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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.
5
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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
3
votes
1
answer
143
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Ascending chain of vertex-transitive graphs
For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x, y\in X\text{ and } x\neq y\big\}$.
Let $G=(V,E)$ be a simple, undirected graph, and suppose ${\cal V}$ is a collection of subsets of $V$ such that f …
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3
votes
0
answers
259
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"Matryoshka" sets and the Axiom of Choice
Consider the following two very similar statements in ${\sf ZF}$:
(Mat_1) There is a set $A$ a map $\alpha: \omega \to {\cal P}(A)$ such that for all $n\in \omega$ we have $\alpha(n+1) \subseteq \alp …
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1
vote
1
answer
273
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Strictly descending sequences of sets, the Partition Principle, and the Boolean Prime Ideal ...
In ${\sf ZFC}$ it can be easily proved that we cannot have infinitely descending sequences of cardinalities, that is, the following statement does not hold:
(DescSeq) There is a set $A$ a map $\alpha …
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3
votes
1
answer
442
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Choice sets and the axiom of choice
Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a cover if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a choice set for ${\cal C}$ if $|D\cap c| …
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3
votes
1
answer
181
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Injections without fixed-points and the Axiom of Choice
Consider the following statement in $\sf ZF$:
(I) Whenever $X$ is a set with more than $1$ element, there is an injective map $\iota: X\to X$ such that $\iota(x) \neq x$ for all $x\in X$.
The Axiom …
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4
votes
0
answers
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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 …
- 48.9k
7
votes
1
answer
320
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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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6
votes
1
answer
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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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3
votes
0
answers
207
views
Weak Power Hypothesis with injections instead of bijections
Let $x,y$ be sets. We use the following notation:
$x\simeq y$ means that there is a bijection $\varphi:x\to y$, and
$x\leq y$ means that there is an injection $\iota:x\to y$.
The Weak Power Hypothes …
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7
votes
0
answers
196
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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
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 $ …
- 48.9k
13
votes
1
answer
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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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4
votes
1
answer
207
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Weak Power Hypothesis and Dependent Choice
Consider in $\newcommand{\ZF}{{\sf (ZF)}}\ZF$ the following statement:
Weak Power Hypothesis (WPH): if $X,Y$ are sets and there is a bijection between $\newcommand{\P}{{\cal P}}\P(X)$ and $\P(Y)$, th …
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4
votes
Accepted
Proof of the axiom of choice for finite sets in ZF
It holds vacuously for $A = \emptyset$: then $\bigcup A = \emptyset$ and the empty function $\emptyset: \emptyset \to \emptyset$ trivially fulfills $f(a) \in A$ for all $a\in A$ -- because it is impos …
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