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Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis — a set $X$ is large, if a fixed nonstandard number is in $X^*$ — and one can then show that the ultrapower of $\mathbb{R}$ by this ultrafilter is a submodel of that particular instance of the hyperreals.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.

• Voting theory. Consider a group of individuals with individual preferences, which we want to unify into an amalgamated group preference relation, such as in the case of voting. Both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem can be seen as the claim that if certain very reasonable assumptions on these individual and group preference relations hold, then the group preference relation is determined by an ultrafilter on the individuals. In effect, the group preference must be the ultraproduct of the individual preferences. In the case of a finite population, since all ultrafilters on a finite set are principal, this means one individual serves as dictator.

Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis — a set $X$ is large, if a fixed nonstandard number is in $X^*$ — and one can then show that the ultrapower of $\mathbb{R}$ by this ultrafilter is a submodel of that particular instance of the hyperreals.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.

• Voting theory. Consider a group of individuals with individual preferences, which we want to unify into an amalgamated group preference relation, such as in the case of voting. Both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem can be seen as the claim that if certain very reasonable assumptions on these individual and group preference relations hold, then the group preference relation is determined by an ultrafilter on the individuals. In effect, the group preference must be the ultraproduct of the individual preferences. In the case of a finite population, since all ultrafilters on a finite set are principal, this means one individual serves as dictator.

In a sense the ultraproduct construction provides a powerful means by which to amalgamate many different structures into one, and this aspect of ultraproducts is manifested in the proof of the compactness theorem, in nonstandard analysis (where the amalgamation of increasing tiny numbers becomes an infinitesimal), and in voting theory. So these different applications are unified by the idea of amalgamation of structure.

Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis — a set $X$ is large, if a fixed nonstandard number is in $X^*$ — and one can then show that the ultrapower of $\mathbb{R}$ by this ultrafilter is a submodel of that particular instance of the hyperreals.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.

Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis — a set $X$ is large, if a fixed nonstandard number is in $X^*$ — and one can then show that the ultrapower of $\mathbb{R}$ by this ultrafilter is a submodel of that particular instance of the hyperreals.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.

• Voting theory. Consider a group of individuals with individual preferences, which we want to unify into an amalgamated group preference relation, such as in the case of voting. Both Arrow's impossibility theorem and the Gibbard-Satterthwaite theorem can be seen as the claim that if certain very reasonable assumptions on these individual and group preference relations hold, then the group preference relation is determined by an ultrafilter on the individuals. In effect, the group preference must be the ultraproduct of the individual preferences. In the case of a finite population, since all ultrafilters on a finite set are principal, this means one individual serves as dictator.

Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis---a set $X$ is large, if a fixed nonstandard number is in $X^*$.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.

Here are a few common uses that come to mind:

• Large cardinals. Ultrapowers are used pervasively in large cardinal set theory. Most of the familiar large cardinal concepts can be characterized by their relation to certain kinds of ultrapowers of the set-theoretic universe.

• Nonstandard analysis. Ultrapowers provide one of the robust ways to construct hyperreal fields $\mathbb{R}^*$, a real-closed field with infinitesimal elements and the transfer principle. The nonstandard analogue $f^*$ of every function $f$ on the reals is simply the ultrapower of it. Meanwhile, this answer shows a sense in which ultrafilters are inherently part of nonstandard analysis — a set $X$ is large, if a fixed nonstandard number is in $X^*$ — and one can then show that the ultrapower of $\mathbb{R}$ by this ultrafilter is a submodel of that particular instance of the hyperreals.

• Compactness theorem. One can prove the compactness theorem in model theory by the use of a suitable ultraproduct. If a theory $T$ is finitely satisfiable, then take the set of all finite subsets $t \subset T$, and a model $M_t\models t$, and find an ultrafilter $\mu$ such that every sentence is in $\mu$-almost every $t$. So the ultrapower $\prod M_t/\mu$ will satisfy every sentence in $T$.

• Alternative to König's lemma. If a set of tiles has the property that it tiles arbitrarily large squares in the plane, then it also tiles the whole plane. One can prove this by König's lemma, by setting up a certain tree. But one can also prove it by ultraproducts — one takes the ultraproduct of increasing large finite tilings, and the standard part of the ultraproduct provides a full tiling of the plane. And there are many similar instances where ultraproducts provide an alternative to König's lemma.