Questions tagged [ultrapowers]

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Ultraproduct reflexive

Hello I have the following construction: Let $(E,\|\cdot\|_E)$ be a Banach space, $E_n:=E^n$ and $\|x_n\|_n:=\frac{1}{n}\sum_{k=1}^n \|x_n(k)\|_E$ for all $x_n=(x_n(1),...,x_n(n))\in E^n$ and $n\in\...
Michaeli's user avatar
1 vote
1 answer
211 views

Seeking clarification of ultrapower nonstandard model of arithmetic

I've read that one nonstandard model of arithmetic is: take $\mathbb{N}^\mathbb{N}$, the set of countably infinite sequences of natural numbers take a quotent that gives the ultrapower: identify ...
Dave Pritchard's user avatar
7 votes
0 answers
287 views

A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$

Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have $M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
Noah Schweber's user avatar
6 votes
0 answers
177 views

Interest in the size of ultrapowers in model theory

It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set ...
Monroe Eskew's user avatar
  • 18.1k
20 votes
6 answers
3k views

What are some nice uses of ultraproducts/ultrapowers?

Motivated by a recent post (Non-definability of graph 3-colorability in first-order logic), I was wondering: what are some nice arguments based on ultraproducts? I don't mind definability results, but ...
3 votes
1 answer
256 views

Ultra*powers* in the category of structures and elementary embeddings

This is based on a few previous questions. Can one characterize ultrapowers in the category of L-structures (modeling a fixed complete theory, say) and elementary embeddings? Previous posts showed ...
Pteromys's user avatar
  • 151
16 votes
3 answers
1k views

Ultraproducts of Banach spaces versus model theoretic ultraproduct

Reading about ultraproducts in model theory and in Banach spaces leads to two distinct definitions. E.g., for an ultrapower given by an ultrafilter $\mu$ on $\mathbb{N}$, both notions of ultrapower ...
Anthony D'Arienzo's user avatar
2 votes
0 answers
122 views

An ultrafilter on $\omega_1$ with a nice Fubini product with an ultrafilter on $\omega$

Fix an ultrafilter $U$ on $\omega$ (that is, $U$ is an ultrafilter on the Boolean algebra of all subsets of $\omega$). Let $(f_\alpha \mid \alpha < \omega_1)$ be an increasing sequence in $\mathbb{...
Gawr Gura's user avatar
  • 153
4 votes
0 answers
67 views

The embedding of a Banach lattice in an ultrapower

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...
M.González's user avatar
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4 votes
0 answers
245 views

ultrapower(ultrapower)=ultrapower

Is there a nonprincipal ultrafilter $\omega$ on $\mathbb N$ such that for any metric space $M$ there is an isometry $$(M^\omega)^\omega\to M^\omega?$$ (In other words, the $\omega$-power of $\omega$-...
Anton Petrunin's user avatar
14 votes
2 answers
645 views

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

It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}...
Neil Barton's user avatar
4 votes
1 answer
169 views

The locale of morphisms vs a morphism to an ultrapower?

I'm fixing some type of structure $\Sigma$ (possibly multi-sorted, with functions symbols and relation symbols, though assuming it single sorted with only relation symbols wouldn't change anything). ...
Simon Henry's user avatar
  • 40.1k
0 votes
0 answers
74 views

Is $(\omega+1)^\omega/{\cal U}$ "unique"?

If ${\cal U}_i$ free ultrafilters on $\omega$ for $i = 1,2$ , are the ultrapowers $(\omega+1)^\omega/{\cal U}_i$ necessarily isomorphic as lattices for $i = 1,2$?
Dominic van der Zypen's user avatar
17 votes
0 answers
1k views

Non-rigid ultrapowers in $\mathsf{ZFC}$?

Originally asked and bountied at MSE: Question: Can $\mathsf{ZFC}$ prove that for every countably infinite structure $\mathcal{A}$ in a countable language there is an ultrafilter $\mathcal{U}$ on $\...
Noah Schweber's user avatar
4 votes
1 answer
318 views

On a completeness property of hyperreals

Let $\mathbb{R}_*=\mathbb{R}^\omega/\mathcal U$ for some ultrafilter $\cal U$. In the definitions of this question and assuming ZFC + CH there are only three types of cuts in $\mathbb{R}_*$: $(\omega,\...
ar.grig's user avatar
  • 1,133
6 votes
1 answer
512 views

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

Let $F$ be a field, let $E$ be a field extension of $F$, and let $U$ be an ultrafilter. Then my question is, what is the relationship between the Galois groups $Gal(\Pi_U E/\Pi_U F)$ and $Gal(E/F)$? ...
Keshav Srinivasan's user avatar
12 votes
1 answer
438 views

Maximal ideals of ultraproducts of full matrix algebras

Let $\mathscr U$ be a non-principal ultrafilter over the natural numbers. Let $M_{\mathscr U}$ be the ultraproduct of all full matrix algebras $M_n$ along $\mathscr U$. This is a C*-algebra that is ...
Tomasz Kania's user avatar
  • 11.3k
2 votes
1 answer
242 views

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

Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a strictly increasing ...
Sergey Grigoryants's user avatar
12 votes
1 answer
638 views

When do two ultrafilters yield isomorphic ultrapowers?

Fix a cardinal $\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}$. Consider the equivalence relation on $\beta\lambda$ given by $\cU\sim \cV$ when for all first-order structures $M$ ...
tomasz's user avatar
  • 1,216
6 votes
0 answers
190 views

Isomorphism of hyperreal fields viewed as extensions of the field of reals

I asked this question on Mathematics Stackexchange but got no answer. Question. Does $ZFC$ prove that there are non-principal ultrafilters $\mathcal U$ and $\mathcal V$ over $\mathbb N$ such that the ...
Pierre-Yves Gaillard's user avatar
8 votes
2 answers
285 views

Stationary correctness of ultrapowers by low order measures

Suppose $U$ is a normal ultrafilter on $\kappa$ of Mitchell order zero, and let $j_U : V \to M$ be the associated embedding. Does there exist a nonstationary $X \subseteq \kappa^+$ such that $X \in M$...
Monroe Eskew's user avatar
  • 18.1k
6 votes
0 answers
210 views

Influence of cardinal characteristics on nonstandard analysis?

As I understand, nonstandard analysis usually proceeds by taking a ultrapower of the universe by some nonprincipal ultrafilter on $\mathbb N$. There are continuum many “integers” of this model, but ...
Monroe Eskew's user avatar
  • 18.1k
3 votes
0 answers
246 views

Iterated ultrapowers vs limit ultrapowers

Does anyone know an example of a limit ultrapower of a structure that is not isomorphic to an iterated ultrapower of that structure? I scoured Chang-Keisler but without any luck. Here are some ...
Isaac's user avatar
  • 771
2 votes
1 answer
528 views

Ultrapower of an ultrapower of von Neumann algebras

Let $M$ be a $\mathrm{II}_1$ factor, fix a ultrafilter $\omega$, we know the ultrapower $M^{\omega}$, is again $\mathrm{II}_1$ factor. The question is that what is the ultrapower of $M^{\omega}$, i.e.,...
sibani's user avatar
  • 181
1 vote
0 answers
166 views

Why is $\widetilde{W}$ closed?

We consider $(x _{n})$ a sequence of almost fixed points for $T$ in $C _{0}$. Since $C _{0}$ is weakly compact, we can assume that $(x _{n})$ is weak compact. Also, since the problem of the fixed ...
G. P's user avatar
  • 23
16 votes
1 answer
722 views

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

Does there exist a pair of non-isomorphic structures $\mathfrak{A}$ and $\mathfrak{B}$ as well as sets $I$ and $J$ and ultrafilters $\mathcal{U}$ on $I$ and $\mathcal{F}$ on $J$ such that $\mathfrak{A}...
James Hanson's user avatar
  • 10.3k
4 votes
1 answer
226 views

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?

More specifically, if $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, does $κ$ necessarily have some form of $μ$-compactness? Is it related to strong compactness in ...
Keith Millar's user avatar
  • 1,242
5 votes
1 answer
247 views

Model theory of Banach algebras

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...
Tomasz Kania's user avatar
  • 11.3k
2 votes
1 answer
96 views

A Question about an irreducible ultra-power II,

Let $E$ be an irreducible Banach $A$-module, for a Banach algebra $A$. One can easily show that for an ultra filter $\mathcal U$, $(E)_\mathcal U$ is a Banach $(A)_\mathcal U$-module. Is it possible ...
Meisam Soleimani Malekan's user avatar
0 votes
0 answers
70 views

A question about an irreducible ultra-power

Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
Meisam Soleimani Malekan's user avatar
1 vote
1 answer
255 views

About reflexivity of ultrapower

It is obvious that for a Banach space $E$, $E$ is reflexive iff $\ell^2(E)$ is reflexive. Let $\mathcal U$ be an ultrafilter. Is the reflexivity of $(E)_\mathcal U$ equivalent to refelxivity of $(\ell^...
Meisam Soleimani Malekan's user avatar
0 votes
1 answer
212 views

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

What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess? Ok, I already posted this question, but a couple of notational errors and assumptions were made in the ...
Keith Millar's user avatar
  • 1,242
3 votes
1 answer
194 views

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

Let $I$ be a set, $\mathcal{U}$ be an ultrafilter on $I$ and $1\leq p<\infty$. Let $X_{i}=L_{p}(\mu_{i})(i\in I)$, where $\mu_{i}$ is a probability measure for each $i\in I$. Relying on standard ...
Dongyang Chen's user avatar
8 votes
1 answer
483 views

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

I asked Turing degree of a turing machine with access to an (arbitrary) nonstandard integer, not thinking about the possiblity that this could depend on the model used. The question was not formulated ...
Christopher King's user avatar
-1 votes
1 answer
471 views

On ultraproducts of topological spaces

Intuitively, I understand the construction of the hyperreals by ultraproducts as a process of turning the limit operation into an algebraic object. More precisely, to check the existence of the limit $...
André Porto's user avatar
4 votes
1 answer
147 views

Biduals of Banach algebras

For a Banach algebra $A$ the bidual $A^{**}$ may be given two natural products called the Arens products. By local reflexivity, there is an ultrafilter $U$ so that $A^{**}$ embeds into the ultrapower $...
user512365's user avatar
2 votes
0 answers
111 views

Ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$

I would like to know if there exist an explicit decription of ultrapowers of $c_0(\ell_1)$ and $\ell_1(c_0)$. The best option would be -- "they are complemented subspaces of $C(K, L_1(\mu))$ and $L_1(\...
Norbert's user avatar
  • 1,687
5 votes
0 answers
136 views

Banach spaces complemented in their ultrapowers

By the principle of local reflexivity, the second dual $X^{**}$ of a Banach space $X$ is complemented in some ultrapower $X^U$ of $X$. Even when $X$ is separable, the index set of $U$ cannot be ...
Tomasz Kania's user avatar
  • 11.3k
5 votes
2 answers
528 views

Statements that Could be Forced by Ultrapowers

Ultrapower of a structure is a very flexible mathematical creature in comparison with the ground structure and its ordinary products. Depending on the nature of ground structure and the good ...
Morteza Azad's user avatar
2 votes
0 answers
153 views

Ultraproducts and subobjects of projectives

Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
87 views

Reduced products of (abelian and triangulated) categories: references?

For a filter $U$ on a set $X$ and for a family of categories $C_x$ indexed by $X$ I would like to consider the (corresponding categorical version of) reduced product of $C_x$ (for $x\in X$) with ...
Mikhail Bondarko's user avatar
3 votes
0 answers
195 views

Which compact topological spaces are homeomorphic to their ultrapower?

It is well known that for any compact metric space $(X, d)$, and any ultrafilter $\mu$ there is a map $i_\mu:\prod_\mu (X, d) \to (X_d)$ in the category of metric spaces and Lipschitz maps where $i_\...
Nate Ackerman's user avatar
4 votes
2 answers
612 views

References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat ...
Kyle Gannon's user avatar
6 votes
1 answer
499 views

Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation: $M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$. $\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...
user avatar
10 votes
1 answer
454 views

K-theory of ultrapowers

It may well be a trivial question but I was wondering if there is any relation between $K$-groups and ultrapowers of $C^*$-algebras. For instance, if $A$ is a $C^*$-algebra does $K_0(A^U)$ depend on ...
TMK's user avatar
  • 103
6 votes
0 answers
175 views

Ultrapowers and complemented subspaces

Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers. It is not difficult to see that if $Y$ is ...
M.González's user avatar
  • 4,301
5 votes
1 answer
241 views

Complemented subspaces of ultrapowers

It's a famous result of Maurey that a Banach space $E$ is finitely representable in $X$ if and only if it is a subspace of some ultrapower of $X$. Is there an analogous result for complemented ...
Mailer's user avatar
  • 105
2 votes
1 answer
325 views

How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall M,N\...
user47544's user avatar
  • 141
14 votes
4 answers
2k views

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

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
Noah Schweber's user avatar
1 vote
0 answers
247 views

Ultrapowers of ultrapowers

Suppose that you have some structure $S$, and you want to construct an ultrapower of cardinality $\kappa$ to obtain $S^*_\kappa$. Then, say you want to construct a new ultrapower from $S^*_\kappa$, ...
Mike Battaglia's user avatar