In
set theory, the critical point of an
elementary embedding of a
transitive class 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 subdivis ...
, 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 on ...
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(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 \rangl ...
of
ZFC and
has no critical point, i.e. every ordinal maps to itself, then
is trivial.
References
Large cardinals
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