# Tagged Questions

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.

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### How to show that the event that a prisoner does no go free is not measurable

I was reading this webpage a few months ago about the following problem- A countable infinite number of prisoners are placed on the natural numbers, facing in the positive direction (ie, everyone ...
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### In $ZF+ \neg C$ can we always find a bijection from a given set $A$ to a transitive set?

Let us write $|A| = |B|$ iff there is a bijection $f \colon A \rightarrow B$. Working in $\operatorname{ZF + \neg C}$, can we prove that for any $A$ there is a transitive $B$ with $|A| = |B|$? This ...
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### The generalized Axiom of Dependent Choice (DC) - is this a valid generalization?

After studying the axiom of dependent choice, I tried to think of a possible generalization of the axiom that would work in a similar way on infinite uncountable sets: by replacing the binary relation ...
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### Is there a turing machine for which halting is equivalent to the Axiom of Choice or its negation?

As seen in "A Turing machine for which halting is outside ZFC", Gödel's incompletness theorem can that there a turing machines for which halting can not be decided. My question is, is there a turing ...
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### Topological properties of $[0,\omega_1)$ without choice.

Reading this wikipedia article, I arrived at the fact that $\omega_1$ can exist without choice. Since the proof I know of the fact that $[0,\omega_1)$ is sequentially compact depends on the fact that ...
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### Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice?

Why does a proof of $\exists f: X\to Y$ injection $\iff \exists g: Y \to X$ surjection requires the axiom of choice? This question is answered here: There exists an injection from $X$ to $Y$ if and ...
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### Can it be proved without the axiom of choice that every cardinal is comparable with every finite cardinal?

Can it be proven in ZF, without using the axiom of choice, that every finite set is a universal size comparator, meaning, is comparable with every set in terms of size? And what is the proof?
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### Over ZF does “every non-seperable Hilbert space has an orthonormal basis” imply “there exists a non-Lebesgue measurable set”?

I know from this question that it's an open problem whether or not the existence of a dense orthonomral basis for every real or complex Hilbert space $(\text{B}_\text{orth})$ implies the axiom of ...
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### Does construction of infinite product measure require axiom of choice?

I am learning about infinite (countable) product measure, which the exact statement of the theorem I write below. I was wondering if the theorem requires axiom of choice or not. I would appreciate any ...
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### Where is the axiom of choice used in Rudin's proof of “the countable union of countable sets is countable”?

Baby Rudin proves that the countable union of countable sets is countable. From reading other proofs online, the axiom of choice has to be invoked; however, I'm not seeing immediately where that is ...
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### W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: (...
### If a set $S$ has a choice function, does $\bigcup S$ have one too?
I have an exercise in a book that asserts that if a set $S$ has a choice function on it, then so does the union of all its elements $\bigcup S$ (without assuming the axiom of choice). I, however, have ...