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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.
31
votes
Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?
There are models of ZF+DC in which every subset of every Polish space has the property of Baire (I can try to add references later, I think to Solovay and Shelah, but these are pretty well known). Th …
- 29.7k
17
votes
Accepted
Is it possible to formulate the axiom of choice as the existence of a survival strategy?
It is certainly true that some choice is required.
An isomorphic game (mapping (giraffes, scarves, lion) to (prisoners, hats, warden)) was considered by Hardin and Taylor in their stimulating (and el …
- 29.7k
20
votes
Accepted
Generalized limits on $\ell^\infty(\mathbb{N})$
Yes. In fact, if you work in $\mathsf{ZF}+\mathsf{DC}+$ "all sets of reals have the property of Baire" ($\mathsf{BP}$), say the Solovay or Shelah models, you can prove that $(\ell^\infty)^* = \ell^1$ …
- 4,713
5
votes
Most 'unintuitive' application of the Axiom of Choice?
Since this question has been resurrected...
One of my favorite things about the hat-guessing problem in the question is what happens when you think about it probabilistically. Let's say the hats are …
- 29.7k