i<k\}$ tovary, let us denote by $n(\{B_{i}|i<k\})$ the least that suffices for
int $ok$ pieces can be identified$i<k\}$ . The space of partitions ofand thus when endowed with the product topology is awithcompact topological space. Moreover, with this topology on thespace of partitions, the function sending $\{B_{i}|i<k\}$ to $n(\{B_{i}|$being sufficient for$i<k\})$ is clearly pseudocontinuous, sincefrom , which$i<k\}$ is witnessed by finitely many finite tuples. Thus by Lemma 3 the image ofof course define an open set in$s\backslash t_{0}\subseteq B_{i}\cap G’$$n_{t}$$n$$k$$\{B_{i}|$$n$$A_{j}$$k^{A_{j}}$$n$$\{B_{i}|$$n$$s\backslash t$$k^{A_{j}}$$B_{i}$

23
J\"ORG BRENDLE AND ANDREW D. BROOKE-TAYLOR

the function is bounded below . The least such upper bound will
be $f_{G’,k,j,T}(m)$ , and it follows that $f_{G’,k,j,T}$ is well-defined.
$n$$\omega$$\dashv$$\omega$

LEMMA 9. There exists an
and$T\in[[\omega]^{<\omega}]^{<\omega},$$\alpha<\lambda$

such that for all$G’\in \mathcal{G}_{\beta}’,$$k,j\in$$f_{\alpha}\not\leq^{*}f_{G’,k,j,T}.$PROOF. We first note that, given $j,$ , and compact
the function $f_{k,j,T}$ sending $G’$ to $f_{G’,k,j,T}$ is pseudocontinuous from
$K$ to, by much the same argument as in the proof of Lemma 8.
Indeed, fixing $m$ and $n,$ $\{G’|f_{G’,k,j,T}(m)\leq n\}$ is open in $K.$
$f_{k,j,T}K$ is
We thus have from Lemma 3 that for eachbounded in , say by . Since
has fewer than many elements,
there is ansuch thatis not eventually dominated by any, and hence not by any $f_{G’,k,j,T}.$of theWe now show that as given by Lemma 9 will make (SEAL) holdfor an appropriate choice of, and. Given
, let
$k,$

$T$

$K\in \mathcal{K}_{\beta},$$\omega^{\omega}$$K\in \mathcal{K}_{\beta},$$h_{K}$$\omega^{\omega}$$\alpha<\lambda$$\lambda$$\mathcal{K}_{\beta}$$f_{\alpha}$$h_{K}$$\dashv$$\alpha$$t\in[\omega]^{<\omega},$$\mathcal{H}_{\beta}$$G\in \mathcal{P}(\omega)$$m\in\omega$$g_{t,G}^{\beta}(m)= \min\{n|\exists s\supseteq_{ee}t(s\backslash t\subseteq G\wedge\tau_{\beta}(s, m)=n)\}$if the set on the right hand side is non-empty, and otherwise put. Thus,is a function inLetbe such$f_{G’,k,j,T}$
that
is not dominated by any, as given by Lemma 9, and$g_{t,G}^{\beta}(m)=\omega$$\alpha<\lambda$$(\omega+1)^{\omega}$$g_{t,G}^{\beta}$$f_{\alpha}$

define$\mathcal{H}_{\beta}=\{H\subseteq\omega|\exists t\in[\omega]^{<\omega}(g_{t,\omega\backslash H}^{\beta}\geq^{*}f_{\alpha})\}.$

Note that given$t\in[\omega]^{<\omega}$and$m_{0}\in\omega$, the set$\{H\subseteq\omega|\forall m\geq m_{0}(g_{t,\omega\backslash H}^{\beta}(m)\geq f_{\alpha}(m))\}$

is closed in, and hence compact. Therefore,
To see that this set is an appropriate choice ofabove, we check the following.
$\mathcal{P}(\omega)$

CLAIM 10. Any filter
PROOF. Let$\mathbb{M}(\mathcal{H})$$\mathbb{M}(\mathcal{H})$

-name
and

$\dot{g}$is a$\mathcal{H}_{\beta}$$\mathcal{H}\supseteq \mathcal{H}_{\beta}$$\mathcal{H}_{\beta}$set.
as called for$K_{\sigma}$satisfies (SEAL).

be a filter, and assume
for some
Suppose there were $(t, G)\in$
for a function insuch that$\mathcal{H}\supseteq \mathcal{H}_{\beta}$$\tau_{\beta}=\tau_{\dot{g}}$$\omega^{\omega}$$m_{0}\in\omega$$(t, G)|\vdash_{\mathbb{M}(\mathcal{H})}\forall m\geq\check{m}_{0}(\dot{g}(m)\geq\check{f}_{\alpha}(m))$

By the definition of
for all $m\geq m_{0}$ . So
is a filter.

$9_{t,G}^{\beta}$.

, we must then also have, contradicting the fact that$g_{t,G}^{\beta}(m)\geq f_{\alpha}(m)$$\omega\backslash G\in \mathcal{H}_{\beta}\subseteq \mathcal{H}$$\mathcal{H}\dashv$

24
A VARIANT PROOF OF Con

CLAIM 11.

$\mathcal{H}_{\beta}\cup \mathcal{G}_{\beta}$$(\mathfrak{b}<a)$generates a filter.

, and for some $k<\omega,$
PROOF. We takehas cardinality . Assume
for $i<k$ , and argue that
. For eachfor the sake of contradiction that
. Also fix such that
$i<k$ , fixsuch that. Without loss of generality, we may take $a<\omega$ such thatand $\max(t_{i})\geq a$ for every$i<k$ (if necessary by extending each with a sufficiently large). Fixelement of: this can only increase the values offor all $m\geq m_{0}$ and $i<k.$such that
Let $T=\{t_{i}|i<k\}$ and let $\{B_{i}|i<k\}$ be a partition ofsuch
for all $i<k$ . By the definition ofthat
there is some $m>m_{0}$ such that $f_{\alpha}(m)>f_{G’,k,j,T}(m)$ ; take such a$m$ , and denote $f_{G’,k,j,T}(m)$ by, there. By the definition ofis an such that for all $t\in T$ , there issuch that$G’$andis a subset of the intersection ofand . In particular,$\min(s\backslash t_{i})>\max(t_{i})\geq a,$, and,, from which we haveThus
contradicting the choice offrom , and hence theThis completes the construction of
proof of Theorem 6.
. Recall from the
We are now ready for the consistency ofisbeginning of this section that our ground model satisfies, and:is unbounded -well-ordered.
regular,$F\in \mathcal{F},$$H_{i}\in \mathcal{H}_{\beta}$$G’\in \mathcal{G}_{\beta}’$$F \cap G’\cap\bigcap_{i<k}H_{i}$$\omega$$F \cap G’\subseteq^{*}\bigcup_{i<k}\omega\backslash H_{i}$$t_{i}\in[\omega]^{<\omega}$$j$$g_{t_{i},\omega\backslash H_{i}}^{\beta}\geq^{*}f_{\alpha}$$A_{j}\subseteq^{*}F$$A_{j}\backslash a\subseteq F,$$F \cap G’\backslash a\subseteq\bigcup_{i<k}\omega\backslash H_{i}$$t_{i}$$\omega\backslash H_{i}$$g_{t_{i},\omega\backslash H_{i}}^{\beta}$$g_{t_{i},\omega\backslash H_{i}}^{\beta}(m)\geq f_{\alpha}(m)$$m_{0}\in\omega$$A_{j}$$B_{i}\cap G’\backslash a\subseteq\omega\backslash H_{i}$$f_{\alpha},$$f_{G’,k,j,T}$$n$$\tau_{\beta}(s, m)\leq n$$s\supset_{ee}t$$i$$s\backslash t$$B_{i}$$s\backslash t_{i}\subset B_{i}\cap G’$$\tau_{\beta}(s, m)\leq n.$$g_{t_{i},\omega\backslash H_{i}}^{\beta}(m)\leq n<f_{\alpha}(m)$$s\backslash t_{i}\subseteq\omega\backslash H_{i}$$\dashv$$m_{0}.$$\mathcal{G}_{\beta+1}$$\mathcal{G}_{\beta}$$\dashv$$\mathfrak{b}<\mathfrak{a}$$V$$2^{\lambda}=\lambda^{+}$$\langle f_{\alpha}$$c=\lambda$$<^{*}$$\alpha<\lambda\rangle$

THEOREM 12. There is a $ccc$ forcing such thatis still unbounded.
:and$\mathbb{P}$$|\vdash_{\mathbb{P}}\mathfrak{a}=\lambda^{+}$

In particular,$\langle f_{\alpha}$$b\leq\lambda<\lambda^{+}=\mathfrak{a}$$\alpha<\lambda\rangle$

is consistent.

PROOF. Perform a finite support iteration of orderings of typeof length , going through all (names for) mad families along
the way by a bookkeeping argument (this is possible by the assump). The unboundedness of:is preserved intion
the successor step of the iteration by Theorem 6 and in the limit
step, by standard preservation results.
$\mathbb{P}(\mathcal{A})$$2^{\lambda}=\lambda^{+}$$\lambda^{+}$$\langle f_{\alpha}$$\alpha<\lambda\rangle$$\dashv$25$J\dot{O}$RG BRENDLE AND ANDREW D. BROOKE-TAYLORREFERENCES[1] ANDREAS BLASS, TAPANI HYTTINEN, and YI ZHANG, Mad families and theirneighbors, preprint.
[2] J\"ORG BRENDLE, Mob families and mad families, Archive for Mathematical
Logic, vol. 37 (1998), pp. 183-197.
[3] OSVALDO GUZM\’AN, MICHAEL HRU\v{s}\’AK, and ARTURO ANTONIO MATiNEZ, Canjar filters, preprint.

GROUP OF LOGIC, STATISTICS AND INFORMATICS,GRADUATE SCHOOL OF SYSTEM INFORMATICS,
KOBE UNIVERSITYROKKO-DAI 1-1,
NADA, KOBE, 657-8501
JAPAN-mail: brendle@kurt.scitec.kobe-u.ac.jp-mail: andrewbt@kurt.scitec.kobe-u.ac.jp$E$$E$