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.
5,270
questions
4
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Reference request: destroying saturation at an inaccessible?
An ideal $I$ on $P(\kappa)$ is said to be $\kappa^+$-saturated if there is no sequence $\langle S_\alpha \mid \alpha<\kappa^+\rangle$ of $I^+$ sets such that $\alpha<\beta<\kappa^+\implies S_\...
6
votes
3
answers
436
views
How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?
A set $\mathscr{U}$ is a universe if the following conditions are met:
For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$
For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{U}$,
...
12
votes
1
answer
577
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Translating Grothendieck axiom UB into ZFC
In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...
7
votes
1
answer
594
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Action of infinite symmetric groups on iterated power sets
Let $X$ be an infinite set, and $k \ge 1$ be a natural number. We work without the axiom of choice.
Let $G_0$ be the full symmetric group on $X$, and let $G_1$ be the
full symmetric group on ${\cal ...
2
votes
1
answer
170
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Enumeration Hierarchies
A set $M$ of real values is said to be enumerable if there is a bijection between the elements of $M$ the elements of $\mathbb{N}$.
That definition does however not impose any restrictions on the ...
11
votes
0
answers
512
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What happens when you iterate Cohen reals?
There are a few classical theorems in set theory:
The finite support iteration of ccc forcing is ccc.
The countable support iteration of proper forcing is proper.
The finite support iteration of ...
10
votes
1
answer
487
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Looking for “Set theory for a small universe” by Ketonen
In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not ...
10
votes
0
answers
185
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stationary reflection in $[\kappa]^\omega$
It is well-known that the following reflection principle is consistent relative to a supercompact:
For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
6
votes
0
answers
162
views
Singular strong generator
Let $V$ be a model of $\mathrm{ZFC}$ and let $j\colon V \to M$ be an elementary embedding with a critical point $\kappa$ ($M$ is transitive). A strong generator of $j$ is an ordinal $\zeta \geq \kappa$...
1
vote
1
answer
161
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A proof of recontruction of Sacks generic filter from it's Sacks real (M[G] = M[f])
Given the Sacks forcing $ (\mathbb{S} = \{T \subset 2^{<\omega} : T \text{ is perfect}\},\subset) $ and $G$ generic over M, we have $f = \bigcup \bigcap G = \bigcup_{T \in G}stem(G) $ a path ...
11
votes
1
answer
385
views
The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?
Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
3
votes
1
answer
109
views
Function $f:\kappa\to\alpha$ with small fibers where $\alpha\in\kappa$
Is it consistent in $\mathsf{ZF}$ that there is an infinite cardinal $\kappa$, cardinals $\alpha, \beta\in\kappa$ and a function $f:\kappa\to \alpha$ such that for each $x\in\alpha$ there is an ...
14
votes
0
answers
392
views
O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
0
votes
2
answers
209
views
Are Regularity schema and $\in$-induction schema equivalent in intuitionistic logic?
In posting "Does Regularity schema imply $\in$-induction when added to first order Zermelo set theor?"
the answer was that they are equivalent in classical first order logic with membership "$\in$".
...
32
votes
1
answer
2k
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Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?
On page 205 of his Topology textbook, James Munkres made an interesting remark:
It is not known whether $\mathbb{R}^\omega$ is normal in the box topology. Mary-Ellen Rudin has shown that the answer ...
6
votes
1
answer
748
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A ridiculous combinatorial cardinal characteristic of the continuum?
This question assumes familiarity with combinatorial cardinal characteristics of the continuum. It is abstracted out of a question in a joint research with Jialiang He. I hope we've got the ...
7
votes
3
answers
620
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Characterizing "bounded" distributivity in terms of dense open sets
Prikry forcing is a well-known example of a forcing which adds an $\omega$-sequence to some cardinal $\kappa$, but does not add bounded subsets to $\kappa$.
There are other examples of forcings which ...
4
votes
1
answer
302
views
bijections and order types
Suppose $\kappa$ is an infinite cardinal and $\alpha$ is an ordinal of cardinality $\kappa$. Is it possible to find a bijection $f : \kappa \to \alpha$ such that for all $x \subseteq\kappa$, $\mathrm{...
1
vote
2
answers
284
views
Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?
That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...
-1
votes
1
answer
122
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Injective choice function for non-separable $T_2$-spaces
For any set $X$ and cardinal $\kappa$ let $[X]^\kappa$ be the collection of all subsets of $X$ of cardinality $\kappa$.
I was looking for $T_2$-spaces $(X,\tau)$ with the property that
$(P)$ ...
0
votes
1
answer
191
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 ...
7
votes
1
answer
485
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 ...
4
votes
0
answers
338
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.
...
8
votes
1
answer
441
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 $n\in\mathbb N$...
15
votes
1
answer
699
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
votes
1
answer
378
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 ...
7
votes
1
answer
270
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 $\...
3
votes
0
answers
179
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 ...
17
votes
3
answers
978
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 $A^{6640}$ for any uncountable ...
4
votes
0
answers
120
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
votes
1
answer
954
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 $\...
0
votes
0
answers
369
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 ...
2
votes
1
answer
200
views
A possible characterization of regular cardinals?
For a cardinal $\kappa$ by $[\kappa]^{<\kappa}$ we denote the family of all subsets of cardinality $<\kappa$ in $\kappa$.
Question. Assume that for an infinite cardinal $\kappa$ there exists ...
4
votes
2
answers
334
views
The cofinality of the poset $[\kappa]^{<\kappa}$ for a singular cardinal $\kappa$
For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of ...
11
votes
0
answers
255
views
A ZFC-example of a countably compact paratopological group which is not a topological group
Problem. Does there exist a ZFC-example of a countably compact Hausdorff paratopological group which is not a topological group?
(The problem posed 27 May 2015 by Alexander Ravsky on page 9 of Volume ...
8
votes
1
answer
356
views
A combinatorial property of uncountable groups, II
Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that
1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and
2) for any function $\...
6
votes
1
answer
242
views
A combinatorial property of uncountable groups
Let $A,B$ be two uncountable sets in a group $G$ such that for any elements $x,y\in G$ the intersection $xA\cap yB$ is finite. Let $\Phi:G\to 2^G$ be a function assigning to each element $x\in G$ some ...
7
votes
1
answer
638
views
Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe
A Grothendieck's universe is such a set $U$ so that
$\forall x \in U, x \subseteq U$,
$\forall x,y \in U, \{x,y\} \in U$,
$\forall x \in U, \mathcal{P}(x) \in U$,
given a family $(X_i)_{i \in I}$ ...
11
votes
3
answers
605
views
Cardinality of families of subsets of $\mathbb{N}$ whose intersections are finite
Does there exist an uncountable $P \subset \mathcal{P}(\mathbb{N}) $ with the property that for any distinct $x,y \in P$, $|x \cap y|$ is prime?
A more general, but likely harder, question: is it ...
7
votes
1
answer
280
views
Can $\Delta^{1}_{2}$ separate degrees of constructibility?
Suppose that $\phi(x)$ is a $\Delta^{1}_{2}$-formula (without parameters) and let $A:=\{x\subseteq\omega:\phi(x)\}$. It is clear that, e.g. if there are Cohen-generics over $L$, then $A$ cannot be the ...
3
votes
0
answers
138
views
If any satisfiable $\mathcal{L}_{κ,κ}(Q_{=κ})$-theory remains satisfiable when replacing $Q_{=κ}$ with $Q_{=μ}$, is $κ$ huge?
Recently, I have asked a model-theoretic question concerning a weakening of different forms of compactness. I now present another model-theoretic question as a weakening of hugeness.
If any ...
7
votes
1
answer
283
views
Can we inductively define Wadge-well-foundedness?
For a topological space $X$ (which I'll identify with its underlying set of points), we define the Wadge preorder $Wadge(X)$: elements of the preorder are subsets of $X$, and the ordering is given by $...
2
votes
0
answers
144
views
C.c.-ness of a forcing notion based on an atomless complete Boolean algebra
Given $\mathbb{B} = \langle B, \wedge, \vee, \neg, 0, 1 \rangle$ an atomless complete Boolean algebra that has a $< \mkern-4mu \kappa$-closed dense subset and is $\kappa^+$-c.c., we define a ...
0
votes
1
answer
584
views
What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?
The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and ...
6
votes
2
answers
294
views
Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?
Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances?
Formal version of question. If $X$ is a set, let $[X]...
9
votes
3
answers
487
views
A property of an ultrafilter
Let $\mathcal U$ be a free ultrafilter on a set $X$. For $n\in\mathbb N$ let $\mathcal F$ be a family of $n$-element subsets of $X$ such that $\bigcup\mathcal F\in\mathcal U$.
Question. Is there a ...
4
votes
1
answer
259
views
Generalizing the $T_0$-axiom
The starting point of this question is a slight reformulation of the $T_0$ separation axiom: A topological space $(X,\tau)$ is $T_0$ if for all $x\neq y\in X$ there is a set $U\in \tau$ such that $$\{...
3
votes
1
answer
71
views
$|V|$ and $|E|$ in hypergraphs with a separation property
Let $H=(V,E)$ be a hypergraph. We call it $T_0$ if for all $x\neq y \in V$ there is $e\in E$ with $\{x,y\}\not\subseteq E$ and $\{x,y\}\cap e\neq \emptyset$ (i.e., $e$ contains exactly one of $x,y$).
...
4
votes
1
answer
677
views
Hereditarily indecomposable groups
Question. Is it true that each uncountable group $G$ contains an uncountable subgroup $A$ and an infinite subgroup $B$ such that $A\cap B=\{1\}$? What will be the answer if we additionally require ...
2
votes
1
answer
136
views
Encoding sets in locally generic sets
Let $\alpha$ be an ordinal, and let $a\subseteq\alpha$ such that $\alpha$ is countable in $L[a]$. Moreover, let $\beta>\alpha$ be an ordinal such that, in $L[a]$, $\alpha$ and $\beta$ have the same ...