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 ...
, the critical point of 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 ...
of a
transitive class
In set theory, a branch of mathematics, a set A is called transitive if either of the following equivalent conditions hold:
* whenever x \in A, and y \in x, then y \in A.
* whenever x \in A, and x is not an urelement, then x is a subset of A.
Sim ...
into another transitive class is the smallest
ordinal which is not mapped to itself.
[ p. 323]
Suppose that
is an elementary embedding where
and
are transitive classes and
is definable in
by a formula of set theory with parameters from
. Then
must take ordinals to ordinals and
must be strictly increasing. Also
. If
for all
and
, then
is said to be the critical point of
.
If
is ''
V'', then
(the critical point of
) is always a
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 subdivisio ...
, i.e. an uncountable
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. Th ...
''κ'' such that there exists a
-complete, non-principal
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 ...
over
. Specifically, one may take the filter to be
. Generally, there will be many other <''κ''-complete, non-principal ultrafilters over
. However,
might be different from the
ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factor ...
(s) arising from such filter(s).
If
and
are the same and
is the identity function on
, then
is called "trivial". If the transitive class
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 ...
of
ZFC and
has no critical point, i.e. every ordinal maps to itself, then
is trivial.
References
Large cardinals
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