24 votes

What are some nice uses of ultraproducts/ultrapowers?

One of my favourite applications is proving that $\sqrt2$ is irrational using ultraproducts. This requires knowing a nontrivial fact, that there are infinitely many prime numbers $p$ such that $x^2\...
21 votes

What are some nice uses of ultraproducts/ultrapowers?

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 ...
18 votes
Accepted

Ultraproducts of Banach spaces versus model theoretic ultraproduct

The ultraproduct of Banach spaces is the ultraproduct in the sense of metric structures in continuous logic. For a nice survey on this topic, see Model theory for metric structures by Ben Yaacov, ...
Alex Kruckman's user avatar
17 votes
Accepted

Maximal ideals of ultraproducts of full matrix algebras

I think Nik Weaver is right that the ideal mentioned is the unique maximal ideal. This simultaneously answers both questions (since the quotient is clearly infinite dimensional). Let $\tau$ be the ...
Caleb Eckhardt's user avatar
16 votes

Ultraproducts of Banach spaces versus model theoretic ultraproduct

As a logician, I take the model-theoretic notion of ultraproduct as the primary one, so the following formal connection describes how to get the Banach-space ultraproduct from the model-theoretic one. ...
Andreas Blass's user avatar
13 votes

What are some nice uses of ultraproducts/ultrapowers?

Ultraproducts are useful in certain applications to combinatorics, with a famous example being Hrushovski's work on finite approximate groups, followed by the structure theorem of Breuillard, Green, ...
11 votes
Accepted

What is the Turing degree associated with an ultrafilter $U$?

Great question! This is something that Uri Andrews, Mingzhong Cai, David Diamondstone, and I looked at in a recent (still unpublished) paper. First of all, let's note that there's an important ...
Noah Schweber's user avatar
11 votes

Ultraproducts of Banach spaces versus model theoretic ultraproduct

The "Banach space ultraproduct" is also referred to as the nonstandard hull, precisely in order to distinguish it from the model theoretic ultraproduct (which I shall simply call "...
Terry Tao's user avatar
  • 109k
11 votes

What are some nice uses of ultraproducts/ultrapowers?

The paper Chromatic homotopy theory is asymptotically algebraic (by Barthel, Schlank and Stapleton) is an interesting example. In chromatic homotopy theory we study the category $\mathcal{L}(p,n)$ of ...
10 votes

What are some nice uses of ultraproducts/ultrapowers?

One common slogan is that ultrapower arguments rely on the axiom of choice. This isn't as true as it may sound, though. The technique still has purchase in other contexts, such as inner models where ...
9 votes
Accepted

Stationary correctness of ultrapowers by low order measures

It is consistent that $\kappa$ has a unique normal measure (so, in particular, Mitchell minimal) and that measure's ultrapower $M$ satisfies $\mathcal{P}(\kappa^+)\subseteq M$, so, by your note, the ...
Miha Habič's user avatar
  • 2,289
9 votes
Accepted

When do two ultrafilters yield isomorphic ultrapowers?

If one allows an arbitrary signature, the answer to this question is fairly well-known. Consider the structure $(\lambda,A)_{A\subseteq \lambda}$. Let $(M,R_A)_{A\subseteq \lambda}$ be the common ...
Gabe Goldberg's user avatar
9 votes
Accepted

What is the Galois group of one ultrapower over another ultrapower?

$\newcommand{\Gal}{\operatorname{Gal}}$If $E/F$ is a finite Galois extension, then $\Gal(\prod_UE/\prod_UF)$ is canonically isomorphic to $\Gal(E/F)$. Indeed, by the primitive element theorem, $E=F(\...
Wojowu's user avatar
  • 27.4k
8 votes

Statements that Could be Forced by Ultrapowers

Cichoń's maximum, arXiv:1708.03691, uses 4 consecutive ultrapowers of a finite support ccc iteration.
Goldstern's user avatar
  • 13.9k
8 votes
Accepted

Is there a pair of non-isomorphic structures each of which is isomorphic to an ultrapower of the other?

An example assuming large cardinals: suppose $U$ and $W$ are normal ultrafilters on a measurable cardinal $\kappa$ such that $(V_{\kappa+2})^{M_U}\neq (V_{\kappa+2})^{M_W}$, where $M_U$ denotes the ...
Gabe Goldberg's user avatar
7 votes
Accepted

About reflexivity of ultrapower

$\newcommand{\mc}{\mathcal}$The confusion seems to be over the following claim: Claim: Let $\mc U$ be a countably incomplete ultrafilter, and let $E$ be a Banach space. If $(E)_{\mc U}$ is ...
Matthew Daws's user avatar
  • 18.5k
7 votes
Accepted

On a completeness property of hyperreals

This is also called Cauchy-completeness, and it coincides for non-Archimedean ordered fields with the natural valuation to the valuation-theoretic notion of completeness. Also, this is the same as ...
nombre's user avatar
  • 2,367
6 votes
Accepted

Are there interesting examples of theorems proved using ‘height’ extensions?

Here is another instance, which appears in my recent paper with Bokai Yao on second-order reflection in the context of KMU with abundant urelements. Joel David Hamkins and Bokai Yao, Reflection in ...
Joel David Hamkins's user avatar
6 votes
Accepted

Biduals of Banach algebras

In general, no: for certain $A$ one can get non-zero $c\in A$ and sequences $(a_n)$ and $(b_n)$ in $A$ that converge weakly to zero, such that $a_n b_n=c$ for all $n$; these will show that $\ker\sigma$...
Yemon Choi's user avatar
  • 25.5k
6 votes

What are some nice uses of ultraproducts/ultrapowers?

A ring $R$ (commutative, with identity) is said to be solid if the unique homomorphism $\mathbb{Z} \to R$ is an epimorphism in the category of rings, or equivalently if $r \otimes 1 = 1 \otimes r$ in $...
5 votes

An Extender is a Generalization of an Ultrafilter?

I know this question is four years old, but if anyone is stumbling across this question, this might be something worth noting. Maybe it has a way simpler proof, but this should work. Proposition. A ...
Dan Saattrup Nielsen's user avatar
5 votes

Is every field extension of an ultrafield an ultrafield?

I have mentioned in a comment that the accepted answer is not correct, although the argument is correct when the index set is countable. Here's a result with no algebraic closedness assumption, which ...
YCor's user avatar
  • 60.2k
5 votes
Accepted

A question on ultraproducts of $L_{p}(\mu)$-spaces

As for the first question the answer is no even if $\mu$ is assumed to be $\sigma$-finite as by a simple change of measure you may reduce the problem to the case where $\mu$ is finite. Let us fix a ...
Tomasz Kania's user avatar
  • 11.3k
5 votes

Stationary correctness of ultrapowers by low order measures

I once tried and failed to answer this question in the canonical (forgive me) inner models, and your really nice observation about strong cardinals and Mitchell rank helped me finally make some ...
Gabe Goldberg's user avatar
4 votes

Unbounded $\omega_1$-sequence in $^*\mathbb{N}$

If HC (continuum hypothesis in French) holds, then some of those sequences are cofinal whereas some are not. Indeed, HC implies that the corresponding ultrapower$\ ^*\mathbb{R}$ of $\mathbb{R}$ is a ...
nombre's user avatar
  • 2,367
4 votes
Accepted

What strengthenings of measurability do the Mostowski collapses of ultrapowers possess?

Let's adopt Miha's interpretation of your question: which kinds of large cardinal properties can be witnessed by ultrapower embeddings? Here, there are a variety of things one can say. If one ...
Joel David Hamkins's user avatar
4 votes
Accepted

If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?

Let me start by observing that this property is equivalent to a more standard property: Claim: Let $M\subseteq V$ be a transitive model of $\mathrm{ZFC}$ and let $\mu \in M$ be a cardinal in $V$. The ...
Yair Hayut's user avatar
  • 5,192
4 votes

Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

You may also look at Lelek’s conjecture, where Shelah's theorem is used in an essential way (the questions at the end of the paper ask if one can give an easier (more direct) proof of the results ...
Mohammad Golshani's user avatar
3 votes

On ultraproducts of topological spaces

Much of the work of Paul Bankston is about ultraproducts of topological spaces, so I suggest that you look at his work. See the paper [1] for an introduction of the ultraproduct construction of ...
Joseph Van Name's user avatar
3 votes

Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy

Alec, I'm afraid your construction does not work. You claim you obtain the surreals by forming the Dedekindean Completion of what you call the surrationals. However, since No--the surreals-- is not ...
Philip Ehrlich's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible