Axiom of Choice
One of the most well-known and debated axioms in modern set theory, proposed by Zermelo in 1904. It asserts that for any family of non-empty sets there exists a choice function selecting one element from each set. ZFC set theory includes this axiom, while ZF set theory does not.
Timeline (last 90 days)
The Partition Principle cannot be proven without the Axiom of Choice.
The Axiom of Choice was proposed by Ernst Zermelo in 1904.
The Axiom of Choice asserts that for any family of non-empty sets, there exists a choice function that selects one element from each set.
The Banach–Tarski paradox, which allows a ball to be decomposed and reassembled into two balls of equal size, is a counterintuitive consequence derived from the Axiom of Choice.
ZF set theory does not include the Axiom of Choice, while ZFC set theory does.
维塔利集是不可测集，其构造依赖选择公理
All Facts (6)
The Partition Principle cannot be proven without the Axiom of Choice.
50%UnverifiedThe Axiom of Choice was proposed by Ernst Zermelo in 1904.
50%UnverifiedThe Axiom of Choice asserts that for any family of non-empty sets, there exists a choice function that selects one element from each set.
50%UnverifiedThe Banach–Tarski paradox, which allows a ball to be decomposed and reassembled into two balls of equal size, is a counterintuitive consequence derived from the Axiom of Choice.
50%UnverifiedZF set theory does not include the Axiom of Choice, while ZFC set theory does.
50%Unverified维塔利集是不可测集，其构造依赖选择公理
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