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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.
20
votes
1
answer
534
views
Almost orthogonal maps $f:\omega \to \{-1,1\}$
Let $\omega$ denote the set of non-negative integers. For sets $A,B$, let $B^A$ denote the set of maps $f:A\to B$. For $f,g\in\{-1,1\}^\omega$ we say that $f,g$ are almost orthogonal if there is $C_0\ …
- 48.9k
15
votes
1
answer
1k
views
Parity and the Axiom of Choice
Motivation. The three-dimensional cube can be formalized by $\mathcal P(\{0,1,2\})$ where vertices $x,y\in\mathcal P(\{0,1,2\})$ are connected by an edge if and only if their symmetric difference $x\m …
- 48.9k
13
votes
4
answers
729
views
Cardinality of the set of functions commuting with $f:X\to X$
If $X$ is an infinite set and $f:X\to X$, do we have $$\big|\{g:X\to X: g\circ f =f \circ g\}\big|\geq |X| ?$$
- 48.9k
13
votes
1
answer
512
views
"Drinking number" of a graph
Motivation. A while ago I attended a party and I only knew some, but not all, of the attendees. There were 2 kinds of drinks: beer and soda. I noticed that amongst my acquaintances, more than half dra …
- 48.9k
13
votes
1
answer
274
views
Can any poset of cardinality $\leq 2^{\aleph_0}$ be embedded in ${\cal P}(\omega)/(\text{fin...
We endow ${\cal P}(\omega)$ with an equivalence relation by saying that $A\simeq_{\text{fin}} B$ iff the symmetric difference $A\Delta B$ is finite. The resulting set of equivalence classes is denoted …
- 48.9k
13
votes
1
answer
932
views
Cantor-Bernstein with "weakly injective" functions
Let us call a map $f: X \to Y$ between non-empty sets a "weak injection" if $f^{-1}(\{y\})\subseteq X$ is finite for every $y \in Y$.
Recall that the (Schroeder-)Cantor-Bernstein-Theorem (sometimes ab …
- 48.9k
13
votes
1
answer
631
views
$T_2$-spaces where all non-empty open sets are homeomorphic
We say that a $T_2$-space $(X,\tau)$ has homeomorphic open sets if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$.
The rationals with the Euc …
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12
votes
1
answer
678
views
Graphs $G$ with $G \cong \text{Aut}(G)$
Let $G=(V,E)$ be a simple, undirected graph. By $\newcommand{\Aut}{\text{Aut}}\Aut(G)$ we denote the collection of graph isomorphisms $\varphi:G\to G$. We let $$E(\Aut(G)) =\big\{\{\varphi, \psi\}:\va …
- 48.9k
12
votes
1
answer
291
views
Can the cardinal $2^{\aleph_0}$ be order-embedded in ${\cal P}(\omega)/(\text{fin})$?
For $A,B\in{\cal P}(\omega)$ we say $A\subseteq^* B$ if $A\setminus B$ is finite (that is, $A$ is "almost contained" in $B$). We write $A\simeq_{\text{fin}} B$ if $A\subseteq^* B$ and $B\subseteq^* A$ …
- 48.9k
11
votes
1
answer
446
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?
- 48.9k
11
votes
1
answer
746
views
Generalized limits on $\ell^\infty(\mathbb{N})$
Let $\ell^\infty(\mathbb{N})$ denote the set of bounded real sequences $(a_n)_{n\in\mathbb{N}}$. The $\lim$ operator is a partial linear operator from $\ell^\infty(\mathbb{N})$ to $\mathbb{R}$. With t …
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10
votes
1
answer
2k
views
Does the axiom of choice follow from the statement "Every simple undirected graph is either ...
Using the Well-Ordering Principle, which is equivalent to the Axiom of Choice, it can be proved that
(S): for every simple, undirected graph $G$, finite or infinite, either $G$ or its comple …
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10
votes
2
answers
579
views
Maximal Abelian subgroups of $S_\omega$
Let $S_\omega$ be the group of permutations (bijections) $\varphi:\omega\to\omega$, together with composition as binary operation.
Zorn's Lemma implies that every commutative subgroup of $S_\omega$ is …
- 48.9k
10
votes
2
answers
243
views
Minimal refinements of open covers of $T_2$-spaces
Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if
$\bigcup {\cal U} = X$, and
$X\notin {\cal U}$.
${\cal U}$ is minimal if for all $U_0\in {\cal U}$ we ha …
- 48.9k
10
votes
2
answers
973
views
Size of maximal intersecting families
Let $X$ be a non-empty set, and let ${\cal S}\subseteq {\cal P}(X)$ be family of non-empty subsets of $X$. We say that ${\cal S}$ is intersecting if any two members of ${\cal S}$ have non-empty inters …
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