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 […]

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?

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 $\mathcal{F}$, $\bigcap_{i \in N}{F_{i}} \in \mathcal{F}$?

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 […]

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) […]

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 […]

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 […]

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

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, […]

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 […]

Intereting Posts

Cauchy sequence is convergent iff it has a convergent subsequence
Frobenius Norm Triangle Inequality
$x^{2000} + \frac{1}{x^{2000}}$ in terms of $x + \frac 1x$.
In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False?
Show that $(f_n)$ is equicontinuous, given uniform convergence
Reflection across a line?
The non-existence of non-principal ultrafilters in ZF
Prove for every $n,\;\;$ $\sum\limits_{k=1}^{\infty}\left\lfloor \frac{n}{2^{k}}+\frac{1}{2} \right\rfloor=n $
Can a limit of an integral be moved inside the integral?
Why is the group action on the vector space of polynomials naturally a left action?
Probability Theory – Fair dice
Geometric justification for the prime spectrum and “generic points”
Topology of the ring of formal power series
induced representation, dihedral group
Why is this inclusion of dual of Banach spaces wrong?