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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 …
- 47.4k
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votes
1
answer
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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 …
- 39.8k
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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 …
- 48.9k
3
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1
answer
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A possible ${\sf (ZF)}$-theorem in the spirit of the $3$-set-lemma
The number $3$ plays an interesting role in the following statement:
$\newcommand{\S}{\sf(S_3)}\S$ 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 …
- 13.4k
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 …
- 39.8k
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 …
CommunityBot
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1
answer
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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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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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3
votes
2
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Posets such that the collection of principal down-sets does not have property ${\bf B}$
We say that a hypergraph $H=(V,E)$ has property ${\bf B}$ if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$.
Let $(P,\leq)$ be a pa …
- 11.4k
4
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0
answers
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The monochromatic principle and the axiom of choice
For any set $A\neq\emptyset$, denote by $[A]^A$ the collection of sets $B\subseteq A$ such that there is a bijection $\varphi:B\to A$. If ${\cal S}\subseteq [A]^A$, we say that $B\in[A]^A$ is monochro …
- 48.9k
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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 …
- 236k
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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 …
- 13.4k
3
votes
0
answers
207
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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 …
- 48.9k
9
votes
1
answer
506
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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 …
- 48.9k
3
votes
1
answer
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Does the partition principle imply (DC)?
For sets $x, y$ we write $x\leq y$, if there is an injection $\iota: x \to y$, and we write $x \leq^* y$ if either $x = \emptyset$ or there is a surjection $s: y \to x$. In ${\sf (ZF)}$ we have that $ …
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