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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.

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        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 $\kappa$ ...
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        1answer
        119 views

        Relation between ultrafilters ${\scr U}$ and ${\scr U} \otimes {\scr U}$

        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\...
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        77 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 ...
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        1answer
        52 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 $\...
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        votes
        1answer
        172 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
        131 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 ...
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        42 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 ...
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        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^\...
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        votes
        2answers
        208 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 ...
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        votes
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        165 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 ...
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        1answer
        115 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 ...
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        53 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\...
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        1answer
        91 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, ...
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        86 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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        votes
        1answer
        109 views

        Amorphous proper classes in MK

        Working in $ZFC$ every cardinal is either finite or in bijection with a proper subset of itself (Dedekind infinite). Without Choice it is consistent that there are infinite sets which can't be ...

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