Editing Ultrafilter (section)

==Applications==

Ultrafilters on [[power set]]s are useful in [[topology]], especially in relation to [[Compact space|compact]] [[Hausdorff space|Hausdorff]] spaces, and in [[model theory]] in the construction of [[Ultraproduct|ultraproducts and ultrapowers]]. Every ultrafilter on a compact Hausdorff space converges to exactly one point. Likewise, ultrafilters on Boolean algebras play a central role in [[Stone's representation theorem for Boolean algebras|Stone's representation theorem]]. In [[set theory]] ultrafilters are used to show that the [[axiom of constructibility]] is incompatible with the existence of a [[measurable cardinal]] {{mvar|κ}}. This is proved by taking the ultrapower of the set theoretical universe modulo a {{mvar|κ}}-complete, non-principal ultrafilter.<ref> Kanamori, The Higher infinite, p. 49.</ref>

The set <math>G</math> of all ultrafilters of a poset <math display="inline">P</math> can be topologized in a natural way, that is in fact closely related to the above-mentioned representation theorem. For any element <math>x</math> of <math display="inline">P</math>, let <math>D_x = \left\{ U \in G : x \in U \right\}.</math> This is most useful when <math display="inline">P</math> is again a Boolean algebra, since in this situation the set of all <math>D_x</math> is a base for a compact Hausdorff topology on <math>G</math>. Especially, when considering the ultrafilters on a powerset <math>{\mathcal P}(S)</math>, the resulting [[topological space]] is the [[Stone–Čech compactification]] of a [[discrete space]] of cardinality <math>| S |.</math>

The [[ultraproduct]] construction in [[model theory]] uses ultrafilters to produce a new model starting from a sequence of <math>X</math>-indexed models; for example, the [[compactness theorem]] can be proved this way.
In the special case of ultrapowers, one gets [[elementary extension]]s of structures. For example, in [[nonstandard analysis]], the [[hyperreal number]]s can be constructed as an ultraproduct of the [[real numbers]], extending the [[domain of discourse]] from 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 the functions and relations are defined "[[Ultraproduct#Definition|pointwise modulo]]" <math>U</math>, where <math>U</math> 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 <math>U</math> is nonprincipal, then the extension thereby obtained is nontrivial.

In [[geometric group theory]], non-principal ultrafilters are used to define the [[Ultralimit#Asymptotic cones|asymptotic cone]] of a group. This construction yields a rigorous way to consider {{em|looking at the group from infinity}}, that is the large scale geometry of the group. Asymptotic cones are particular examples of [[ultralimit]]s of [[metric space]]s.

[[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 (for example, Kirman and Sondermann, 1972).<ref>{{Cite journal|last1 = Kirman|first1 = A.|last2 = Sondermann|first2 = D.|title = Arrow's theorem, many agents, and invisible dictators|journal = Journal of Economic Theory|volume = 5|issue = 2|pages = 267–277|year = 1972|doi = 10.1016/0022-0531(72)90106-8}}</ref> Mihara (1997,<ref name=mihara97>{{Cite journal|last1 = Mihara|first1 = H. R.|title = Arrow's Theorem and Turing computability|journal = Economic Theory|volume = 10|issue = 2|pages = 257–276|year = 1997|postscript = Reprinted in K. V. Velupillai, S. Zambelli, and S. Kinsella, ed., Computable Economics, International Library of Critical Writings in Economics, Edward Elgar, 2011.|doi = 10.1007/s001990050157|url = https://econwpa.ub.uni-muenchen.de/econ-wp/pe/papers/9408/9408001.pdf|citeseerx = 10.1.1.200.520|s2cid = 15398169 }}</ref> 1999)<ref name=mihara99>{{Cite journal|last1 = Mihara|first1 = H. R.|title = Arrow's theorem, countably many agents, and more visible invisible dictators|journal = [[Journal of Mathematical Economics]]|volume = 32|issue = 3|pages = 267–277|year = 1999|doi = 10.1016/S0304-4068(98)00061-5|url= http://econpapers.repec.org/paper/wpawuwppe/9705001.htm|citeseerx = 10.1.1.199.1970}}</ref> shows, however, such rules are practically of limited interest to social scientists, since they are non-algorithmic or non-computable.