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      Questions tagged [set-theory]

      forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

      2
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      33 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 ...
      2
      votes
      0answers
      115 views

      Sunflower lemma in a more general poset?

      The sunflower lemma may be viewed as a statement about the poset $P_\omega(\omega_1)$, and the generalized sunflower lemma may be viewed as a statement about the poset $P_\lambda(\kappa)$ for $\kappa$ ...
      5
      votes
      1answer
      141 views

      Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$ [on hold]

      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\...
      -1
      votes
      0answers
      124 views

      Can we get rid of the primitive symbol $V$ in Ackermann's set theory this way?

      I want to get rid of the primitive $V$ in Ackmerann set theory, without changing the axioms so much. I have the following try in my mind, but I'm not sure if it works. So we instead work in the pure ...
      1
      vote
      1answer
      58 views

      Dense subfilter of selective ultrafilter

      Given selective ultrafilter $\mathcal{U}$ on $\omega$ and dense filter $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$, where $\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$ if the limit exists. Let $\...
      3
      votes
      1answer
      182 views

      Models of $\mathsf{ZFC}$ with neither $P$- nor $Q$-points

      A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or finite-to-one. A $Q$...
      3
      votes
      1answer
      138 views

      Finite covers of Boolean algebras by their subalgebras

      It is a student exercise that no group can be represented as a set-theoretic union of its two proper subgroups. The same also can be shown for Boolean algebras. On the other hand, it's not hard to ...
      -1
      votes
      0answers
      43 views

      Largest subset of the powerset of a countable set in which no set includes another [duplicate]

      Let S be a set that has countably-infinitely many members. Let a subset of $\mathcal{P}(S)$ (the power-set of S) have the Sperner-family-property iff no two of its members are such that one of them is ...
      0
      votes
      0answers
      121 views

      Subsets of the unit interval [migrated]

      I have been working on a problem and I need to answer the following question: Is there a family $\{F_\alpha: \alpha \in \omega_1\}$ of subsets of the interval $]0,1[$ such that: (a) $F_\alpha=\{x_1^\...
      2
      votes
      2answers
      213 views

      (Types of) induction on infinite chains

      This question may be trivial, or overly optimistic. I do not know (but I guess the latter...). I am a group theorist by trade, and the set-up I describe cropped up in something I want to prove. So ...
      5
      votes
      0answers
      168 views

      Metrically Ramsey ultrafilters

      On Thuesday I was in Kyiv and discussed with Igor Protasov the system of MathOverflow and its power in answering mathematical problems. After this discussion Igor Protasov suggested to ask on MO the ...
      2
      votes
      1answer
      126 views

      The property of the dense subfilter of a selective ultrafilter

      Let us define the density of subset $A\subset\omega$ : $$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$ if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...
      0
      votes
      0answers
      55 views

      Some kind of idempotence of dense filter

      In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...
      0
      votes
      1answer
      94 views

      Semi-rigid boolean algebras

      A boolean algebra is rigid if it has no nontrivial automorphisms. Call it semi-rigid if none of its nontrivial automorphisms has any fixed points other than 0 and 1.* The four-element algebra $\{0, b, ...
      4
      votes
      0answers
      87 views

      Is Ackermann's set theory minus class comprehension equal to ZF?

      Ackermann in 1956 proposed an axiomatic set theory. Reinhard proved that Ackermann's set theory equals ZF It's clear that Zermelo set theory can be interpreted in Ackermann's set theory minus class ...

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