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        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,...

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        29 views

        Sørensen-Dice coefficient equivalence [on hold]

        I'm working on a open source custom NLP engine. Trying to figure out some similarity measures, I wrote this formula (SIM) used to compare the similarity of two sets: $SIM = 1 - \frac{|A-B|+|B-A|}{|A|+...
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        votes
        1answer
        86 views

        Support of a regular measure Reg

        Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...
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        0answers
        31 views

        How to prove Quantified statements with different variables [on hold]

        I would like to know how could i prove this quantified statment knowing that the universal set={a,b} Prove that ?x?yP(x,y) impiles ?y?xP(x,y) is always true (=>)
        2
        votes
        1answer
        141 views

        Is $\in$-induction provable in first order Zermelo set theory?

        Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail? I asked this ...
        3
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        0answers
        104 views

        Ordinal analysis and nonrecursive ordinals

        Ordinal analysis is typically described as characterizing recursive ordinals in a theory $T$, but there is a sense in which it can characterize all $T$-ordinals, even those that are nonrecursive. ...
        5
        votes
        1answer
        270 views

        Can a Shelah semigroup be commutative?

        A semigroup $S$ is called $\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$; $\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\...
        10
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        1answer
        375 views

        Completeness number of ultrafilters

        In what I write below, by "ultrafilter" I mean a non-principal ultrafilter. Given an ultrafilter $U$ on some set $S$, let $\mu$ be the least cardinal such that $U$ is $\mu$-complete but not $\mu^+$-...
        5
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        1answer
        226 views

        $\Delta^{1}_{2}$ and degrees of constructibility $\textbf{on sets}$

        This is a follow-up to my question on $\Delta^{1}_{2}$ and degrees of constructibility of real numbers that was answered by the user "William", see here: Can $\Delta^{1}_{2}$ separate degrees of ...
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        0answers
        138 views

        On the number $n_0$ in Shelah's construction of a Jonsson group

        In the paper "On a Problem of Kurosh, Jonsson Groups, and Applications", Shelah proved the following Theorem 2.1. There exists a number $n_0$ such that for every infinite cardinal $\lambda$ with $\...
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        0answers
        404 views

        Turning a category into a semigroup

        Consider a small (that is objects and morphisms belonging to a Grothendieck universe) category $\mathcal{C}$, whose objects are all small sets, and an identity-on-objects functor ${\uparrow}: \mathbf{...
        3
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        0answers
        144 views

        On Khelif's example of a group of countable cofinality having the Bergman property

        A group $G$ is defined to have the Bergman property if for any subset $X=X^{-1}$ generating $G$ there exists $n$ such that $X^n=G$. By a result of Bergman, the permutation group of any set has the ...
        12
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        1answer
        492 views

        A Shelah group in ZFC?

        In his famous paper "On a problem of Kurosh, Jonsson groups, and applications" of 1980, Shelah constructed a CH-example of an uncountable group $G$ equal to $A^{6641}$ for any uncountable subset $A\...
        3
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        0answers
        100 views

        Completely I-non-measurable unions in Polish spaces

        Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...
        10
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        1answer
        748 views

        Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector

        In a lot of problems in linear algebra one uses the existence, for each $E$ vector space over a field $k$, and each $x\in E$, of a Hyperplane $H$ such that $E=k\cdot x \oplus H$ (Let us denote $\...
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        218 views

        Is there a known shorter axiomatization of NF than this?

        Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...

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