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.
5,812 questions
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Size of cutsets in ${\cal P}(\omega)$ having infinite and co-infinite members only
A chain ${\cal C}\subseteq {\cal P}(\omega)$ is a set such that for all $A, B\in {\cal C}$ we have $A\subseteq B$ or $B\subseteq A$. Using Zorn's Lemma one can show that every chain is contained in a ...
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Boolean ultrapower - set-theoretic vs algebraic/model-theoretic
I've been looking through the Hamkins/Seabold paper "Well-founded Boolean ultrapowers as large cardinal embeddings".
The Boolean ultrapowers are defined there in two different ways:
in ...
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How far does Cantor-Bendixson rank counting let us build computable isomorphisms between ordinals?
This is tangentially related to this old question of mine.
Say that a clean well-ordering is a computable well-ordering $\triangleleft$ of $\mathbb{N}$ such that the following additional data is ...
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Cardinality of the collection of maximal antichains in ${\cal P}(\omega)$
An antichain in $\mathcal P(\omega)$ is a set $\mathcal A\subseteq \mathcal P(\omega)$ such that for all $A, B\in \mathcal A$ with $A\neq B$ we have $(A\setminus B)\neq \emptyset$ and $(B \setminus A)\...
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Set size comparison via non-existence of surjections
If $X, Y$ are sets, let us say that $X$ is strictly smaller than $Y$, in symbols $X \prec Y$, if $Y$ is non-empty and for every map $f:X\to Y$ we have $Y\setminus\text{im}(f) \neq \varnothing$.
Our ...
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Is Zorn's Lemma equivalent to the Axiom of Choice for individual sets?
It is well-known that in $\mathsf{ZF}$, the Axiom of Choice and Well-ordering Theorem are equivalent. What is perhaps less well-known is that there is a "local" version of this equivalence.
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What was the definition of strongly inaccessible in 1958?
I'm reading Erdős and Hajnal's paper "On the structure of set-mappings" from 1958 and also a companion paper "Some remarks concerning our paper…". In it they define a partition ...
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For what cardinality is the cofinite topology on a set symmetrizable?
A symmetric on a set $X$ is any function $d:X\times X\to[0,\infty)$ such that
for every $x,y\in X$ the following two conditions are satisfied:
$d(x,y)=0$ if and only if $x=y$;
$d(x,y)=d(y,x)$.
A ...
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Comparability of power sets and (AC)
For sets $X, Y$ we write $X \leq Y$ if there is an injective map $f:X\to Y$.
Let (S) be the statement:
For any sets $X, Y$, either ${\cal P}(X) \leq {\cal P}(Y)$, or ${\cal P}(Y) \leq {\cal P}(X)$, ...
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Axiom of Foundation and Proper Class epsilon chains
In Kunen, it is emphasized that Axiom of Foundation only requires all non-empty subSETS to have an epsilon-least element. But what about proper classes that (might) have infinite-descending epsilon ...
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Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
This is a sequel to my MSE question about elementary equivalences between the $V_α$.
Given that there are only $ℶ_1$ first-order theories in the language of set theory, by pigeonhole principle, there ...
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$0 = 1$ and (nigh-)inconsistent LCAs
This question is twofold. For one I would like to know which large cardinal notions which got any (at least minimal) traction have been known to be inconsistent. I know, for example, of Berkeley and ...
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Disjoint maximal chain and maximal antichain in ${\cal P}(\omega)$
If $(P,\leq)$ is a partially ordered set, we say that $C\subseteq P$ is a chain if $a\leq b$ or $b\leq a$ for all $a,b\in C$. An antichain is a set $A\subseteq P$ with $a\not \leq b$ and $b\not\leq a$ ...
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Pure buttons in the modal logic of forcing
I've been trying to understand what it means for a button to be pure in the context of the modal logic of forcing, and it would help to have an example of a button which is not a pure button. Based on ...
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Chromatic number of the antichain hypergraph on $\mathcal P(\omega)$
If $H=(V, E)$ is a hypergraph, the its chromatic number $\chi(H)$ is the smallest non-empty cardinal $\kappa$ such that there is a map $c:V \to \kappa$ such that for every $e\in E$ containing more ...