The ultrafilters tag has no wiki summary.

**0**

votes

**2**answers

263 views

### ultrafilters' succession

hi
I'n looking for a increasing and bounded ultrafilters' succession in natural numbers with Rudin-Keisler order, actually I need to prove there is that succession the idea is
$U_1,U_2,....$ with ...

**5**

votes

**0**answers

313 views

### Boolean Prime Ideal Theorem and non-principal ultrafilters

Somewhat related to my other question Existence of non-principal ultrafilters on sets,
is it known whether it is consistent with ZF that every infinite set has a free (non-principal) ultrafilter, but ...

**6**

votes

**1**answer

455 views

### Existence of non-principal ultrafilters on sets

Is it known to be consistent with ZF that there is no non-principal ultrafilter on any infinite set? (Feel free to use your favorite interpretation of "infinite" in this context.
If infinite just ...

**4**

votes

**2**answers

250 views

### Are there q-filters which are not ultrafilters?

I have just read that a selective ultrafilter must necessary be an ultrafilter. Is this also true for q-filters?
Im not sure if using CH for instance, we can follow the construction of a q-point but ...

**4**

votes

**2**answers

829 views

### Are all countable, nonstandard models of arithmetic given by ultrapowers?

Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...

**2**

votes

**2**answers

699 views

### Product of ultrafilters, is it an ultrafilter?

Let $a$ and $b$ are filters. The product $a\times b$ is defined as the filter (on the set of pairs) induced by the base $\{ A\times B | A\in a, B\in b \}$.
It is simple to show that product of a ...

**9**

votes

**3**answers

724 views

### A unique ultrafilter extending a union of filters?

Original Question:
Let $\mathcal{P}(\omega)/fin$ denote the Boolean algebra formed from $\mathcal{P}(\omega)$ by modding out by the ideal $fin$ of finite subsets of $\omega$. As a first pass at the ...

**1**

vote

**0**answers

205 views

### Defining filters in closure algebras: reference request

A closure algebra C is a boolean algebra B together with a unary closure operator, and additional axioms, the Kuratowski axioms, that the closure operator must satisfy. (The Wikipedia article prefers ...

**19**

votes

**3**answers

1k views

### An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request

There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...

**7**

votes

**2**answers

370 views

### Destroying the P-filter-property

It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., Shelah, Proper and Improper Forcing, VI.5)
On the other hand, if we ...

**0**

votes

**1**answer

293 views

### About ordering and equivalence of filters

Let $U$ is a set. I will speak about filters on this set.
If $f$ is a function and $a$ is a filter then I define $f \left[ a \right]$ as
the filter whose base is $\lbrace f[A] | A \in a \rbrace$.
I ...

**16**

votes

**2**answers

584 views

### How “much” does (Grigorieff) forcing destroy an ultrafilter?

Introduction. I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction ...

**10**

votes

**4**answers

802 views

### Is every p-point ultrafilter Ramsey?

A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is a p-point (or weakly selective) iff for every partition $\omega = \bigsqcup _{n < \omega} Z_n$ into null sets, i.e each $Z_n \not \in ...

**8**

votes

**6**answers

760 views

### Spaces of filters

This question arose more from curiosity than from an actual problem. There are situations when you embed some space $X$ in a set of filters on $X$, which inherits properties of $X$ or has even better ...

**5**

votes

**1**answer

309 views

### Does a generic normal measure extend the club filter?

This question is related to this one. The setup is as follows:
In $V$, $\kappa$ is supercompact and $\mathbb{P} = \mathrm{Coll}(\kappa, \aleph_2)$ is the Levy collapse. $G$ is $(V,P)$-generic, and ...

**4**

votes

**1**answer

300 views

### Normal measures on $P_{\kappa }(\lambda )$ extend the club filter

This is (a variation on) exercise 20.4 in Jech's "Set Theory." Let $j : V \to M$ witness $\lambda$-supercompactness of $\kappa$, and consider the normal measure $U$ on $P_{\kappa}(\lambda)$ ...

**4**

votes

**1**answer

678 views

### Questions on ultrafilters

Hi, I know that there are models in ZFC where don't exist p-points, but I can't find (neither on internet) a proof that I understand, some of you could help me please?
Another question: I know that ...

**6**

votes

**3**answers

670 views

### Reference Request: Independence of the ultrafilter lemma from ZF

I'm looking for references for the following facts concerning the ultrafilter lemma (~ "there exist non-principal ultrafilters"):
The ultrafilter lemma is independent of ZF.
ZF + the ultrafilter ...

**10**

votes

**2**answers

552 views

### “Probabilistic ultrafilters?”

A naive question.
Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$.
Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...

**0**

votes

**3**answers

723 views

### Is the Rudin-Keisler order of ultrafilters linear?

Is the Rudin-Keisler order of ultrafilters linear?
Is it a well ordering?

**3**

votes

**1**answer

157 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 ...

**7**

votes

**3**answers

718 views

### Construction of a maximal ideal

Hello,
Let R denote the ring of continuous functions defined on the real line, let I in R be the ideal consisting of functions with compact support. Obviously, I is not maximal, and by Zorn's Lemma ...

**7**

votes

**2**answers

1k views

### Direct construction of the Stone-Čech compactification using ultrafilters?

If $X$ is a set (regarded as a discrete space), its Stone-Čech compactification can be identified with the set of ultrafilters on $X$ with its natural (Stone) topology. If $X$ is a general ...

**4**

votes

**1**answer

365 views

### Which properties of ultrafilters on countable sets hold for filters in general?

Background/motivation: I'm investigating the construction of models for a first-order modal system (S5) as products of classical models. Since ultraproducts are all classical models and I need ...

**4**

votes

**1**answer

278 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 ...

**9**

votes

**10**answers

2k views

### Are nets and filters useful in geometry and topology?

Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...

**3**

votes

**4**answers

551 views

### How much are reduced powers different?

Given two infinite sets $X$ and $I$, and a filter ${\cal F}$ on $I$, one defines as usual the equivalence relation $\approx_{\cal F}$ on $X^I$ and obtains the reduced power $Y = X^I / \approx_{\cal ...

**20**

votes

**8**answers

2k views

### Connections between ultrafilters in topology and logic

I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is ...

**12**

votes

**2**answers

833 views

### Ultrafilters arising from Keisler-Shelah ultrapower characterisation of elementary equivalence

In model theory, two structures $\mathfrak{A}, \mathfrak{B}$ of identical signature $\Sigma$ are said to be elementarily equivalent ($\mathfrak{A} \equiv \mathfrak{B}$) if they satisfy exactly the ...

**1**

vote

**3**answers

288 views

### Ultrafilters containing the image of a filter

Suppose $f:X \to Y$ is a map of sets and $F$ a filter on $X$ such that its image filter is contained in an ultrafilter $G$ on $Y$. Can I find an ultrafilter $H$ on $X$ whose image is $G$?
If this ...

**120**

votes

**0**answers

10k views

### Ultrafilters and automorphisms of the complex field

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that ...

**2**

votes

**1**answer

603 views

### Ultrafilters containing a principal filter

If X is a set and A is a subset of X containing at least two elements, then certainly for any element $a \in A$, the principal ultrafilter of $a$ contains the principal filter of A (which is NOT an ...

**24**

votes

**2**answers

3k views

### “Transitivity” of the Stone-Cech compactification

Let $\beta \mathbb{N}$ be the Stone-Cech compactification of the natural numbers $\mathbb{N}$, and let $x, y \in \beta \mathbb{N} \backslash \mathbb{N}$ be two non-principal elements of this ...

**3**

votes

**2**answers

239 views

### Behavior of externally-infinite elements in ultrapowers of $\langle HF,\epsilon\rangle$

Consider the structure $\langle HF,\epsilon\rangle$ (the hereditarily finite sets with the epsilon-relation). An ultrapower of this structure will have externally-infinite elements -- elements not ...

**11**

votes

**3**answers

2k views

### Non-principal ultrafilters on ω

I thought I had heard or read somewhere that the existence of a non-principal ultrafilter on $\omega$ was equivalent to some common weakening of AC. As I searched around, I read that this is not the ...

**25**

votes

**3**answers

2k views

### product of all F_p, p prime

Let $R$ be the ring $$R = \prod_{p\ \text{prime}} \mathbb{F}_p$$ where $\mathbb{F}_p$ is the field having $p$ elements.
Is it true that $R$ has a quotient by a maximal ideal which is a field of ...

**4**

votes

**1**answer

686 views

### How can an ultrapower of a model of ZFC be “ill-founded” yet still satisfy ZFC?

My understanding (please correct me if I'm wrong) is that if you have some transitive set M which is an $\epsilon$-model of ZFC, and you take an ultrapower of it using an approprate ultrafilter, you ...

**7**

votes

**1**answer

397 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 ...

**13**

votes

**2**answers

607 views

### What is the prime spectrum of a Cauchy series ring?

Let $k$ be a field, and let $| \ |$ be a norm on $k$. The norm induces a metric. To construct the completion $\hat{k}$ as a normed field, the standard recipe is to take the quotient of the ring ...

**3**

votes

**2**answers

302 views

### Definition modifications without choice

What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...

**8**

votes

**1**answer

511 views

### Controlling Ultrapowers

Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. ...