# Tagged Questions

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### A basic question on Stone-Cech compactification of $\mathbb{Z}$

Can the identity isomorphism on the additive group $\mathbb{Z}$ be extended to a non-identity semigroup isomorphism on $\beta\mathbb{Z}$, and still preserves $\beta\mathbb{Z}\setminus\mathbb{Z}$? ...
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### Whether the result that an ultraproduct which satisfies ACCP is automatically a field generalizes to ultrafilters on larger indexing sets

Let $F$ be an ultrafilter on some set $X$, $R$ an integral domain and $R_F$ the resulting ultraproduct ring. For an element $(a)$ in the product ring of $R$ indexed by $X$, denote its equivalence ...
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### Characterization of Stone-Cech compactifications

Suppose I have an infinite discrete topological space $X$ of cardinality $\kappa$. Then I know some things about the Stone-Cech compactification, $\beta X$: it is Hausdorff and compact but not ...
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### Existence (or non) of “definable” ultrafilters

This is a question which I suspect has an absurdly easy answer, but I'm not seeing it. Let $\langle\cdot,\cdot\rangle:\omega^2\rightarrow\omega$ be your favorite pairing map (for me, this is the ...
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### Lattice of differences between ultrafilters

Consider two ultrafilters, $U$ and $V$, on the same cardinal $\kappa$. Let $D(U, V)=\lbrace X\subseteq \kappa: X\in U-V\rbrace$; clearly $D(U, V)$ is a lattice under $\subseteq, \cap, \cup$ since the ...
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### A Q-point not Ramsey

May I ask where can I read the example of a q-point which is not Ramsey. I'm especially looking for a coloring of the set of two element subsets of $\mathbb{N}$ without a homogeneous set in the q-...
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### Mathias forcing with Ramsey ultrafilters, and Cohen reals [closed]

Edit/update: One reason this question never received an answer is because it was founded on a faulty premise! The Blaszczyk-Shelah paper I mentioned below does not prove that Mathias forcing with a ...
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### Ultrafilter and contracting maps

I was trying to construct some element with specific properties in an ultraproduct and it boils down to a question which seems relatively natural but leaves me perfectly clueless. $\textbf{Question:}$...
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### An Extender is a Generalization of an Ultrafilter?

I'm not sure whether this belongs here or on math stackexchange, but I'll give a try here. I've heard that an extender is a generalization of an ultrafilter. This is not to say that an ultrafilter ...
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### Does ultrafilter have measure one?

Define a new product measure on cantor space as follows:u({0})=a,u({1})=1-a,where a$\in$(0,1/2]. Does any ultrafiter U hasn't measure one? When a=1/2,I know U hasn't measue one.I guess neither when ...
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### Ultrafilters over vector spaces

Perhaps my question is naive, but let me try. Take a (real or complex) vector space $V$ and consider an ideal $\mathcal{I}$ of subsets of $V$ with the following property (call it (*)): for each ...
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### closed set and z-ultrafilter on normal space

Let $X$ be a completely regular, Hausdorff topological space and let $\cal F$ be a $z$-ultrafilter on $X$. Then for each zero set $W$ in $X$, either $W\in \cal F$ or there exists $Z\in \cal F$ such ...
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### 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 ...
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### 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 ...
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### 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 $P_{\kappa}(\lambda)$ ...
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### 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 ...
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### 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 ...
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### “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 ...