Questions tagged [ultrapowers]
The ultrapowers tag has no usage guidance.
78
questions
0
votes
0
answers
74
views
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\...
- 1
1
vote
1
answer
212
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 ...
- 1,258
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 ...
- 225k
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 ...
- 19.1k
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 ...
- 18.1k
11
votes
4
answers
1k
views
Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hierarchy
Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\...
- 9,312
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 ...
- 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 ...
- 60.2k
7
votes
1
answer
594
views
A question on ultrapower
Suppose $\kappa_0$ is a measurable cardinal and $\mu_0$ is a normal measure on $\kappa_0$. $M_1$ is the transitive collapse of $Ult(V,\mu_0)$, $j_{0,1}:V\rightarrow{M_1}$ is the elementary embedding ...
- 225k
5
votes
1
answer
378
views
Representation of elements in ultrapowers
Maybe this is an elementary question.
Suppose that $U$ is a non-principal $\kappa$-complete ultrafilter on $\kappa$ and consider the standard ultrapower $M\cong \textrm{Ult}_U(V)$ along with the ...
- 225k
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). ...
- 19.1k
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{...
- 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}(...
- 4,301
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 $\omega$-...
- 43.7k
14
votes
2
answers
647
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}...
- 225k
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 $\...
- 19.1k
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 $i = 1,2$?
- 45.4k
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,\...
- 2,367
12
votes
4
answers
1k
views
ultrapowers and higher order logic
One of the reasons I think ultrapowers are interesting is the following corollary of Łoś's theorem:
Let $V$ be a relational structure and $^*V$ an ultrapower of $V$. Then a first order statement ...
- 19.1k
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 $Gal(E/F)$?
...
- 27.4k
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 ...
- 2,679
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 ...
- 2,367
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$ ...
- 7,674
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 ...
- 4,336
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$...
- 7,674
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 ...
- 4,618
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 ...
- 4,618
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.,...
CommunityBot
- 1
10
votes
3
answers
814
views
Is every field extension of an ultrafield an ultrafield?
Let $K=\lim(K_{i})$ be an ultrafield (over a non-principal ultrafilter), and let $K\hookrightarrow K'$ be a field extension of $K$.
When the field $K'$ is finite over $K$ it is also an ultrafield by ...
- 19.3k
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 ...
- 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}...
- 7,674
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 ...
- 5,192
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 ...
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 &...
- 10.3k
5
votes
2
answers
529
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 ...
CommunityBot
- 1
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 ...
- 18.5k
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 ...
- 2,118
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^...
- 18.5k
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 ...
- 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 ...
- 11.3k
8
votes
1
answer
484
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 ...
- 19.1k
-1
votes
1
answer
472
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 $...
- 27.8k
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 $...
- 25.5k
4
votes
1
answer
241
views
cardinality of local bases in the non-standard reals
Given a index set $S$ together with a ultrafilter $\mu$ on $S$ (such that no set of cardinality $< |S|$ has measure $1$). Let the ordered field $\mathbb{R}(S,\mu)$ denote the ultrapower of $\mathbb{...
CommunityBot
- 1
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(\...
- 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 ...
- 11.3k
7
votes
2
answers
763
views
An Extender is a Generalization of an Ultrafilter?
I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here.
I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
- 1,182
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 ...
- 11.9k
3
votes
1
answer
327
views
Is an ultrapower of a faithful Banach algebra always faithful?
Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)_{\mathcal U}$ faithful?
- 11.3k
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. ...
- 17.5k