In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, a Woodin cardinal (named for
W. Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals, ...
) is a
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
such that for all functions
:
there exists a cardinal
with
:
and an
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
:
from the
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted by ''V'', is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZF ...
into a transitive
inner model
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
with
critical point and
:
An equivalent definition is this:
is Woodin
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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 In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
with
critical point ,
,
and
. (See also
strong cardinal In set theory, a strong cardinal is a type of large cardinal. It is a weakening of the notion of a supercompact cardinal.
Formal definition
If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and ther ...
.)
A Woodin cardinal is preceded by a
stationary set of
measurable cardinal
In mathematics, a measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal , or more generally on any set. For a cardinal , it can be described as a subdivisi ...
s, and thus it is a
Mahlo cardinal. However, the first Woodin cardinal is not even
weakly compact.
Consequences
Woodin cardinals are important in
descriptive set theory. By a result of
Martin Martin may refer to:
Places
* Martin City (disambiguation)
* Martin County (disambiguation)
* Martin Township (disambiguation)
Antarctica
* Martin Peninsula, Marie Byrd Land
* Port Martin, Adelie Land
* Point Martin, South Orkney Islands
Austr ...
and
Steel, existence of infinitely many Woodin cardinals implies
projective determinacy, which in turn implies that every projective set is
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
, has the
Baire property (differs from an open set by a
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
, that is, a set which is a countable union of
nowhere dense sets), and the
perfect set property In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a p ...
(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
In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''.
Definition
Let L = \langle \in \rangle be ...
containing a Woodin cardinal in which there is a
-well-ordering of the reals,
◊ holds, and the
generalized continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
holds.
[W. Mitchell]
Inner models for large cardinals
(2012, p.32). Accessed 2022-12-08.
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
.
Hyper-Woodin cardinals
A
cardinal is called hyper-Woodin if there exists a
normal measure on
such that for every set
, the set
:
is in
.
is
-
-strong if and only if for each
there is a
transitive class and an
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
:
with
:
:
, and
:
.
The name alludes to the classical result that a cardinal is Woodin if and only if for every set
, the set
:
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 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 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.
Notes and references
Further reading
*
* For proofs of the two results listed in consequences see ''Handbook of Set Theory'' (Eds. Foreman, Kanamori, Magidor) (to appear).
Draftsof some chapters are available.
* Ernest Schimmerling, ''Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model'', Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002
online*
{{DEFAULTSORT:Woodin Cardinal
Large cardinals
Determinacy