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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.

6
votes
3answers
358 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{...
12
votes
1answer
376 views

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 ...
0
votes
0answers
62 views

Can MK be interpreted in a class theory about an abstract hierarchy principle + an accessibility principle?

The following is a first order MONOSORTED class theory, that is primarily motivated by an abstract hierarchy principle. It extends first order logic with equality, its language has only two extra-...
6
votes
1answer
457 views

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 ...
1
vote
1answer
123 views

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 ...
9
votes
0answers
345 views

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
1answer
345 views

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
0answers
158 views

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
0answers
144 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
1answer
147 views

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
1answer
282 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
1answer
104 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
0answers
297 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
2answers
143 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 "$\in$". ...
23
votes
0answers
498 views

Is it still an open problem whether $\mathbb{R}^\omega$ is normal in the box topology?

On page 205 of his Topology book, James Munkres makes 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 ...
7
votes
1answer
618 views

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
3answers
534 views

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
1answer
260 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
2answers
170 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
1answer
101 views

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
1answer
141 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 ...
2
votes
1answer
256 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
0answers
159 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. ...
7
votes
1answer
318 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\...
11
votes
1answer
427 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
1answer
298 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 ...
6
votes
0answers
159 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
0answers
157 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 ...
12
votes
1answer
530 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 $A^{6641}$ for any uncountable subset $A\...
3
votes
0answers
109 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
1answer
796 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
0answers
234 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
1answer
162 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
2answers
207 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 ...
9
votes
0answers
126 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
1answer
316 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
1answer
199 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
1answer
408 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 \...
11
votes
3answers
376 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
1answer
236 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
0answers
119 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
1answer
243 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
0answers
127 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
1answer
357 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 ...
5
votes
2answers
222 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
3answers
429 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
1answer
241 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
1answer
67 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
1answer
208 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
1answer
122 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 ...