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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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### 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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### 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 ...
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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### 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 ...
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### $\mathcal{C}$-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 ...
1answer
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### 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 ...
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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 $... 0answers 195 views ### Weakly compact cardinals In Exercises 17.17 and 17.18 of Jech's Set Theory book, he shows that if the language$L_\kappa$satisfies the Weak Compactness Theorem, then$\kappa$is a weakly inaccessible cardinal. Also, in ... 2answers 398 views ### Set of perfect subsets of a Borel set Let$\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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