# Tagged Questions

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,...

**8**

votes

**0**answers

115 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 $A^{6643}$ for any uncountable subset $A\...

**2**

votes

**0**answers

31 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 ...

**9**

votes

**1**answer

576 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

164 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

134 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

161 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 ...

**7**

votes

**0**answers

82 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 zeroth ...

**7**

votes

**1**answer

295 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

181 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 ...

**6**

votes

**1**answer

350 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 \...

**10**

votes

**3**answers

302 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 ...

**6**

votes

**1**answer

198 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

101 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 ...

**3**

votes

**0**answers

153 views

+200

### 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

104 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

226 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

**2**answers

200 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

397 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

193 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 $$\{...

**0**

votes

**0**answers

63 views

### Condition for the existence of minimal subcover

I have a question about the correctness of the following statement:
Given a set $S$, and $\mathcal{F} \subseteq 2^S$ being a
$\textbf{Sperner family}$(antichain), with additional property:
$\...

**3**

votes

**1**answer

60 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

194 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 ...

**1**

vote

**1**answer

111 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 ...

**1**

vote

**1**answer

131 views

### Definition of ineffability behind reflection principles in set theory

A prominent idea found e.g. in Koellner (http://logic.harvard.edu/koellner/ORP_final.pdf) is that reflection principles in set theory are motivated by the idea that proper classes are so large as to ...

**4**

votes

**1**answer

125 views

### If $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, what properties does $κ$ have?

More specifically, if $j:V\prec M$ has critical point $κ$ and for any $X\in M$ with $|X|=μ$, $|X|^M=μ$, does $κ$ necessarily have some form of $μ$-compactness? Is it related to strong compactness in ...

**2**

votes

**1**answer

451 views

### Is there anything against this function j being injective?

Language (first order logic with equality "$=$" and membership "$\in$", and constant symbol "$j$")
Axiom: ID axioms +
There exists a set $A$, such that:
Field: $\forall x \in j \ \exists a \in A \ \...

**2**

votes

**2**answers

237 views

### About the existence of a particular kind of “splitting” function on atomless complete Boolean algebras

Let $\mathbb{B} = \langle B, \wedge, \vee, \leq, \neg, 0, 1 \rangle$ be an atomless complete Boolean algebra.
We call $f$ a splitting function on $\mathbb{B}$ iff
$f : B-\{1\} \longrightarrow B \...

**7**

votes

**2**answers

261 views

### Surjective order-preserving map $f:{\cal P}(X)\to \text{Part}(X)$

Let $X$ be a set, and let $\text{Part}(X)$ denote the collection of all partitions of $X$. For $A, B\in \text{Part}(X)$ we set $A\leq B$ if $A$ refines $B$, that is for all $a\in A$ there is $b\in B$ ...

**4**

votes

**1**answer

173 views

### Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean.
Let $...

**5**

votes

**1**answer

157 views

### Random reals preserving Cohen reals

Suppose we have a model (of $\mathsf{ZFC}$) $M$, and that $x\in 2^\omega$ is random over $M$, and that $y\in 2^{\omega}$ is Cohen over $M$. My question is whether $y$ is also Cohen over $M[x]$. In ...

**2**

votes

**1**answer

176 views

### A variant of Radin forcing

Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties:
$(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $\...

**3**

votes

**0**answers

451 views

### “Antiforcing” - Is there a method to 'remove' sets from a model of ZF?

Forcing is a method of "adding sets" to a model $M$ of ZF by making a new set $M^{(\mathbb{P})}$ consisting of every set of $M$, but you have the option to add certain sets out of $M^{(\mathbb{P})}$ ...

**0**

votes

**1**answer

75 views

### Connected infinite graphs in which all matchings are “small”

Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all matchings $M\subseteq E$ the set $V\setminus (\bigcup M)$ is ...

**7**

votes

**5**answers

368 views

### Ideals on $\mathbb N$ and large sets that have small intersection

Let $\mathcal I$ be a (non-principal) ideal of subsets of $\mathbb N$. Suppose that every family $\mathcal{A} \subset \wp(\mathbb N)\setminus \mathcal I$ with the following property is countable:
$$A,...

**12**

votes

**0**answers

265 views

### Cardinality vs. isomorphism type of vector spaces without choice

One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem:
If $V$ is an infinite vector space over a field $F$, and $...

**5**

votes

**0**answers

146 views

### Decomposition of forcing iterations

One of the great things about a finite support iteration $\Bbb P_\delta$, is that if $\alpha<\delta$, we can write $\Bbb P_\delta$ as the iteration of $\Bbb{P_\alpha\ast\dot Q_\alpha\ast P_\delta/...

**4**

votes

**1**answer

147 views

### Embedding ordinals with the order topology into connected $T_2$-spaces

Is there a limit ordinal $\kappa_0$ with $\kappa_0 \lt 2^{\aleph_0}$ and such that for every limit ordinal $\lambda$ with $\kappa_0\leq \lambda\lt 2^{\aleph_0}$ there is a connected $T_2$-space $X_\...

**5**

votes

**0**answers

195 views

### Bad forcing permutations

Let $P$ be the finite-support product of the Cohen forcing. It adjoins a sequence of Cohen-generic reals say $a_n$, $n<\omega$, which one naturally calls a $P$-generic sequence. Suppose that $\pi$ ...

**16**

votes

**1**answer

347 views

### What non-standard model of arithmetic does Hofstadter reference in GEB?

Following some of the coolest bits of Hofstadter's Gödel, Escher, Bach, extensions of the standard model of arithmetic are described. A ways in, the paragraph "Supernatural Addition and Multiplication"...

**0**

votes

**0**answers

73 views

### Would Singletons+ Boolean union+ Relative complements+ Composition terminate over sets $N, P(N), P(P(N))$?

The following question is related to question asked at
How to decide if a recursive addition of subsets after certain formula would terminate?
But here it will be asked about a specific situation.
...

**5**

votes

**1**answer

298 views

### Buying more absoluteness for countable transitive models?

Let $M$ be a countable transitive model of (enough of) ZFC. Mostowski's Absoluteness Theorem says that $\Pi^1_1$ statements are absolute between $M$ and larger models, in particular, between $M$ and ...

**3**

votes

**1**answer

197 views

### Collapse an inaccessible cardinal to a successor of a singular cardinal

Is it possible to turn an inaccessible cardinal in $V$ to a successor of a singular cardinal in some forcing extension?

**7**

votes

**1**answer

270 views

### Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any “logic”?

According to Cantor's attic, Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic". But I can't find a definition of what a "logic" is either there or in ...

**22**

votes

**4**answers

1k views

### Does Zorn's Lemma imply a physical prediction? [duplicate]

A friend of mine joked that Zorn's lemma must be true because it's used in functional analysis, which gives results about PDEs that are then used to make planes, and the planes fly. I'm not super ...

**3**

votes

**1**answer

239 views

### A weakening of cardinal compactness - is it equivalent?

I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak ...

**7**

votes

**1**answer

254 views

### Stationarity and Fodor's lemma for a (nice) poset?

The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the ...

**11**

votes

**2**answers

222 views

### Countable support product of Sacks forcings and selective ultrafilters

If $U$ is a selective ultrafilter on $\omega$, then $U$ generates an ultrafilter in $V^{\mathbb S}$, where ${\mathbb S}$ is Sacks forcing. The same is true with ${\mathbb S}$ being replaced by ${\...

**2**

votes

**0**answers

66 views

### How to decide if a recursive addition of subsets after certain formula would terminate?

Lets call a definable property $\phi(y,z_1,..,z_n)$ as terminating over a set $A$ if and only if recursive successive additions of every set $\{y \in A| \phi(y,z_1,..,z_n)\}$ from parameters $z_1,..,...

**3**

votes

**0**answers

77 views

### Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...

**4**

votes

**2**answers

264 views

### Problem understanding a passage of the proof of $\mathfrak{p}=\mathfrak{t}$ involving forcing

I've a problem with a passage of the proof of Claim 14.7 of the paper "Cofinality spectrum theorems in model theory, set theory, and general topolgy" by Malliaris and Shelah, or equivalently ...