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9
votes
3answers
384 views

A property of an ultrafilter

Let $\mathcal U$ be a free ultrafilter on a set $X$. For $n\in\mathbb N$ let $\mathcal F$ be a family of $n$-element subsets of $X$ such that $\bigcup\mathcal F\in\mathcal U$. Question. Is there a ...
0
votes
0answers
58 views

Reference sought: maximal ideal space of $\ell_\infty$ modulo sequences that go to 0 along a filter

It is not too hard to show the following. Suppose that $\mathcal{F}$ is a non-principal filter on $\mathbb N$. Denote by $c_0^{\mathcal{F}}$ the subspace of $\ell_\infty$ consisting of sequences that ...
1
vote
1answer
76 views

Are separability and ccc equivalent for closed subspaces of $\beta N$?

Let $\beta \mathbb N$ be the Stone-Cech compactification of the integers. Then $\beta \mathbb N\setminus \mathbb N$ is non-separable because if fails the ccc condition, that is, it has an uncountable ...
7
votes
5answers
366 views

Ideals on $\mathbb N$ and large sets that have small intersection

Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable: $$A,...
3
votes
0answers
115 views

Identification of ultrafilters with measures

We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$. Now my question is which ...
3
votes
0answers
92 views

Selectors for bases of ultrafilters

If $\mathcal{U}$ is a selective (Ramsey) ultrafilter on $\omega$ and $\mathcal{B}$ is its base, then for every sequence $A_0\supsetneq A_1\supsetneq A_2\supsetneq\ldots$ in $\mathcal{B}$ there exists $...
3
votes
2answers
222 views

Selective ultrafilter on $\omega$ is normal. Clear proof

In this question I had asked about proof of the property of selective ultrafilter. As was answered, the proof is trivial if we know that ultrafilter is selective iff it is Ramsey ultrafilter. The ...
3
votes
2answers
245 views

What are the components of the Stone-Cech Remainder?

Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a ...
4
votes
0answers
183 views

Set of subsequences with the same ultrafilter limit of the original sequence

Let $\mathscr{U}$ be a free ultrafilter on the positive integers $\mathbf{N}$ and fix $U \in \mathscr{U}$ such that $U$ is not cofinite (thanks J.D.Hamkins for the correction.) Consider the natural ...
22
votes
2answers
929 views

Ultrafilters and diagonal arguments

Is there a diagonal argument to show that if $x$ is infinite then ${\cal P}(x)$ (the power set of $x$) is smaller than $\beta x$ (the set of ultrafilters on $x$)? (Added later. I tried commenting ...
8
votes
0answers
200 views

Is there a 'local' version of Near Coherence of Filters?

The axiom Near Coherence of Filters (NCF) is known to be independent of ZFC. Axiom (NCF I): For any two free ultrafilters $\mathcal D$ and $\mathcal E$ on $\mathbb N$, there exist finite-to-one ...
5
votes
1answer
283 views

Permutation on $\omega$ and Ramsey ultrafilter

let $\pi:\omega\to\omega$ be permutation and $\mathcal{F}$ is Ramsey selective ultrafilter on $\omega$. There are uncountable many growing increasing subsequences of $\pi$. Can one proof that one of ...
7
votes
0answers
178 views

Possible cofinalities of cuts of ultraproducts

Suppose $\kappa$ is a regular cardinal, $\bar{\mu}=(\mu_i: i<\kappa)$ is an increasing sequence of regular cardinals ($>\kappa$) and set $pcut(\bar \mu)=\{ (\lambda_1, \lambda_2):$ for some ...
3
votes
1answer
129 views

The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder

If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
4
votes
1answer
105 views

“Gaps” in the Rudin-Keisler ordering

If $(P,\leq)$ is a poset and $p\in P$, then we say that $p$ is the lower part of a gap there is $q \in P$, $q>p$ such that $[p,q] = \{p,q\}$. (This is equivalent to the statement that $(\uparrow p) ...
3
votes
1answer
200 views

Infima in the Rudin-Keisler ordering

Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by $${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
2
votes
1answer
163 views

Ultraproduct of metric spaces

Let $I$ be a set and $\mathcal{U}$ be an ultrafilter on $I$. Suppose that $(X_{i}, d_{i})_{i\in I}$ is a family of pointed metric spaces with a distinguished point $e_{i}$ for each $i\in I$. We set $$...
7
votes
1answer
194 views

Are free ultrafilters as posets product-irreducible?

Let $\kappa\geq \aleph_0$ be a cardinal, and suppose that ${\cal U}$ is a non-principal ultrafilter on $\kappa$. We regard ${\cal U}$ as a poset $({\cal U}, \subseteq)$. Suppose that there are posets ...
7
votes
1answer
206 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 ...
2
votes
1answer
68 views

Products of spaces with an underlying free ultrafilter as topology

If ${\cal U}$ is any ultrafilter on $\omega$, the pair $(\omega,{\cal U}\cup \{\emptyset\})$ is a connected topological space. Is there a non-principal ultrafilter ${\cal U}$ on $\omega$ such that we ...
3
votes
1answer
122 views

Cardinality of a set of pairwise non-order-isomorphic ultrafilters on $\omega$

It is well known that there are $2^{2^{\aleph_0}}$ many non-principal ultrafilters on $\omega$. Is there a set ${\frak U}$ of non-principal ultrafilters on $\omega$ with $|{\frak U}| = 2^{2^{\aleph_0}}...
5
votes
1answer
100 views

Extending ground model ultrafilters

Work over a model of $\sf GCH$. Suppose that $\kappa$ is some regular cardinal, and $U$ is a uniform ultrafilter on $\kappa^+$. Does $U$ have some canonical extension after forcing with $\...
3
votes
1answer
236 views

Existence of maximal analytic P-ideal

An ideal $\mathcal{I}$ on the positive integers $\mathbf{N}$ is a P-ideal if for every sequence $(A_n)$ of sets in $\mathcal{I}$ there exists $A \in \mathcal{I}$ such that $A_n\setminus A$ is finite ...
8
votes
0answers
129 views

Non-wellorderable ultrafilters with wellorderable bases

There are some models in which $2^\omega$ is not wellorderable but there is a free ultrafilter over $\omega$. What about the consistency of: $2^\omega$ is not wellorderable + AC for countable sets of ...
1
vote
1answer
247 views

Defining cones and Turing cones

In Set Theory Jech defines a cone to be a subset of the Baire Space $\mathcal{N}$ of the form $$\operatorname{cone}(x_0)= \{x : x_0 \in L[x]\}$$ where $x_0 \in \mathcal{N}$. Jech then defines the ...
6
votes
1answer
319 views

$\operatorname{AD}$ and the measurability of $\omega_1$

Are there proofs of the measurability of $\omega_1$ (under $\operatorname{AD}$) that do not use Turing degrees nor the $\Sigma_1^1$ boundedness lemma? I've been struggling to find an "elementary" ...
1
vote
1answer
163 views

can every Gromov-Hausdorff limit be realized as a metric ultralimit?

Let $(X_i)$ be a countable collection of bounded metric spaces, pre-compact in the Gromov-Hausdorff metric. It is well-known that for any choice of non-principal ultrafilter $U$ on $\mathbb N$, the ...
2
votes
1answer
144 views

Notion of non-selectivity of ultrafilters

A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is called selective (or Ramsey) if for every partition $\mathcal{P}$ of $\omega$ disjoint with $\mathcal{U}$ there is $A\in\mathcal{U}$ such that $...
8
votes
2answers
293 views

Ultrafilters preserved by $\mathbb{P}$ but not by products?

Let $U\in V$ be an ultrafilter on $\omega$. We say $U$ is preserved under forcing with $\mathbb{P}$ if $\Vdash \forall x\subset \omega \ \exists Z\in U \ Z\subset x \vee Z\subset x^c$. In other words,...
9
votes
2answers
470 views

Extracting subsequences in Banach spaces, along an ultrafilter?

There are various principles in Banach space theory that allow one to pass from a given sequence of vectors $(x_n)$, to a subsequence $(x_{n_k})$ with some desired property. I'm thinking here, in ...
5
votes
0answers
137 views

Preservation of Baumgartner's I-ultrafilters under various forcings

For $I\subset \mathcal{P}(2^\omega)$, an ultrafilter $U$ on $\omega$ is said to be an I-ultrafilter if for all $f:\omega \to 2^\omega$, there exists $A\in U$ such that $f''[A]\in I$ [Baumgartner]. In ...
3
votes
2answers
176 views

What are the minimal requirements for the definable hyperreal field plus transfer?

It is interesting that to prove the transfer principle for the definable hyperreal field, one requires no more choice than for proving, for instance, the countable additivity of the Lebesgue measure. ...
5
votes
1answer
159 views

Can There be Rudin-Keisler Immediate Sucessors?

There are several well-studied orderings on the set $\omega^*$ of ultrafilters on the natural numbers. Three popular ones are $\le_i$ for $i = 1,2,3$. We define $\mathcal U \le_i \mathcal V$ to mean ...
10
votes
3answers
403 views

Description of $\big(\ell^\infty(\mathbb N)\big)^{\!*}$ via ultrafilters

Let $\beta\mathbb N$ is the set of ultrafilters on $\mathbb N$ and $\mathscr F\in\beta\mathbb N$. Assume that $l_{\mathscr F}\in\big(\ell^\infty(\mathbb N)\big)^{\!*}$ is the functional which assigns ...
2
votes
1answer
133 views

Representation of the elements of $c_0^\perp$ as integrals over ultrafilters

Let $$ X=\big\{\varphi\in\ell_\infty^{\,*}(\mathbb N) : \varphi(\{a_n\})=0\,\,\text{whenever $a_n\to 0$}\big\}. $$ If $\varphi_{\mathscr F}(\{a_n\})$ is the limit of $\{a_n\}$ with respect to the non-...
6
votes
0answers
141 views

Does Łoś's theorem imply choice given a free ultrafilter?

In the paper "Łoś's theorem and the boolean prime ideal theorem imply axiom of choice" Howard has shown that Łoś's theorem and the boolean prime ideal theorem imply axiom of choice. At the end of the ...
3
votes
1answer
262 views

When can you canonically extend an ultrafilter after forcing?

Suppose that $V$ is a model of $\sf ZFC$, and fix some regular $\kappa$, say $\omega_1$ for practical purposes. Let $\cal U$ be an ultrafilter on $\omega_1$ in $V$ which is non-principal and even ...
8
votes
0answers
304 views

What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) (...
9
votes
0answers
293 views

Embedding $\beta\mathbb{N}$ into a product of Cantor sets

Let us consider $\beta\mathbb{N}$, the Stone-Čech compactification of the natural numbers (where we do not take $0$ to be a natural number, so the only idempotent elements are nonprincipal ...
4
votes
0answers
170 views

Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let $(M_i,p_i)$ be a sequence of $n$-dimensional Riemannian manifolds with lower Ricci curvature bound $-1$. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there ...
6
votes
1answer
201 views

Comparing cardinalities of the spectrum of two masas in $B(H)$

If I imagine that (the self-adjoint part of) a C*-algebra $A$ represents the algebra of observables of some quantum system, then certain perspectives on algebraic quantum theory would ask me to ...
9
votes
0answers
156 views

Is it possible that all ultrafilters are determined by the meet-semilattice of sub-ultrapowers?

Suppose that $\mathcal{Z}$ is a filter on a set $X$. Let $\Pi(X)$ denote the lattice of all partitions of the set $X$. Then $(\Pi(X),\wedge)$ is a meet-semilattice where $P\wedge Q=\{R\cap S|R\in P,S\...
-1
votes
1answer
199 views

Lowering from filters to ultrafilters for an infinitary relation

Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$). I denote $\mathcal{L}\in \upuparrows f \...
-2
votes
1answer
240 views

Expressing a value related to an infinitary relation through ultrafilters

Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and ...
4
votes
1answer
147 views

Ultrafilters of weight $\aleph_2$ in Sacks model

It is well-known that in Sacks model there are P-points and even Ramsey ultrafilters, but what the usual (i.e. findable in the literature) proofs for these facts do is proving that ground model P-...
6
votes
1answer
379 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
308 views

Preservation of ultrafilters by Sacks forcing

It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ...
2
votes
1answer
224 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\...
2
votes
3answers
343 views

Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure. Q1. Is there any important notion of structure on an ultrafilter? Q2. Is there any non-trivial notion of structure on ...
1
vote
0answers
228 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 $S^*_\kappa$, ...