# Questions tagged [ultrafilters]

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### Ultrafilters of closed sets

The following definition should be standard, but let me state it just in case there is some ambiguity: If $\mathscr{L}$ is a set of subsets of a set $X$ that is closed under finite unions and ...
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### Silver-like forcing preserves p-points (reference request)

A Silver forcing "below $2^n$" is defned e.g. in Definition 7.4.11 of [Tomek Bartoszyński and Haim Judah, Set Theory: on the structure of the real line, A. K. Peters, Wellesley, 1995.]. It ...
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### Is $(\omega+1)^\omega/{\cal U}$ complete for ${\cal U}$ free ultrafilter?

Let ${\cal U}$ be a free ultrafilter on $\omega$. Is the linearly ordered set $(\omega+1)^\omega/{\cal U}$ complete?
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### Are all free ultrafilters 'the same' in some sense?

Consider the set of ultrafilters $\beta(\mathbb N)$ on $\mathbb N$. Any function $f\colon\mathbb N\to\mathbb N$ extends to a function $\beta f\colon \beta \mathbb N \to \beta\mathbb N$. We say that ...
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### Can there be no "surprisingly averageable" second-order sentences?

Say that a second-order sentence $\varphi$ is averageable iff there exists some infinite cardinal $\kappa$ and some nonprincipal ultrafilter $\mathcal{U}$ on $\kappa$ such that for every $\kappa$-...
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### "Completion property" in $(\beta\omega,+)$

Let $\beta\omega$ be collection of all ultrafilters on $\omega$ (principal and non-principal). We endow $\beta\omega$ with an operation $+$ in the following way. For ${\bf a}, {\bf b}\in \beta\omega$, ...
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### Unbounded $\omega_1$-sequence in $^*\mathbb{N}$

Let $\mathcal{F}$ be a non-principal ultrafilter on $\omega$. Let $^*\mathbb{N}$ = $\mathbb{N}^\omega/\mathcal{F}$ be an ultrapower. Let $\{n_\alpha\}_{\alpha\in\omega_1}$ be a strictly increasing ...
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I read the following result in an article. Let $X$ be a regular space. Let $\mathcal{M}$ be free closed ultrafilter on $X$. Set $\mathcal{U=}\left\{ U:U\text{ is open and there exists a }F\in \mathcal{... 12 votes 1 answer 584 views ### When do two ultrafilters yield isomorphic ultrapowers? Fix a cardinal$\lambda$$\newcommand{\cU}{\mathcal U}\newcommand{\cV}{\mathcal V}. Consider the equivalence relation on \beta\lambda given by \cU\sim \cV when for all first-order structures M ... 2 votes 1 answer 201 views ### External ultrafilters definitions I am reading a paper by Goldberg, in which he defines ultrafilter over a model of set theory (not transitive). These are the definitions: I get the definition of an M-ultrafilter, it is a real subset ... 4 votes 2 answers 264 views ### Multiplicative and additive groups of the field (\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z})/\simeq_{\cal U} Let {\cal U} be a non-principal ultrafilter on \omega, and for each n\in\omega, let p_n denote the nth prime, that is p_0 = 2, p_1=3, \ldots Next we introduce the following standard ... 2 votes 0 answers 125 views ### Ultralimit of metric spaces vs. inductive limits of underlying topological spaces Let \{(X_n,d_n)\}_{n =1}^{\infty} be a sequence of bounded metric spaces such that: X_n \subseteq X_{n+1} is a metric subspace of X_n. Let \omega denote a non-principal ultrafilter (i.e.: ... 6 votes 1 answer 151 views ### Regular limit points of possible cofinalities Let A be a non-empty set of regular cardinals such that \vert A\vert <\text{min}\ A, and \{\nu_i\mid i<i_0\}\subseteq \text{pcf}\ A be a strict increasing sequence having limit length i_0... 2 votes 1 answer 110 views ### What is the smallest cardinality of a base of an ultrafilter on \omega related to an almost disjoint family of cardinality \mathfrak c? Let (A_\alpha)_{\alpha\in\mathfrak c} be an almost disjoint family of infinite subsets of \omega. The almost disjointness of the family means that A_\alpha\cap A_\beta is finite for any ordinals ... 0 votes 0 answers 43 views ### Notation of P^+-families - bibliography searching have you ever met with notation of P^+-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd? Thank you in advance 5 votes 1 answer 267 views ### NCF, P-points, weak P-points, and cardinalities The post is a bit long, but all the questions are similar or concern the same topic. Let \omega^*=\beta\omega\setminus\omega. A well-known topological definition of a P-point (on \omega) is as ... 0 votes 1 answer 200 views ### A nonprincipal ultrafilter that is not a p-point On pg 76 of Jech's Set Theory, he proves the existence of a nonprincipal ultrafilter on \omega that is not a p-point. Given a partition \{A_n\} of \omega into \aleph_0 infinite pieces, ... 7 votes 1 answer 261 views ### Pseudo-intersections, splitting families, and ultrafilters Suppose U is a non-principal ultrafilter on \omega, and let us define \tau(U) to be the minimum cardinality of a family \mathcal{X}\subseteq U such that \mathcal{X} does not have an infinite ... 9 votes 1 answer 446 views ### Is the product of commuting ultrafilters an ultrafilter? If U is a filter on X and W is a filter on Y, their product is the filter U\times W on X\times Y generated by rectangles A\times B where A\in U and B\in W. In certain circumstances ... 6 votes 1 answer 242 views ### A question on simple P_{\aleph_2}-points This question is motivated by discussion surrounding this MO question. An ultrafilter U on \omega is a simple P_{\aleph_2}-point if it is generated by a sequence \langle X_\alpha:\alpha<\... 3 votes 0 answers 119 views ### The existence of T-ultrafilters in ZFC Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on T-ultrafilters hoping that somebody on MO can solve it. First I recall the necessary ... 7 votes 1 answer 388 views ### Ultrafilter on the ordinal \omega^\omega For any ultrafilter \mathcal{U} on \omega and any finite k we can construct tensor power \mathcal{U}^{\otimes k} which is ultrafilter on \omega^k. Does there exist some natural extension of ... 3 votes 1 answer 210 views ### The intersection of all normal ultrafilters on a measurable cardinal Suppose \kappa is a measurable cardinal. Let W be the intersection of all normal ultrafilters on \kappa. I am interested in a precise characterization of the filter W. One sure way to ... 5 votes 1 answer 183 views ### Club filter basis in \omega_1 My question is about existing of basis of club filter club(\omega_1) with cardinality c. Does it exist? 3 votes 1 answer 186 views ### Ultrapower of amenable group Let \Gamma be an amenable group. Consider its ultrapower ^*\Gamma. It is known that ^*\Gamma need not be amenable. In fact, there is a stronger notion of uniform amenability for \Gamma (... 1 vote 1 answer 123 views ### Trying to construct the ultrafilter 2-monad on \mathbf{Cat} By which I mean, following Bôrger's paper Coproducts and Ultrafilters, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed. So far, what I have is, ... 12 votes 0 answers 184 views ### What is the "right" notion of exponentiation in \beta \mathbb N? The Stone–Čech compactification \beta \mathbb N of the positive integers has extensive applications in combinatorial number theory. A feature of \beta \mathbb N that makes these applications ... 1 vote 1 answer 109 views ### Non-existence of countable base of arbitrary ultrafilter \scr{B} is the base of a nonprincipal ultrafilter \scr{U} on \omega if 1. \forall U,V\in\mathscr{B}~\exists T\in\mathscr{B}:~T\subset U\cap V, 2. \forall X\in\mathscr{U}~\exists U\in\mathscr{... 5 votes 1 answer 317 views ### Number of ultrafilters in an extender Assume GCH. Suppose j: V\to M is an elementary embedding such that \mathrm{crit}(j)=\kappa, {}^\kappa M \subset M, M\supset V_{\kappa+2}. We can assume j is defined from some extender of ... 7 votes 0 answers 163 views ### Asymptotically discrete ultrafilters Definition 1. A ultrafilter \mathcal U on \omega is called discrete (resp. nowhere dense) if for any injective map f:\omega\to \mathbb R there is a set U\in\mathcal U whose image f(U) is a ... 5 votes 1 answer 196 views ### Relation between ultrafilters {\scr U} and {\scr U} \otimes {\scr U} [closed] If {\scr U} and {\scr V} are ultrafilters on non-empty sets A and B respectively, then the tensor product {\scr U}\otimes{\scr V} is the following ultrafilter on A\times B:$$\big\{X\...
Let $F$ be a field, $I$ a set, and $U$ an ultrafilter on $I$. Is the ultrapower $\prod_U F$ a purely transcendental field extension of $F$? According to Chapter VII, Exercise 3.6 from Barnes, Mack "...