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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.

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 ...

asked 10 hours ago

domotorp
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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&...

asked 11 hours ago

M. Winter
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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 ...

asked yesterday

M. Winter
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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 $...

asked 2 days ago

domotorp
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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 \...

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?

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 "$...

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 ...

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 $C$ ...

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 ...

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 ...

asked Aug 7 at 18:46

Sean Cox
1,5061111 silver badges1515 bronze badges

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 ...

asked Aug 7 at 15:11

Carlos
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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 $...

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 ...

asked Aug 1 at 2:31

Isaac
11844 bronze badges

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 ...