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Results tagged with axiom-of-choice
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user 26809
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.
10
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270
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Strength of claims about extensions of partial preorders and orders to linear ones
Consider these two axioms:
Every partial order extends to a linear order.
Every partial preorder (reflexive and transitive relation) extends to a linear preorder while preserving strict orderings: i. …
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20
votes
0
answers
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Hahn-Banach and the "Axiom of Probabilistic Choice"
Stipulate that the Axiom of Probabilistic Choice (APC) says that for every collection $\{ A_i : i \in I \}$ of non-empty sets, there is a function on $I$ that assigns to $i$ a finitely-additive probab …
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Accepted
Probabilities in a riddle involving axiom of choice
The probabilistic reasoning depends on a conglomerability assumption, namely that given a fixed sequence $\vec u$, the probability of guessing correctly is $(n-1)/n$, then for a randomly selected sequ …
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Unique existence and the axiom of choice
In A definable nonstandard model of the reals, Kanovei and Shelah surprisingly managed to prove the existence of a ZFC-definable (i.e., specifiable via an explicit ZFC construction) nonstandard model …
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3
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Strength of some claims about finitely additive measures on infinite sets?
A quick note that (2) doesn't follow from ZF (assuming ZF is consistent) or even ZF+DC.
(2)+Countable Choice for Finite Sets (CC(fin)) implies that every uncountable set has a non-principal (finite …
- 2,555
6
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4
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609
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Strength of some claims about finitely additive measures on infinite sets?
Assume ZF. Consider the claim:
(1) For any infinite set $\Omega$, there is a finitely additive probability measure $\mu:2^\Omega\to[0,1]$ with $\mu(A) = 0$ whenever $|A|<|\Omega|$.
Then (1) is impl …
- 2,555
10
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Full conditional probabilities and versions of AC?
A probability is a finitely additive measure on a boolean algebra with total measure $1$.
A function $P:\scr B \times (\scr B - \{ 0 \})$ is a full conditional probability on $\scr B$ (for a boolean …