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2
votes
1answer
150 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 ...
5
votes
1answer
273 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 ...
8
votes
1answer
190 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 ...
4
votes
0answers
141 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 ...
5
votes
1answer
168 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 ...
2
votes
1answer
125 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 ...
10
votes
3answers
869 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 ...
1
vote
0answers
177 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 $S^*_\kappa$, ...
2
votes
1answer
195 views

Can we flex the rigid models by enough power?

Definition (1): ‎An ‎‎$‎‎‎\mathcal{L}‎$ -‎ ‎structure ‎‎$‎‎‎\mathcal{M}‎$ ‎called ‎"‎‎rigid" ‎iff ‎‎there ‎is ‎no ‎non-trivial automorphism on ‎$‎\mathcal{M}‎$.‎ ‎‎ Definition (2): ‎An ...
5
votes
1answer
142 views

Is there a truth approximation on a‎ cumulative hierarchy‎‎‎‎?

‎‎‎Note ‎to ‎the ‎following ‎well known theorem:‎ Theorem (1): ‎If ‎‎$‎‎\kappa‎‎$ ‎be a ‎‎"measurable" ‎cardinal ‎and ‎‎$‎‎‎\mathcal{F}‎$ be a‎ ‎"non-principal ‎$‎‎‎\kappa‎$-complete normal‎" ...
2
votes
0answers
65 views

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 ...
13
votes
2answers
388 views

Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?

I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..." i ...
4
votes
1answer
111 views

Galois action on ultrapowers

Let $K$ be a char $0$ field with algebraic closure $\bar K$ and absolute Galois group $G$. Let $\mathcal U$ be an ultrafilter on $\mathbb N$ and $F=\bar K^\mathbb N/\mathcal U$ be the ultrapower of ...
1
vote
1answer
197 views

When can we “displace” an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal ...
5
votes
0answers
199 views

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 ...
1
vote
1answer
170 views

an elementary substructure of a natural numbers ultrapower

Hi I'm looking for an elementary substructure of a natural numbers ultrapower with a free ultrafilter over a numerable set also must not be isomorphism between the elementary substructure and any ...
5
votes
2answers
310 views

Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter): (Woodin) Let $V[G]$ be a ...
6
votes
2answers
329 views

Ultrapowers of operators

Can we prove that for each infinite dimensional Banach space $X$ and any free ultrafilter (possibly over uncountable set of indices) $\mathcal{U}$ the obvious embedding ...
2
votes
1answer
145 views

Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets

Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence ...
10
votes
1answer
360 views

Do ultrapowers of classical Banach spaces have unconditional bases?

I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$. Since the ...
6
votes
1answer
334 views

iterating ultrapowers of C*-algebras: the Calkin algebra

Elsewhere I asked about ultrapowers of the C*-algebra $A$ of compact operators on separable infinite-dimensional Hilbert space. My question was whether the process of taking ultrapowers of ultrapowers ...
5
votes
2answers
355 views

iterating ultrapowers of C*-algebras

Let $A$ be something interesting like the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space and let $A^1$ be an ultrapower of $A$. Then $A^1$ is a primitive C*-algebra ...
5
votes
1answer
420 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 ...
5
votes
2answers
316 views

Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?

The (model-theoretic) ultrapowers had been used for studying elementary equivalnce of first-order structures. Then, they have been adapted to Banach spaces, which are, let me say, second-order ...
8
votes
1answer
369 views

Homomorphic images of a Cartesian product of finite groups

What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...
8
votes
2answers
451 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 ...
14
votes
0answers
455 views

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 ...
3
votes
1answer
154 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 ...
9
votes
4answers
746 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 ...
4
votes
1answer
277 views

What is known about the ultra-inverse limit?

Given a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ and a sequence of groups $G_i$, one can define its ultraproduct as: $$ ^*\prod_{i\in \mathbb{N}}G_i:=\{(x_i)_{i \in \mathbb{N}}| x_i\in ...
14
votes
4answers
1k views

Ultraproducts of finite cyclic groups

Let G be the (non-principal) ultraproduct of all finite cyclic groups of orders n!, n=1,2,3,... . Is there a homomorphism from G onto the infinite cyclic group?
5
votes
1answer
298 views

The closure of a generic ultrapower

Let $I$ be a normal ideal on $P_{\kappa} (\lambda)$. Let $V$ denote our ground model. Now we force with the $I$-positive sets, and if $G$ is the resulting generic filter, it can be shown that $G$ is a ...