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Results tagged with axiom-of-choice
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user 50073
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.
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Is Axiom of Choice for convex sets of distributions on naturals necessary?
Take any family $(S_i)_{i∈I}$ such that each $S_i$ is a convex set of functions $f : ℕ→[0,1]$ where $\sum_{k∈ℕ} f(k) = 1$. By "convex" we mean that for any $f,g∈S_i$ and any $a,b∈[0,1]$ such that $a+b …
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Unnecessary uses of the axiom of choice
It is easy to prove the following in Z+CC (Zermelo plus countable choice):
Every uncountable closed set of reals is in bijection with the reals.
I was informed by Asaf that it can be proven in ZF (n …