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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,...

4
votes
0answers
37 views

Is each cover of the plane by lines minimizable?

A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called $\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$; $\bullet$ minimizable if $\...
4
votes
2answers
210 views

Counting without one-to-one correspondence?

Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
-1
votes
0answers
58 views

Existence of a disjoint set of same cardinality [on hold]

Let $E$ be a set. In a lot of proof it is useful to take another set $S$ such thar there is a bijection $E\rightarrow S$ and $E\cap S= \emptyset$. But how to create a set like that ? I tried to take $...
0
votes
1answer
70 views

Can the union of difference sets in towers equal $\omega$?

We write $A\subseteq^* B$ if $A\setminus B$ is finite. Let $(A_n)_{n\in\omega}$ be a sequence of subsets of $\omega$ such that for all $n\in\omega$ we have $A_n \subseteq^* A_{n+1}$ and $A_{n+1}\not\...
3
votes
1answer
120 views

On filters possessing a countable network

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$ A family $\mathcal N$ of subsets of $\omega$ is called a network ...
4
votes
0answers
156 views

$MK+CC$ as a foundation for category theory

Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner? According to wikipedia, any category ...
7
votes
2answers
1k views

“Mächtigkeit” versus “Kardinalität”?

In Cantor's set theory, is there any difference between the terms Mächtigkeit and Kardinalität ?
7
votes
0answers
639 views

Examples of set theory problems which are solved using methods outside of logic

The question is essentially the one in the title. Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?
12
votes
1answer
239 views

Does this consequence of measurability in terms of games of length $\omega+1$ imply measurability?

For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of ...
3
votes
1answer
78 views

Partition theorems for located words

In this paper Bergelson, Blass, and Hindman prove the following Theorem 1.2 Let $W(\Sigma; v)$ be colored with finitely may colors and let $\bar s$ be an infinite sequence from $W(\Sigma; v)$. ...
4
votes
1answer
115 views

Minimal covers in hypergraphs with finite edges

Let $H=(V,E)$ be a hypergraph. We say that $C\subseteq E$ is a cover if $\bigcup C = V$. Let $H$ be a hypergraph with the following properties: $\bigcup E = V$, all members of $E$ are finite, and $d,...
10
votes
0answers
305 views

On the Number of Parallel Automorphism Lines

Given a group $G$, one can define the transfinite line of iterative automorphisms of $G$ to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the ...
2
votes
1answer
148 views

Maximality with respect to the splitting property

Let $X$ be a set and ${\cal P}(X)$ its powerset. We say that ${\cal F} \subseteq {\cal P}(X)$ has the splitting property (SP) if there is $A\in {\cal P}(X)$ such that for all $F\in {\cal F}$ we have $$...
0
votes
1answer
100 views

Upward generators of $[\omega]^\omega$

If $(P,\leq)$ is a poset and $S\subseteq P$ we let $$\uparrow S = \{p\in P: p\geq s\text{ for some }s\in S\}.$$ Let $([\omega]^\omega,\subseteq)$ denote the collection of infinite subsets of $\omega$,...
9
votes
2answers
432 views

Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets?

Is the following consistent with $\text{ZF}$? There exists a set $S=\{x_1,x_2,x_3,...\}$ such that: $|x_{i+1}| < |x_i|$ $\forall m,n \in S (|P(m)|=|P(n)|)$ Where cardinality $``||"$ is ...
34
votes
7answers
3k views

Paradoxical Mathematical Objects Pending for Construction [duplicate]

The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
6
votes
1answer
182 views

Interpreting a space in Baire space: how many facts do I need to understand the whole thing?

Below I'm working in ZF+DC+AD or similar; I want enough choice that things don't explode, but I also want the Wadge hierarchy to be well-behaved everywhere. Since this question is a bit long, I've put ...
6
votes
2answers
433 views

Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?

The question of existence of sets $x,y$ such that $$|x|<|y| \wedge |P(x)|=|P(y)|$$ is known to be independent of $\text{ZFC}$! But are there known examples of sets fulfilling the above condition ...
2
votes
1answer
60 views

Example of self-dual hypergraph with infinite edges

What is an example of a hypergraph $H=(V,E)$ with $|e|\geq \aleph_0$ for all $e\in E$ and the property that $H\cong H^*$ where $H^*$ is the dual hypergraph of $H$?
8
votes
1answer
460 views

Whence “Durchschnitt” and “Vereinigung”?

Today the set-theoretic operations of intersection $\cap$ [German: Durchschnitt] and union $\cup$ [German: Vereinigung] are standard. The modern notations are present in the first edition of van der ...
8
votes
0answers
175 views

Universally meager spaces and large cardinals

Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...
11
votes
1answer
347 views

Fixed points of injective self-maps

Is it consistent in $\mathsf{ZF}$ that there is a set $X$ with more than $1$ point such that every injective map $f:X\to X$ has a fixed point?
4
votes
0answers
170 views

Can there be a segment of regular cardinals with the tree property capped by an almost-strongly-compact?

Recall that a cardinal $\kappa$ is $(\lambda,\infty)$-almost-strongly-compact if every $\kappa$-complete filter can be refined to a $\lambda$-complete ultrafilter. A cardinal $\mu$ has the tree ...
-2
votes
0answers
50 views

Set models of ZFC and their perspectives on themselves and on other models [migrated]

Say $L$ is a PL1 language and $T$ is an $L$ theory. Then we have: (Gödel) Completeness. $T$ consistent $\Rightarrow$ $T$ has a (set) model Say $T = ZFC$ and $M \subset N$ for two (set) ...
13
votes
1answer
305 views

Do choice principles in all generic extensions imply AC in $V$?

It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a ...
32
votes
2answers
3k views

How to find Erdős' treasure trove?

The renowned mathematician, Paul Erdős, has published more than 1500 papers in various branches of mathematics including discrete mathematics, graph theory, number theory, mathematical analysis, ...
10
votes
1answer
424 views

Set-theoretical multiverses and their representation as functors? Why *the* multiverse?

In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...
5
votes
1answer
142 views

“König's theorem” for $T_2$-spaces?

For any topological space $(X,\tau)$ we define a matching to be a collection of non-empty and pairwise disjoint open sets. We define the matching number $\nu(X,\tau)$ to be the smallest cardinal $\...
12
votes
2answers
637 views

Set theory bootstrapping

Let $\mathcal{L}$ be the first order language of ZFC set theory, and let $\mathcal{L}_{\infty,\infty}$ be the usual infinitary extension of the language allowing arbitrary long disjunctions/...
18
votes
2answers
1k views

Mathematical Evidence Backing $|\mathbb{R}|=\aleph_2$

The "true" size of the real line, $\mathbb{R}$, has been the subject of Hilbert's first problem. Due to the Goedel and Cohen's work on the inner and outer models of $\text{ZFC}$, it turned out to be ...
7
votes
2answers
195 views

Non-separable metric probability space

Let us say a metric probability space $(X,\rho,\mu)$ has property (*) if: the support of $\mu$ is contained in a separable subspace of $X$. Questions: 1. Is there a standard name for this property? ...
9
votes
2answers
371 views

What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
3
votes
1answer
267 views

What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?

The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators. In 1977, ...
9
votes
1answer
232 views

Embeddings of Boolean algebras in $\wp(\omega)/Fin$

If we assume MA+¬CH, then every boolean algebra with cardinality smaller than the continuum embeds in ℘(ω)/Fin. A proof of this result can be found in Theorem 1.1, Chapter 8 of the book "Hausdorff ...
4
votes
1answer
424 views

What are examples of non-equivalent virtualizations of a large cardinal?

This is a follow up to my previous question concerning virtual large cardinals, that are generally weaker axioms of infinity obtained from ordinary large cardinals through the so-called virtualization ...
3
votes
1answer
275 views

On the Actual Potential of Virtual Large Cardinals

Virtual large cardinals belong to a relatively new breed of strong axioms of infinity. They often appear as statements of the following form: Definition. Suppose $A$ is a large cardinal property ...
3
votes
1answer
98 views

Connected spaces where every dense set is large

Let $\kappa >\aleph_0$ be a cardinal. Is there a connected space $(X,\tau)$ with $|X| = \kappa$ such that for every dense set $D\subseteq X$ we have $|D|=|X|$?
6
votes
0answers
174 views

Does $0^{\#}$ imply that $L$'s set generic multiverse has irreconcilable theories?

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P}...
9
votes
0answers
415 views

Which finite sets could be packed into a square?

This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space. The problem starts with a two-...
9
votes
1answer
563 views

Is every complete Boolean algebra isomorphic to the quotient of a powerset algebra?

Is every complete Boolean algebra isomorphic to a quotient, as a Boolean algebra, of some powerset algebra $\wp(X)$? It is not true for arbitrary Boolean algebras, see the comments, or see my MathSE ...
5
votes
0answers
178 views

Co-cones in the Turing degrees

Let the cocone of a Turing degree ${\bf d}$ be the set $cc({\bf d}):\{{\bf c}: {\bf c}\not\ge_T {\bf d}\}$. I'm curious what's known about the various partial orders (isomorphic to ones) of the form $...
1
vote
2answers
159 views

Bipartite subgraphs with lots of edges

Suppose $G=(V,E)$ is a simple, undirected graph with $|V|,|E|$ infinite. Is there $B\subseteq E$ with $|B| = |E|$ such that $(V,B)$ is bipartite?
9
votes
1answer
376 views

On the Large Cardinal Strength of Normal Moore Space Conjecture

In his seminal 1937 paper, Jones [1] proved the following result about Moore spaces: Theorem. (Jones) If $2^{\aleph_0}<2^{\aleph_1}$ then all separable normal Moore spaces are metrizable. Then ...
9
votes
1answer
483 views

A reference to infinite version of the Sunflower Lemma

Please help me to find a proper reference to the following infinite version of the Sunflower Lemma. Lemma. Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite ...
6
votes
1answer
165 views

Cohen generics over the ground model still Cohen over other generic extensions?

Let $M$ be a countable transitive model of (enough of) ZFC. I'm looking for notions of forcing $\mathbb{P}$ such that if $G$ is $M$-generic for $\mathbb{P}$, then $c$ is a Cohen real over $M$ if and ...
14
votes
1answer
1k views

What is the modal logic of outer multiverse?

The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation. The modal logic associated ...
11
votes
1answer
822 views

The Tall Tale of Terminating Transfinite Towers

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct ...
6
votes
2answers
325 views

Paul Mahlo's Original Large Cardinal Paper

It is well known that Paul Mahlo (1883-1971) developed a systematic hierarchy of inaccessible cardinals of the type $\pi_{a,b}$ where $\pi_{1,b}$ enumerates the strongly innacessible cardinals, $\pi_{...
13
votes
2answers
2k views

Who first chose the names Alice and Bob for players A and B? [closed]

Who first chose the names Alice and Bob for the players (or observers) A and B?
2
votes
1answer
157 views

Infinite graph with lots of non-isomorphic induced subgraphs

Given an infinite cardinal $\kappa$, is there a graph on $\kappa$ vertices that contains $2^\kappa$ pairwise non-isomorphic induced subgraphs?