- In the
mathematical field of
order theory, an
ultrafilter on a
given partially ordered set (or "poset") P {\textstyle P} is a
certain subset of P , {\displaystyle...
- In the
mathematical field of set theory, an
ultrafilter on a set X {\displaystyle X} is a
maximal filter on the set X . {\displaystyle X.} In
other words...
- ideals. A
variation of this
statement for
filters on sets is
known as the
ultrafilter lemma.
Other theorems are
obtained by
considering different mathematical...
-
property true
almost everywhere is
sometimes defined in
terms of an
ultrafilter. An
ultrafilter on a set X is a
maximal collection F of
subsets of X such that:...
- prin****l
ultrafilter on X {\displaystyle X} . Moreover,
every prin****l
ultrafilter on X {\displaystyle X} is
necessarily of this form. The
ultrafilter lemma...
- be
extended to an
ultrafilter, but the
proof uses the
axiom of choice. The
existence of a
nontrivial ultrafilter (the
ultrafilter lemma) can be added...
-
ultrafilter is
called the
ultrafilter lemma and
cannot be
proven in Zermelo–Fraenkel set
theory (ZF), if ZF is consistent.
Within ZF, the
ultrafilter...
-
element i ∈ I {\displaystyle i\in I} (all of the same signature), and an
ultrafilter U {\displaystyle {\mathcal {U}}} on I . {\displaystyle I.} For any two...
- to the
following criterion: ****uming the
ultrafilter lemma, a
space is
compact if and only if each
ultrafilter on the
space converges. With this in hand...
- then the
ultrafilter U
witnessing that κ is
measurable will be in Vκ+2 and thus in M. So for any α < κ, we have that
there exist an
ultrafilter U in j(Vκ)...
