Articles of filters

Do Ramsey idempotent ultrafilters exist?

I was studying idempotent ultrafilters when I saw that no principal ultrafilter could ever be idempotent, because $\left\langle n \right\rangle \oplus \left\langle n \right\rangle = \left\langle 2n \right\rangle$. A Ramsey ultrafilter is, by definition, never principal, so the question that arose to me was: do Ramsey idempotent ultrafilters exist? First some explanation: One of the […]

Principal ultrafilter and free filter

I have 2 questions about filter and (ultra-)filters: Which relations are there between free filter, principal filter, ultrafilter, Frechet filter, and co-finite filter? If a filter is free, does it imply that it is a principal ultrafilter?

Existence of non-trivial ultrafilter closed under countable intersection

Under what conditions on $\Omega$ does there exist $\mathcal{F} \subset \mathcal{P}(\Omega)$ such that $\mathcal{F}$ is a non-trivial ultrafilter and, for every sequence $(F_{i})_{i \in N}$ of elements of $\mathcal{F}$, $\bigcap_{i \in N}{F_{i}} \in \mathcal{F}$?

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley’s Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. The solution is very simple using the Frechet filter $\mathcal{F}:=\{A: X\setminus A \text{ finite}\}$ and defining the measure on the ultrafilter […]

Injective function and ultrafilters

An exercise left by my teacher let me think that the following statement is true: Let $\mathcal{U}$ be an ultrafilter on $\mathbb{N}$. Then every injective function $g:\mathbb{N}\rightarrow\mathbb{N}$ is $\mathcal{U}$-equivalent to an increasing function. Is it true? If yes, how can I prove it? EDIT: ok, it’s false. I wanted to prove that (1) implies (2) […]

Questions related to intersections of open sets and Baire spaces

EDIT: I have reposted this question on MathOverflow. (The version posted there is more concise, with some details omitted. I have also added a question about pseudobases with similar property.) MO copy: Intersections of open sets and $\alpha$-favorable spaces I was curious a little about the classes of topological spaces described below. For each of […]

The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that “it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone.” And in Jech (2000, p.81), he mentioned that “[i]t is known that the theorem [the Prime Ideal Theorem] cannot be proved without using […]

Arrow's Impossibility Theorem and Ultrafilters. References

I need some references (far away from Wikipedia) about the proof using Ultrafilters of Arrow’s Impossibility Theorem. Online resources are preferred.

Existence of ultrafilters

STATEMENT: An “ultrafilter” is a filter that is not properly contained in any other filter. Use Zorn’s lemma to show that every filter is contained in an ultrafilter. PROOF: Let $F$ be the set of all filters that are contained in $X$. Let $\mathscr{C}$ be a chain in F. Then let us take $\mathcal{C}’=\bigcup_{C∈\mathscr{C}}C.$ Then, […]

Tychonoff theorem (1/2)

I was trying to prove Tychonoff theorem. First I used (which I showed also): The following are equivalent (a) $X$ is compact (b) every filter of closed set $F$ on $X$ has non-empty intersection (c) every ultrafilter of closed set $F$ on $X$ has non-empty intersection Then my proof is this: Let $F$ be a […]