Questions tagged [ultrapowers]
The ultrapowers tag has no usage guidance.
23
questions with no upvoted or accepted answers
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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 $\...
- 19.1k
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646
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Dual of the Ultraproduct of a Banach Space
Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like":
$(E_i^*)_U$, the ultraproduct of the duals of the ground spaces.
The space made up ...
- 7,135
7
votes
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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 ...
- 19.1k
6
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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 ...
- 18.1k
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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 ...
- 4,336
6
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210
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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 ...
- 18.1k
6
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175
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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 ...
- 4,301
5
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136
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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 ...
- 11.3k
5
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Double ultrapower of the hyperfinite $II_1$-factor
Let $\omega$ be a free ultrafilter on the natural numbers and $R$ be the hyperfinite $II_1$-factor (the definition of $R$ is recalled in the comments).
Question: Does there exist another free ...
- 5,894
4
votes
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67
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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}(...
- 4,301
4
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245
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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 $\omega$-...
- 43.7k
3
votes
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answers
246
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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 ...
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3
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195
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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_\...
- 2,209
3
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167
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What is known about the krull dimension of an ultrapower ring?
Let $R$ be a ring, $F$ a free ultrafilter on a set $X$ which is not countably complete, and $R_F$ the ultrapower of $R$ with $R \not\cong R_F$. The following two results are from a masters thesis ...
- 6,110
2
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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{...
- 153
2
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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(\...
- 1,687
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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. ...
- 17.5k
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vote
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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 ...
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1
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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 ...
- 16.5k
1
vote
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247
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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 $S^*_\kappa$, ...
- 4,525
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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\...
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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 $i = 1,2$?
- 45.4k
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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 ...
- 2,118