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 supercompact cardinal is a type of
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least Î ...
. They display a variety of reflection properties.
Formal definition
If ''λ'' is any
ordinal, ''κ'' is ''λ''-supercompact means that there exists 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 ...
''j'' from the universe ''V'' 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 ...
''M'' with
critical point ''κ'', ''j''(''κ'')>''λ'' and
:
That is, ''M'' contains all of its ''λ''-sequences. Then ''κ'' is supercompact means that it is ''λ''-supercompact for all ordinals ''λ''.
Alternatively, an uncountable cardinal ''κ'' is supercompact if for every ''A'' such that , ''A'', ≥ ''κ'' there exists a
normal measure In set theory, a normal measure is a measure on a measurable cardinal κ such that the equivalence class of the identity function on κ maps to κ itself in the ultrapower construction. Equivalently, if f:κ→κ is such that f(α)<α for most α<κ ...
over
'A''sup>< ''κ'', in the following sense.
'A''sup>< ''κ'' is defined as follows:
:
An
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
''U'' over
'A''sup>< ''κ'' is ''fine'' if it is ''κ''-complete and
, for every
. A normal measure over
'A''sup>< ''κ'' is a fine ultrafilter ''U'' over
'A''sup>< ''κ'' with the additional property that every function
such that
is constant on a set in
. Here "constant on a set in ''U''" means that there is
such that
.
Properties
Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-
huge cardinal In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and
:^M \subset M.\!
Here, ''αM'' is the class of al ...
) that is witnessed by a structure of limited rank exists above a supercompact cardinal ''κ'', then a cardinal with that property exists below κ. For example, if ''κ'' is supercompact 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 ...
(GCH) holds below ''κ'' then it holds everywhere because a bijection between the powerset of ''ν'' and a cardinal at least ''ν''
++ would be a witness of limited rank for the failure of GCH at ''ν'' so it would also have to exist below ''κ''.
Finding a canonical inner model for supercompact cardinals is one of the major problems of
inner model theory In set theory, inner model theory is the study of certain models of ZFC or some fragment or strengthening thereof. Ordinarily these models are transitive subsets or subclasses of the von Neumann universe ''V'', or sometimes of a generic extensio ...
.
See also
*
Indestructibility
*
Strongly compact cardinal
*
List of large cardinal properties
References
*
*
* {{cite book, last=Kanamori, first=Akihiro, authorlink=Akihiro Kanamori, year=2003, publisher=Springer, title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, title-link= The Higher Infinite , edition=2nd, isbn=3-540-00384-3
Large cardinals