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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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      83 views

      Can planar set contain even many vertices of every unit equilateral triangle?

      Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle? I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
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      1answer
      114 views

      Existence of a certain set of 0/1-sequences without the Axiom of Choice

      Is there a set $\mathcal X\subset\{0,1\}^{\Bbb N}$ of 0/1-sequences, so that For any two 0/1-sequences $x,y\in\{0,1\}^{\Bbb N}$ for which there is an $N\in\Bbb N$ with $$x_i=y_i,\;\;\text{for all $i&...
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      vote
      1answer
      68 views

      Measure of real numbers with converging average over binary digits

      Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period. If $x_1,x_2,x_3,...$ are the digits of $x$, then consider the $k$-th ...
      3
      votes
      0answers
      81 views

      Move one element of finite set out from A in plane

      Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
      3
      votes
      1answer
      141 views

      A weak (?) form of Shelah cardinals

      The following definition of a large cardinal property combines parts of the definitions of "Shelah cardinal" and "Woodin cardinal": A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \...
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      0answers
      102 views

      Two questions about the Löwenheim–Skolem theorem [on hold]

      The L?wenheim–Skolem theorem implies that if ZFC is consistent then its countable model M exists. What theory is used to say that M is countable? Is there an uncountable model if ZFC is consistent?
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      1answer
      178 views

      Is the power set axiom essential for constructing L?

      Take ZFC, remove axiom of Power set, and put instead of it the following axiom: Axiom of Successor Cardinals: $\forall \kappa \exists x \forall \alpha ( \alpha \leq \kappa \to \alpha \in x)$ where "$...
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      68 views

      Is second order ordinal arithmetic stronger than the multi-sorted first order based variant of it?

      In posting: Is ZFC interpretable in a kind of an extended form of second order arithmetic? I aimed to prove the consistency of ZFC by it being interpretable in what may be better named as "second ...
      3
      votes
      2answers
      103 views

      Avoiding multiply covered vertices in graph edge coverings

      Let $G=(V,E)$ be a simple, undirected graph with $\bigcup = E$ (that is, there are no isolated vertices). We say that $C\subseteq E$ is an edge cover of $G$ if $\bigcup C = V$. For any edge cover $C$ ...
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      0answers
      166 views

      Absolute formulas with high complexity

      This is a repost of a MSE question. It is a standard result that $\Sigma_1^{\mathsf{ZF}}$-formulas are upward absolute between $\mathsf{ZF}$ $\in$-models, while $\Pi_1^{\mathsf{ZF}}$-formulas are ...
      3
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      0answers
      141 views

      $\mathcal{C}$-filtering of modules inherited by submodules

      I'll state the question about modules, but I'm open to examples in other contexts. I am not an algebraist, so please forgive any non-conventional terminology. DEFINITION: Let $\mathcal{C}$ be a ...
      8
      votes
      1answer
      217 views

      Uncountable disjoint closed coverings of $[0,1]$

      It is well known that the unit interval $[0,1]$ cannot be decomposed as a countable union of pairwise disjoint closed (nonempty) subsets. See for instance this math.stackexchange question. The proof ...
      3
      votes
      1answer
      201 views

      The size of sheafification

      Let $X$ be a small site. Let $\aleph$ be an infinite cardinal, such that $|Ob(X)|\leq \aleph$ and $|Mor(X)|\leq \aleph$, where $Mor(X)$ is the set of all morphisms. We define the size of a presheaf $...
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      0answers
      195 views

      Weakly compact cardinals

      In Exercises 17.17 and 17.18 of Jech's Set Theory book, he shows that if the language $L_\kappa$ satisfies the Weak Compactness Theorem, then $\kappa$ is a weakly inaccessible cardinal. Also, in ...
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      votes
      2answers
      398 views

      Set of perfect subsets of a Borel set

      Let $\mathbb{P}$ be the set of all perfect (i.e., every node has incomparable successors) subtrees of the full binary tree $2^{<\omega}$. We can endow $\mathbb{P}$ with a Borel structure by ...

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