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Results tagged with axiom-of-choice
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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.
9
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2
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875
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Can we have an infinite sequence of decreasing cardinality all terms of which have equal siz...
Is the following consistent with $\text{ZF}$?
There exists a set $S=\{x_1,x_2,x_3,...\}$ such that:
$|x_{i+1}| < |x_i|$
$\forall m,n \in S (|P(m)|=|P(n)|)$
Where cardinality $``||"$ is defined …
- 11.3k
9
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2
answers
1k
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Are there known examples of sets whose power set is equal in size to power set of larger set...
The question of existence of sets $x,y$ such that
$$|x|<|y| \wedge |P(x)|=|P(y)|$$
is known to be independent of $\text{ZFC}$!
But are there known examples of sets fulfilling the above condition
th …
- 11.3k
-1
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1
answer
257
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Is Proper Class Choice equivalent to Global Choice?
Working in "MK-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom:
Axiom of Super-Choice:$$\forall \ relation \ R \ \exists F \subset R \ ( …
- 11.3k
1
vote
0
answers
105
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Can small class choice be weaker than global choice and stronger than set choice + collection?
In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*". However if t …
- 11.3k
0
votes
0
answers
86
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Does existence of those cyclic cardinals imply a kind of Choice?
Working in $\sf ZF - Fnd$, add the following axiom:
AntiFoundation: $\forall x: x \neq \emptyset \to \exists! y: y \in y \land y \sim x$
where "$\sim$" stands for existence of a bijection.
In this the …
- 11.3k
2
votes
1
answer
132
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What would be examples of modified Scotts cardinals with less structure?
The intention behind this posting is to arrive at a definition of cardinality that can work in $\sf ZF$, and at the same time doesn't exhaust the whole of $V$. The known definition uses Scott's trick, …
- 11.3k
1
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0
answers
153
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Can we have a spectrum of intermediate choice properties between set choice and global choice?
Global choice is equivalent to saying $|V|=ON$, while ordinary choice is equivalent to saying that $V= H_{< ON}$. So the relative cardinalities of $V$ and $ON$ seems to affect the degree of choice ava …
- 11.3k
6
votes
2
answers
452
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Can Deep Choice entail Axiom of Choice?
Deep Choice:
$\forall X \ [\forall a,b \in X \, ( a \neq \emptyset \land (a \neq b \to a \cap b = \emptyset)) \to \\ \exists Y \exists f \,(f: X \rightarrowtail Y \land \forall x \in X \,( f(x) \in \ …
- 11.3k
-4
votes
1
answer
266
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Is Nested Selection equivalent to AC?
Nested Selection: For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and each el …
- 11.3k
1
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0
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76
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Is this version of Nested Selection equivalent to AC?
This is an endeavor to salvage "Nested Selection" principle presented in a prior posting.
Define
$ \begin{align} Y \text { is } \Phi \text{-image of } X \iff &\forall a \in X \exists b \in Y: \Phi(b, …
- 11.3k
1
vote
2
answers
166
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Does n-well ordered choice schema imply the axiom of choice?
Define: $\operatorname {wo}^n(x) \iff \forall y (y \in^n x \to \operatorname {wo} (y))$
Where $\operatorname {wo}(y)$ refers to $y$ being well orderable.
Where $y \in^0 x \iff y=x \\ y \in^{n+1} x \if …
- 11.3k
3
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0
answers
147
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Can well-ordering of the universe due to global choice survive extensive failure of Extensio...
That axiom of global choice leads to the well-ordering of the universe given the other axioms of Zermelo set theory is a famous result.
Now, if we weaken the power set axiom to the axiom stating that …
- 11.3k
5
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1
answer
427
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Is there a class choice principle over MK that is equivalent to class well ordering over MK?
$\sf MKCWO$ is the theory obtained by adding a new primitive binary relation $\prec$ to the signature of $\sf MK$ and axiomatize that $\prec$ is a well order on classes, that is:
$\textbf{Transitive: …
- 11.3k
3
votes
1
answer
252
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Is the Class Well Ordering principle "CWO" the maximal choice principle?
In a prior posting, the Class Well-Ordering principle "$\sf CWO$" was presented, which simply states that there is a well-ordering over all classes of $\sf MK$. On the other hand, it is known that the …
- 11.3k
2
votes
1
answer
583
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Is there a strict limit on choice principles in $\sf ZFC$?
Is there a principle $\sf P$ that $\sf ZFC$ [or some suitable extension of it] proves to be a strict limit on choice principles?
By a choice principle I mean a sentence (or scheme) that is equivalen …
- 11.3k