Ultrafilter

In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged (as a filter). An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" (has measure 1) or "almost nothing" (has measure 0). If A is a subset of X, then either A or X \ A is an element of the ultrafilter (here X \ A is the relative complement of A in X; that is, the set of all elements of X that are not in A). The concept can be generalized to Boolean algebras or even to general partial orders, and has many applications in set theory, model theory, and topology.

1 Formal definition
2 Completeness
3 Generalization to partial orders
4 Types and existence of ultrafilters
5 Applications
6 Ordering on ultrafilters
7 Ultrafilters on ω
8 See also
9 Notes
10 References

Formal definition

Given a set X, an ultrafilter on X is a set U consisting of subsets of X such that

The empty set is not an element of U
If A and B are subsets of X, A is a subset of B, and A is an element of U, then B is also an element of U.
If A and B are elements of U, then so is the intersection of A and B.
If A is a subset of X, then either A or X \ A is an element of U. (Note: axioms 1 and 3 imply that A and X \ A cannot both be elements of U.)

A characterization is given by the following theorem. A filter U on a set X is an ultrafilter if any of the following conditions are true:

There is no filter F finer than U, i.e. $U\subseteq F$ implies U = F.
$A\cup B\in U$ implies $A\in U$ or $B\in U$ .
$\forall A\subseteq X\colon A\in U$ or $X\setminus A \in U$ .

Another way of looking at ultrafilters on a set X is to define a function m on the power set of X by setting m(A) = 1 if A is an element of U and m(A) = 0 otherwise. Then m is a finitely additive measure on X, and every property of elements of X is either true almost everywhere or false almost everywhere. Note that this does not define a measure in the usual sense, which is required to be countably additive.

For a filter F which is not an ultrafilter, one would say m(A) = 1 if A ∈ F and m(A) = 0 if X \ A ∈ F, leaving m undefined elsewhere.

Note that an ultrafilter on an infinite set S is non-principal if and only if it contains the Fréchet filter of cofinite subsets of S. This is obvious, since a non-principal ultrafilter contains no finite set, and is closed under complement, which means that it contains all cofinite subsets of S, which is exactly the Fréchet filter.

Completeness

The completeness of an ultrafilter U on a set is the smallest cardinal κ such that there are κ elements of U whose intersection is not in U. The definition implies that the completeness of any ultrafilter is at least $\aleph_0$ . An ultrafilter whose completeness is greater than $\aleph_0$ —that is, the intersection of any countable collection of elements of U is still in U—is called countably complete or $\sigma$ -complete.

The completeness of a countably complete nonprincipal ultrafilter on a set is always a measurable cardinal.

Generalization to partial orders

In order theory, an ultrafilter is a subset of a partially ordered set (a poset) which is maximal among all proper filters. Formally, this states that any filter that properly contains an ultrafilter has to be equal to the whole poset. An important special case of the concept occurs if the considered poset is a Boolean algebra, as in the case of an ultrafilter on a set (defined as a filter of the corresponding powerset). In this case, ultrafilters are characterized by containing, for each element a of the Boolean algebra, exactly one of the elements a and ¬a (the latter being the Boolean complement of a).

Ultrafilters on a Boolean algebra can be identified with prime ideals, maximal ideals, and homomorphisms to the 2-element Boolean algebra {true, false}, as follows:

Maximal ideals of a Boolean algebra are the same as prime ideals.
Given a homomorphism of a Boolean algebra onto {true, false}, the inverse image of "true" is an ultrafilter, and the inverse image of "false" is a maximal ideal.
Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
Given an ultrafilter of a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".

Let us see another theorem which could be used for the definition of the concept of “ultrafilter”. Let B denote a Boolean algebra and F a proper filter^[1] in it. F is an ultrafilter iff:

for all $a,b \in \mathbf B$ , if $a \vee b \in F$ , then $a \in F$ or $b \in F$

(To avoid confusion: the sign $\vee$ denotes the join operation of the Boolean algebra, and logical connectives are rendered by English circumlocutions.) See details (and proof) in.^[2]

Types and existence of ultrafilters

There are two very different types of ultrafilter: principal and free. A principal (or fixed, or trivial) ultrafilter is a filter containing a least element. Consequently, principal ultrafilters are of the form F_a = {x | a ≤ x} for some (but not all) elements a of the given poset. In this case a is called the principal element of the ultrafilter. For the case of filters on sets, the elements that qualify as principals are exactly the one-element sets. Thus, a principal ultrafilter on a set S consists of all sets containing a particular point of S. An ultrafilter on a finite set is principal. Any ultrafilter which is not principal is called a free (or non-principal) ultrafilter.

One can show that every filter of a Boolean algebra (or more generally, any subset with the finite intersection property) is contained in an ultrafilter (see Ultrafilter lemma) and that free ultrafilters therefore exist, but the proofs involve the axiom of choice in the form of Zorn's Lemma. On the other hand, the statement that every filter is contained in an ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory (ZF) and the ZF theory augmented by the axiom of choice (ZFC). Proofs involving the axiom of choice do not produce explicit examples of free ultrafilters. Nonetheless, almost all ultrafilters on an infinite set are free. By contrast, every ultrafilter of a finite poset (or on a finite set) is principal, since any finite filter has a least element.

Applications

Ultrafilters on sets are useful in topology, especially in relation to compact Hausdorff spaces, and in model theory in the construction of ultraproducts and ultrapowers. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on posets are most important if the poset is a Boolean algebra, since in this case the ultrafilters coincide with the prime filters. Ultrafilters in this form play a central role in Stone's representation theorem for Boolean algebras.

The set G of all ultrafilters of a poset P can be topologized in a natural way, that is in fact closely related to the abovementioned representation theorem. For any element a of P, let D_a = {U ∈ G | a ∈ U}. This is most useful when P is again a Boolean algebra, since in this situation the set of all D_a is a base for a compact Hausdorff topology on G. Especially, when considering the ultrafilters on a set S (i.e. the case that P is the powerset of S ordered via subset inclusion), the resulting topological space is the Stone–Čech compactification of a discrete space of cardinality |S|.

The ultraproduct construction in model theory uses ultrafilters to produce elementary extensions of structures. For example, in constructing hyperreal numbers as an ultraproduct of the real numbers, we first extend the domain of discourse from the real numbers to sequences of real numbers. This sequence space is regarded as a superset of the reals by identifying each real with the corresponding constant sequence. To extend the familiar functions and relations (e.g., + and <) from the reals to the hyperreals, the natural idea is to define them pointwise. But this would lose important logical properties of the reals; for example, pointwise < is not a total ordering. So instead we define the functions and relations "pointwise modulo U", where U is an ultrafilter on the index set of the sequences; by Łoś' theorem, this preserves all properties of the reals that can be stated in first-order logic. If U is nonprincipal, then the extension thereby obtained is nontrivial.

In geometric group theory, non-principal ultrafilters are used to define the asymptotic cone of a group. These construction yields a rigorous way to consider looking at the group from infinity, that is the large scale geometry of the group. Asymptotic cones are particular examples of ultralimits of metric spaces.

Gödel's ontological proof of God's existence uses as an axiom that the set of all "positive properties" is an ultrafilter.

In social choice theory, non-principal ultrafilters are used to define a rule (called a social welfare function) for aggregating the preferences of infinitely many individuals. Contrary to Arrow's impossibility theorem for finitely many individuals, such a rule satisfies the conditions (properties) that Arrow proposes (e.g., Kirman and Sondermann, 1972^[3]). Mihara (1997,^[4] 1999^[5]) shows, however, such rules are practically of maximal interest to computability theorists, since they are non-algorithmic or non-computable.

Ordering on ultrafilters

Rudin–Keisler ordering is a preorder on the class of ultrafilters defined as follows: if U is an ultrafilter on X, and V an ultrafilter on Y, then $V\le_{RK}U$ if and only if there exists a function f: X → Y such that

$C\in V\iff f^{-1}[C]\in U$

for every subset C of Y.

Ultrafilters U and V are Rudin–Keisler equivalent, $U\equiv_{RK}V$ , if there exist sets $A\in U$ , $B\in V$ , and a bijection f: A → B which satisfies the condition above. (If X and Y have the same cardinality, the definition can be simplified by fixing A = X, B = Y.)

It is known that $\equiv_{RK}$ is the kernel of $\le_{RK}$ , i.e., $U\equiv_{RK}V$ if and only if $U\le_{RK}V$ and $V\le_{RK}U$ .

Ultrafilters on ω

There are several special properties that an ultrafilter on ω may possess, which prove useful in various areas of set theory and topology.

A non-principal ultrafilter U is a P-point (or weakly selective) iff for every partition of ω, $\{ C_n\mid n<\omega \}$ such that $C_n\not\in U,\forall n<\omega$ , there exists $A\in U$ such that $|A\cap C_n|<\omega,\forall n<\omega$ .
A non-principal ultrafilter U is Ramsey (or selective) iff for every partition of ω, $\{C_n\mid n<\omega\}$ such that $C_n\not\in U,\forall n<\omega$ , there exists $A\in U$ such that $|A\cap C_n|=1,\forall n<\omega$

It is a trivial observation that all Ramsey ultrafilters are P-points. Walter Rudin proved that the continuum hypothesis implies the existence of Ramsey ultrafilters.^[6] In fact, many hypotheses imply the existence of Ramsey ultrafilters, including Martin's axiom. Saharon Shelah later showed that it is consistent that there are no P-point ultrafilters.^[7] Therefore the existence of these types of ultrafilters is independent of ZFC.

P-points are called as such because they are topological P-points in the usual topology of the space βω \ ω of non-principal ultrafilters. The name Ramsey comes from Ramsey's theorem. To see why, one can prove that an ultrafilter is Ramsey if and only if for every 2-coloring of $[\omega]^2$ there exists an element of the ultrafilter which has a homogeneous color.