# Questions tagged [axiom-of-choice]

The axiom of choice is a common set-theoretic axiom with many equivalents and consequences. This tag is for questions on where we use it in certain proofs, and how things would work without the assumption of this axiom.

1,385 questions
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### Products and the axiom of choice

Here: Universal property of the direct product, proof verification Matematleta noted in the comments, that the definition of the product uses the axiom of choice by default. Why is that? The ...
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### Banach-Tarski nonparadox: Reassembling a ball into two balls with a total volume equal to the original volume

The Banach-Tarski paradox states that a ball can be partitioned into finitely many pieces which can be rotated and translated into two balls identical to the original one. But can a ball be ...
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### References for Axiom of Choice and other Axioms of Set Theory [on hold]

I need references regarding the axiom of choice, variations of the axiom of choice, and other set theory axioms which imply or are consequences of some choice axioms. Some or all of these sources ...
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### Sequential compactness $\rightarrow$ limit point compactness and axiom of choice

A space $S$ is sequentially compact if every sequence has a convergent subsequence. A space $S$ is limit point compact is every infinite subset has a limit point in $S$. Proving sequential ...
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### ℩-descriptor and the axiom of choice?

I've recently finished Nederpelt and Geuvers Type Theory and Formal Proof, and I remember somewhere in the book the authors mentioned that their $\iota$-descriptors (which are used to refer to the ...
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### is it consistent with AC that every set is measurable? [duplicate]

Does every set is measurable follow from AC, or from it's negation? I think that by Vitali's construction from the AC follows that some set is not masurable. But here in the 1st comment they claim ...
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### Partition of positive reals with each part closed under addition without choice

It is an easy exercise using transfinite recursion to prove the following (in ZFC): There exists sets $S,T$ that partition $\mathbb{R}_{>0}$ such that each of $S$ and $T$ is closed under ...
### Not $\sigma$-compact set without axiom of choice
Today in measure theory, we introduced the concept of a $\sigma$-compact set, which is a set which can be expressed as the countable union of compact set. Since the set of $\sigma$-compact sets ...