3
votes
1answer
45 views

Is this an equivalent characterization of rapid filters?

A filter $\mathcal F$ on $\omega$ is called rapid filter if for every function $f\in\omega^\omega$ there exists $X\in\mathcal F$ such that $|X\cap f(n)|\le n$ for $n\in\omega$. In Lemma 4.6.2 in the ...
0
votes
0answers
21 views

Choosing from two index families of sets

Let $a$ be a (nontrivial) ultrafilter and $n$ be an infinite set. Let also $U$ be an infinite set. Define $n$-ary relation $\phi$ on $\mathscr{P}U$ by the formula $L\in\phi \Leftrightarrow \forall ...
3
votes
1answer
49 views

Unique ultrafilter on $\omega$

We know that from axiom of choice (or just BPIT) we can deduce ultrafilter lemma, which states that every filter can be extended to an ultrafilter. From this lemma we can derive existence of at least ...
1
vote
1answer
26 views

Splitting Frechet filter into two proper filters

Let $\Omega$ be a Frechet filter (=cofinite filter) on an infinite set. Do there exist proper filters $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}\cap\mathcal{B} = \Omega$ and ...
6
votes
1answer
98 views

Building normal filters around a stationary set

Recently I've been looking at connections between Laver functions on large cardinals and diamonds. While $\diamondsuit$-like principles tend to readily generalize to Laver function-like concepts, I've ...
2
votes
1answer
27 views

Equivalent definitions of P-Points (Ultrafilters)

It is stated in lots of places (including this one) that these two are equivalent definitons for an ultrafilter $u$ to be a p-point. If for every sequence $\left < A_n: n \in \omega \right>$ ...
4
votes
2answers
60 views

Different definitions of P-Points (ultrafilters)

Recently I have been exposed to the concept of P-Points. I have three definitions: A non-principal ultrafilter $u$ is a P-Point if: For every sequence $\left < A_n \right >_{n\in \omega}$ of ...
0
votes
0answers
39 views

An other question about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Conjecture: For every ...
0
votes
1answer
37 views

A problem about filters and ultrafilters

Let $a$ be an ultrafilter on a set $\mho$. Let $\mathcal{L}$ be an $a$-indexed family of filters (that is $\mathcal{L}$ is a function from $a$ to the set of filters on $\mho$). Are the following two ...
0
votes
2answers
28 views

Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...
3
votes
2answers
61 views

How generic ultrafilter is used in forcing

So I just learned what ultrafilter is and generic filter is. As all maths are for beginners, it just looks like pure concepts, and I don't see how they are going to be applied in forcing. I am looking ...
1
vote
1answer
31 views

What does it mean for ultrafilter to be $\kappa$-complete?

What does it mean when ultrafilter is said to be $\kappa$-complete? I cannot find suitable Internet resource, so I am asking here.
0
votes
0answers
39 views

A special filter on cartesian product of sets

The following is inspired by this article in nLab (in attempt to simplify it using my notions of funcoids and reloids, which notation is however outside of the scope of this question). Fix a set $U$. ...
2
votes
0answers
27 views

A totally bounded uniformity and certain filters

Conjecture For every totally bounded uniform space $(U;F)$ and filters $\mathcal{X}$, $\mathcal{Y}$ on $U$ such that $$\forall E\in F,X\in\mathcal{X},Y\in\mathcal{Y}:(X\times Y)\cap E\ne\emptyset,$$ ...
0
votes
0answers
36 views

A relation between certain families of filters and filters on a cartesian product of sets

(By filters, I will mean all filters on a set, including the improper filter.) The product $\mathcal{A}\times\mathcal{B}$ of two filters is the filter defined by the base $\{ A\times B \,|\, ...
2
votes
1answer
47 views

Normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$

I am looking at fine normal ultrafilters on $\mathcal{P}_{\kappa}(\lambda)$ and I have seen that there are two (seemingly) different formulations of normality, in terms of regressive (sometimes also ...
2
votes
1answer
65 views

Countable intersection on an ultrafilter

Does there exist a nontrivial ultrafilter $a$ such that there are no sequence of sets $K_0,K_1,K_2\dots\in a$ such that $$K_0\cap K_1\cap K_2\dots = \emptyset?$$
4
votes
2answers
88 views

How to show that a free ultrafilter cannot have an infinite pseudointersection?

The following text is a quote from p.180 of Halbeisen's book Combinatorial Set Theory. This book is also available on website of a course taught by the author. (As mentioned in Asaf's comment, it is ...
5
votes
1answer
115 views

Injective or constant function on measurable cardinal

I'm working on the following question: Let $\kappa$ regular cardinal, and $f:\kappa \rightarrow \kappa$. Prove that $\{\alpha < \kappa\mid f|_\alpha : \alpha \rightarrow \alpha\}$ is a ...
8
votes
2answers
86 views

Proof of non-existence of non-principal $\kappa$-complete ultrafilter

Let $\lambda$ be a cardinal. I would like to prove that for all cardinals $\lambda < \kappa \leq 2^\lambda$, there can't be a $\kappa$-complete non-principal ultrafilter on $\kappa$. Here is my ...
6
votes
1answer
91 views

If a filter has a unique ultrafilter extending it, then it is that ultrafilter (prove without $\sf{AC}$)

I am not certain if $\sf AC$ (or more conservatively, $\sf UF=$ there is an ultrafilter extending any given filter) is necessary to prove the following statement: For filters $F,G$ with $\bigcup ...
5
votes
1answer
54 views

What is the cardinality of an element of an free ultrafilter?

Let $U$ be a free ultrafilter on a set $X$. I want to prove that the cardinality of every element $u\in U$ is equipotent to $X$. Is that true? Or does it lack some hypothesis?
2
votes
1answer
65 views

Ramsey ultrapowers: any interesting properties?

A Ramsey ultrafilter is a free ultrafilter $D$ on $\omega$ such that for any partition of $\omega$ into subsets, none of which is in $D$, there exists a set in $D$ that intersects each member of the ...
7
votes
1answer
57 views

Existence of a specific type of ultrafilter

Does there exist an ultrafilter $\omega$ on $N$ with the following property: $\forall A \in \omega$ we have that $kA \in \omega$ for a fixed $k \in N$. Where $kA = \{ka : a \in A\}$. I do not ...
3
votes
1answer
122 views

Uniform normal filter on $\kappa$ is $\kappa$-complete and contains the club filter.

The question reads: Edit: The definition for uniform filter I was given is as follows: A filter $\mathcal{F}$ on $\kappa$ is uniform iff $\mathcal{F}$ contains all co-bounded sets in $\kappa$ iff ...
2
votes
2answers
77 views

Prove the Following Property of an Ultrafilter

In her text Introduction to Modern Set Theory , Judith Roitman defined a filter of a set $X$ as a family $F$ of subsets of $X$ so that: (a) If $A \in F$ and $X \supseteq B \supseteq A$ then $B \in ...
1
vote
1answer
99 views

Show that $X$ is separable.

I'm working through Kunen's Set Theory and I'm not sure how to proceed on part of one exercise. Let $X$ be compact Hausdorff and $\mathbb{O}_X$ be the poset of nonempty open sets of $X$ ordered by ...
1
vote
1answer
83 views

Definition of Rudin-Keisler ordering

According to Wikipedia, we can define an order on ultrafilters. However, it seems the definition has an error. It says that: If $U$ and $V$ are ultrafilters on $S$ and $T$, respectively, then ...
2
votes
2answers
109 views

Usage of Filters and Ultrafilters [duplicate]

I don't know why we need the concept of Filters and Ultrafilters. they just seem nothing, and I don't know where to use them can you tell me where do we use those concepts.
2
votes
1answer
135 views

Help with definition: partition mod ultrafilter.

I do not see how the partition is well-defined. By definition $A\neq\varnothing\mbox{ mod }D\iff A\notin I_{D}$. Since D is a maximal filter $A\notin I_{D}\iff A\in D$ . So ...
8
votes
1answer
172 views

If $\kappa$ is measurable, does there exist a normal measure on $\mathcal P_{\kappa}(\kappa)$?

I'm trying to do exercise $10.7$ of Jech's Set Theory: If $\kappa$ is a measurable cardinal, then there exists a normal measure on $\mathcal P_{\kappa}(\kappa)$. For a set $A$, with $|A|\geq ...
4
votes
2answers
215 views

Cardinality of an ultrafilter

If $\mathscr F$ is an ultrafilter on an infinite set $M$, then it can be shown that $|\mathscr F|=2^{|M|}$. We know that for each subset $A\subseteq M$ we have $A\in\mathscr F$ or $M\setminus A \in ...
1
vote
2answers
87 views

Problem on filters

$\mathcal{F}$ is filter on $\mathcal{I}$, but not ultrafilter. Prove that $\exists$$\mathcal{X, Y}\notin\mathcal{F}$ | $\forall\mathcal{Z}\in\mathcal{F}$ $\mathcal{X}\cap\mathcal{Z}\neq$ ...
25
votes
1answer
610 views

Showing a filter on the Power set of $\mathbb{Z}$ is a one point Filter

Let $\mathcal{P}_0(X)$ the Power set of $X$ without the empty set and let $\dot{x}:=\{A\subseteq X: x \in A\}$ the one point filter generated by $x$. Furtermore let $$ \mathcal{A} := \{ f \in ...
1
vote
1answer
34 views

A follow up question on completeness of filter (generalised to p.o.s)

Now I'm looking at the following generalisation of $\kappa$-closedness of a filter on a set to a filter in a partial order: Let $\kappa$ be a regular uncountable cardinal. A partial order $P$ is ...
4
votes
1answer
68 views

Question about definition of $\kappa$-completeness of filter

I am looking at the following definition: Let $\kappa$ be a regular uncountable cardinal and let $\mathcal F$ be a filter on a non-empty set $X$. We say that $\mathcal F$ is $\kappa$-complete if ...
1
vote
1answer
114 views

ultrafilter $\kappa$-complete

Let $A$ an infinite set, $D$ an ultrafilter on $A$ and $\kappa$ an infinite cardinal. I want to show the following : $D$ is $\kappa$-complete iff $\forall\tau<\kappa$ and $\forall$ partition ...
5
votes
1answer
134 views

Ultrafilter on $\omega$

(Show that)If $\mathscr{U}$ is an ultrafilter on $\omega$, then $\mathscr{U}$ contains a subset $A$ of lower density zero. (But) there is an ultrafilter on $\omega$ such that every $A \in ...
4
votes
1answer
86 views

Why is a p-point never the limit of a countable set of non-principal ultrafilters?

There are several descriptions, but I think the following definition of p-point suits for now: A point such that the intersection of countably many neighbourhoods of it, is again a neighbourhood. ...
5
votes
2answers
199 views

Subset of a P-ideal need not be a P-ideal

I was looking for examples showing that subset of a P-ideal is not necessary. I will post below a counterexample I was able to find. (I hope it is correct.) But I'd be glad to see other simple (or ...
3
votes
2answers
100 views

p-point ultrafilters on a countable linear order

Suppose that $(P,\leq)$ is a countably infinite linear ordering, and $U$ is a p-point ultrafilter on $P$. Show that there is an $X\in U$ such that $X$ has order type $\omega$ or $\omega^*$. (This is ...
4
votes
1answer
103 views

Do Ramsey idempotent ultrafilters exist?

I was studying idempotent ultrafilters when I saw that no principal ultrafilter could ever be idempotent, because $\left\langle n \right\rangle \oplus \left\langle n \right\rangle = \left\langle 2n ...
2
votes
1answer
72 views

Where in this argument ultrafilter is used?

http://en.m.wikipedia.org/wiki/Dimension_theorem#section_1 Let's first not assume any choice principle. Let $V$ be a vector space over a field $F$ and $\beta_1,\beta_2$ be bases for $V$. Suppose ...
1
vote
1answer
91 views

Ultra filters are countably complete?

I am reading friends lecture notes. It says "Let $F$ be an ultra filter on a set $X$ such that $F$ is not countably complete. Then there are $Y_n \in F, n \in \omega$ such that $\bigcap_n Y_n = ...
2
votes
1answer
70 views

Uniform ultra filters

I'm trying to show if $X$ is infinite then every uniform filter $F$ on $X$ is contained in a uniform ultra filter $G$ on $X$. A filter is uniform if all the sets in it are of same size. My thoughts: ...
1
vote
2answers
89 views

What does it mean that $f$ is unbounded modulo $D$?

Given an ultrafilter on $\omega$ and a function $f:\omega\longrightarrow\omega$, what does it mean that $f$ is unbounded modulo $D$? Thanks
4
votes
1answer
70 views

Problem with definition of Rudin-Keisler equivalence

I'm trying to do exercise 7.11 of Jech's "Set Theory": If $D$ and $E$ are ultrafilters on $\omega$, then $D\leq E$ and $E\leq D$ implies that $D\equiv E$, where $\leq$ is the Rudin-Keisler ordering, ...
2
votes
2answers
243 views

Using Zorn's Lemma

Background: I am trying to use Zorn's lemma to show the existence of ultrafilters containing an arbitrary filter on a set $X$. My argument goes as follows: Let $\mathcal{F}_0$ be a filter on $X$. If ...
6
votes
1answer
71 views

Does $\beta \mathbb N$ embed into $\beta \mathbb N \setminus \mathbb N$?

Is there a clopen subset of $\beta \mathbb N \setminus \mathbb N$ homeomorphic to $\beta \mathbb N$? If so, is there any plausible description of any such a subspace?
3
votes
1answer
174 views

Self-similarity in ultrafilters over N

First, some notation: Set variables, $X, Y$, range over sets of natural numbers, $\mathbb{N}={1,2,3,..}$. Square brackets represent sets of natural numbers based on a formula. ...