26
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, ...
- 5,031
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 ...
- 2,689
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.
...
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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, ...
12
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 paper.
First of all, let's note that there's an important undefined notion: the ...
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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 "...
- 114k
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
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 ...
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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
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(\...
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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 ...
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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.
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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 ...
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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 ...
- 18.7k
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 ...
- 2,519
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 ...
- 236k
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 $\ker\sigma$...
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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
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 ...
- 63.9k
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 ...
- 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 ...
- 8,099
4
votes
Accepted
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 ...
- 2,519
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 ...
- 5,112
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 ...
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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 ...
- 32.1k
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 ...
- 6,453
3
votes
Accepted
Seeking clarification of ultrapower nonstandard model of arithmetic
Amazing. Thank you to commenter @Mikhail-katz. Looks like my confusion is due to a Wikipedia error from mid-2019 that I will go correct. Thank you!
This is super helpful as now I see that it's quite ...
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