Contemporary Mathematics
Volume 302, 2002
Mad Families and Iteration Theory
Ji::irg Brendle
Dedicated to Saharan Shelah, winner of the Janos Bolyai Prize (2000} and the 2001 Wolf Prize
in Mathematics.
ABSTRACT. This is an expository paper dealing with Shelah's theory of iter-
ated forcing along templates as well as with the consistency of n
()
which is
obtained by iterating along a template.
Introduction
This is an introduction to Shelah's recent theory of iterations along templates
[Sh4],
that is, along wellfounded families of subsets of a (not necessarily well-
founded) linear order. This technique has been developed originally to show the
consistency of D
a
where D is the dominating number and
a
the almost disjoint ness
number, thus solving a long-standing problem in the theory of cardinal invariants.
However, it turns out the template point of view is useful for dealing with other
problems too.
Recall that a family
A~
[w]w
is almost disjoint (a.d.) if
An
B is finite for all
A -j. B
E
A.
If A is maximal with this property, it is said to be a maximal almost
disjoint family (mad family). Every infinite mad family is in fact uncountable. The
almost disjointness number
a
is the size of the smallest infinite mad family. So
N1
::;
a ::;
c
where
c
is the size of the continuum.
For functions f, g
E
ww,
say that f
::;*
g (g eventually dominates
f)
if {n; f(n)
g( n)} is finite. The dominating number D is the size of the least dominating (
=
co-
final) family in
(ww,
::;*) while the unbounding number b is the cardinality of the
smallest unbounded family in
(ww, ::;*).
A simple diagonal argument shows b is
uncountable. And b ::; D ::;
c
is trivial. Furthermore it is easy to see that
b ::; a
in ZFC
[vD].
The (somewhat tricky) consistency of
a
b, obtained by Shelah
two decades ago
[Shl],
can be proved both by finite support iteration of ccc forc-
ing
[Brl]
and countable support iteration of proper forcing
[Shl],
the two standard
2000 Mathematics Subject Classification.
Primary
03E17;
Secondary
03E35, 03E40.
Key words and phrases.
Maximal almost disjoint families, almost disjointness number, dom-
inating number, unbounding number, iterated forcing, ultrapowers of partial orders, preservation
theorems for forcing.
Supported by Grant-in-Aid for Scientific Research
(C)(2)12640124,
Japan Society for the
Promotion of Science.
©
2002 American Mathematical Society
http://dx.doi.org/10.1090/conm/302/05083
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