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Results tagged with set-theory
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user 8628
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.
-1
votes
2
answers
345
views
Subsets of $\mathbb{N}$ whose lower density respects complements
The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: \lambda(\mathbb{N}\se …
- 48.9k
0
votes
Accepted
Surjectivity from union of a set system to the set system
No - let $\mathcal{A} = \{\{0,1\}\} \cup \{\{n\} : n\in \omega\}$.
- 48.9k
0
votes
1
answer
398
views
Almost totally distinct functions
Let us call $f,g:\omega\to \omega$ almost totally distinct if $$|\{n\in \omega: f(n) = g(n)\}| < \aleph_0.\;\;\;\; (\star)$$
It is known that there are uncountable collections of almost totally distin …
- 48.9k
7
votes
2
answers
318
views
Bounding and dominating numbers ${\frak b}, {\frak d}$ via ultrafilters
Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$ and suppose that ${\cal U}$ is a free ultrafilter on $\omega$. We write $f \leq_{\cal U} g$ if $$\{n\in\omega: f(n) \leq g(n)\}\ …
- 48.9k
3
votes
1
answer
529
views
Uncountably many countable graphs with no homomorphism between them
By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such tha …
- 48.9k
2
votes
0
answers
103
views
Strongly minimal covering subsets of $\text{Ind}(G)$
Let $G=(V,E)$ be an undirected, simple graph. Let $\text{Ind}(G)$ be the set of independent subsets of $V(G)$. We say that $K\subseteq \text{Ind}(G)$ is a cover (by independent subsets) if $\bigcup K …
- 48.9k
1
vote
1
answer
141
views
Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$
Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the answe …
- 48.9k
2
votes
2
answers
265
views
Hedetniemi's conjecture for graphs with countable chromatic number
Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph_0$, but $\chi(G\times H) < \aleph_0$?
- 48.9k
-1
votes
2
answers
263
views
Selection problem in a collection of non-empty sets
Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties?
$a\in {\cal F} \implies |a|\geq 2$,
…
- 48.9k
2
votes
1
answer
201
views
Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statement
For any infinite set $X$ there is an injection $\varphi$ from $(X\times\{0\}) \cup (X\times\{1\})$ into $X$.
Does this imply the ${\sf AC}$?
- 48.9k
2
votes
1
answer
198
views
Some very weak statements on choice
This is a follow-up question to Does $|(X\times\{0\}) \cup (X\times\{1\})| \leq |X|$ for $X$ infinite imply ${\sf AC}$?
Consider the statements
$(\text{S}1)$ For any infinite set $X$ there i …
- 48.9k
2
votes
1
answer
151
views
Tileable subsets of $\mathbb{Z}\times\mathbb{Z}$
For $t\in \mathbb{Z}\times\mathbb{Z}$ and $A\subseteq\mathbb{Z}\times\mathbb{Z}$ we set $t+A :=\{t+a: a\in A\}$.
Call $A\subseteq\mathbb{Z}\times\mathbb{Z}$ tileable if there is $T\subseteq\mathbb{Z} …
- 48.9k
4
votes
1
answer
266
views
Does "Every infinite set is splittable" imply $\mathsf{AC}$? [duplicate]
We say an infinite set $X$ is splittable if there are $X_1, X_2\subseteq X$ with $X_1\cap X_2 = \emptyset$, $X_1\cup X_2 = X$ and there are bijections $\varphi:X_1\to X_2$ and $\psi:X_1\to X$.
Does t …
- 48.9k
4
votes
1
answer
274
views
Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ "get larger and lar...
Is the following statement consistent in $\mathsf{ZFC}$?
For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have $2^{\aleph_{\lambd …
- 48.9k
2
votes
0
answers
209
views
Countable non-commutative groups such that all proper subgroups are commutative
Is there an infinite non-commutative group $G$ such that every proper subgroup of $G$ is commutative?
- 48.9k