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Results tagged with axiom-of-choice
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user 8628
An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
23
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Does the "three-set-lemma" imply the Axiom of Choice?
Consider the following curious statement:
$(S)$ $\;$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in X$). Then there are subsets $X_1, X_2, X_3 …
- 48.9k
15
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1
answer
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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 …
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13
votes
1
answer
932
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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 …
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11
votes
1
answer
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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
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1
answer
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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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9
votes
1
answer
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Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$?
For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a …
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9
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1
answer
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Non-Ramsey functions $c:[\omega]^\omega\to\{0,1\}$ and the Axiom of Choice
Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$, and let $c:[\omega]^\omega\to\{0,1\}$ be a function. We say that $a\in [\omega]^\omega$ is monochromatic with respect to $c …
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7
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1
answer
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Notions of infinity in $\mathsf{ZF}$ without choice
Consider the following statements about a given set $X$ in in $\mathsf{ZF}$:
(1) There is $x_0\in X$ such that there is a surjective map $\varphi: X\setminus\{x_0\}\to X$.
(2) There is an injective …
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7
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0
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Does the Weak Power Hypothesis imply the Boolean Prime Ideal Theorem?
If there is a bijection $\varphi:x\to y$ between two sets $x$ and $y$, we use the notation $x\simeq y$. The Weak Power Hypothesis is the following statement:
(WPH) For all sets $x, y$, whenever ${\ca …
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6
votes
1
answer
258
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Does "$|{\cal P}_2(X)| = |X|$ for $X$ infinite" imply ${\sf (AC)}$?
This comes from a comment made by user bof in this thread.
Let $X$ be a set, define ${\cal P}_2(X) = \big\{\{a, b\}: a\neq b\in X\big\}$.
Consider the statement
${\sf (S)}$ If $X$ is an in …
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6
votes
1
answer
279
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The Parity Principle and $\mathbf{C}_2$ (choice for $2$-sets)
The Parity Principle states that
if $X\neq \emptyset$ is a set, then there is $\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in \mathcal P(X)$ with $a\mathbin\Delta b = \{x\}$ for some $ …
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5
votes
1
answer
402
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Existence of surjection vs injection over $\sf ZF$
Consider the following statements in $\sf ZF$:
(S) If $A, B$ are nonempty sets, then there is a surjection $s:A \to B$, or there is a surjection $t:B\to A$.
(I) If $A, B$ are sets, then there is an …
- 48.9k
5
votes
1
answer
166
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The equivalence of Dedekind-infinite and dually Dedekind-infinite as a weak form of (AC)
A set $X$ is Dedekind-infinite if there is an injective map $f: X\to X$ that is not surjective.
A set $X$ is dually Dedekind-infinite if there is a surjective map $f: X\to X$ that is not injective.
In …
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4
votes
0
answers
127
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Graphs without maximal vertex-transivite subgraphs
The axiom of choice is of no use when trying to prove that every vertex-transitive subgraph is contained in a maximal vertex-transitive subgraph, because a union of an ascending chain of vertex-transi …
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4
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0
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276
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Quasi-disjoint subsets of an infinite set and $\neg \mathsf{AC}$
Is it consistent with $\mathsf{ZF}$ (without $\mathsf{AC}$) that there is an infinite set $X$ and a subset $S\subseteq\mathcal P(X)$ of the same cardinality as $\mathcal P(X)$ with the property that w …
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