In
set theory, a Woodin cardinal (named for
W. Hugh Woodin) is a
cardinal number
such that for all functions
, there exists a cardinal
with
and an
elementary embedding
from the
Von Neumann universe
into a transitive
inner model
with
critical point
and
.
An equivalent definition is this:
is Woodin
if and only if
is
strongly inaccessible and for all
there exists a
which is
-
-strong.
being
-
-strong means that for all
ordinals
, there exist a
which is an
elementary embedding with
critical point
,
,
and
. (See also
strong cardinal.)
A Woodin cardinal is preceded by a
stationary set of
measurable cardinals, and thus it is a
Mahlo cardinal. However, the first Woodin cardinal is not even
weakly compact.
The hierarchy
(known as the von Neumann hierarchy) is defined by
transfinite recursion on
:
,
,
, when
is a limit ordinal.
For any ordinal
,
is a set. The union of the sets
for all ordinals
is no longer a set, but a proper class. Some of the sets
have set-theoretic properties, for example when
is an inaccessible cardinal,
satisfies second-order ZFC ("satisfies" here means the notion of
satisfaction from first-order logic).
For a
transitive
class
, a function
is said to be an elementary embedding if for any formula
with free variables
in the language of set theory, it is the case that
iff
, where
is first-order logic's notion of satisfaction as before. An elementary embedding
is called nontrivial if it is not the identity. If
is a nontrivial elementary embedding, there exists an ordinal
such that
, and the least such
is called the critical point of
.
Many large cardinal properties can be phrased in terms of elementary embeddings. For an ordinal
, a cardinal
is said to be
-strong if a transitive class
can be found such that there is a nontrivial elementary embedding
whose critical point is
, and in addition
.
A strengthening of the notion of
-strong cardinal is the notion of
-strongness of a cardinal
in a greater cardinal
: if
and
are cardinals with
, and
is a subset of
, then
is said to be
-strong in
if for all
, there is a nontrivial elementary embedding
witnessing that
is
-strong, and in addition
. (This is a strengthening, as when letting
,
being
-strong in
implies that
is
-strong for all
, as given any
,
must be equal to
,
must be a subset of
and therefore a subset of the range of
.) Finally, a cardinal
is Woodin if for any choice of
, there exists a
such that
is
-strong in
.
[1]
Woodin cardinals are important in
descriptive set theory. By a result
[2] of
Martin and
Steel, existence of infinitely many Woodin cardinals implies
projective determinacy, which in turn implies that every projective set is
Lebesgue measurable, has the
Baire property (differs from an open set by a
meager set, that is, a set which is a countable union of
nowhere dense sets), and the
perfect set property (is either countable or contains a
perfect subset).
The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in
ZF+
AD+
DC one can prove that
is Woodin in the class of hereditarily ordinal-definable sets.
is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see
Θ (set theory)).
Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an
inner model containing a Woodin cardinal in which there is a
-well-ordering of the reals,
◊ holds, and the
generalized continuum hypothesis holds.
[3]
Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on
is
-saturated.
Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an
-dense ideal over
.
A
cardinal
is called hyper-Woodin if there exists a
normal measure
on
such that for every set
, the set
is
-
-
strong![{\displaystyle \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf208e5d370391e66767f13641bd5ee6ad93825)
is in
.
is
-
-strong if and only if for each
there is a
transitive class
and an
elementary embedding
![{\displaystyle j:V\to N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8be63701af8b7d52a27fdd2d468318d41a8a881f)
with
![{\displaystyle \lambda ={\text{crit}}(j),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a7a9d40d4c0be3b09eedd5867e080bbc99db7ed)
, and
.
The name alludes to the classical result that a cardinal is Woodin if and only if for every set
, the set
is
-
-
strong![{\displaystyle \}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cf208e5d370391e66767f13641bd5ee6ad93825)
is a
stationary set.
The measure
will contain the set of all
Shelah cardinals below
.
Weakly hyper-Woodin cardinals
A
cardinal
is called weakly hyper-Woodin if for every set
there exists a
normal measure
on
such that the set
is
-
-
strong
is in
.
is
-
-strong if and only if for each
there is a transitive class
and an elementary
embedding
with
,
, and
The name alludes to the classic result that a cardinal is Woodin if for every set
, the set
is
-
-
strong
is stationary.
The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of
does not depend on the choice of the set
for hyper-Woodin cardinals.
Woodin-in-the-next-admissible cardinals
Let
be a cardinal and let
be the least
admissible ordinal greater than
. The cardinal
is said to be Woodin-in-the-next-admissible if for any function
such that
, there exists
such that
, and there is an extender
such that
and
. These cardinals appear when building models from iteration trees.
[4]p.4