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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.
3
votes
Construction of a maximal ideal
This may be relevant:
Call an ideal on your ring free if there is no point where all functions in this ideal vanish. Then the following holds:
The existence of a definable (in the language of Zerme …
- 9,631
2
votes
Construction of a proper uncountable subgroup of $\mathbb{R}$ without Choice.
There's a Borel example of such groups here
- 9,631
178
votes
Accepted
Does every non-empty set admit a group structure (in ZF)?
In ZF, the following are equivalent:
(a) For every nonempty set there is a binary operation making it a group
(b) Axiom of choice
Non trivial direction [(a) $\to$ (b)]:
The trick is Hartogs' const …
- 9,631