# 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. Use this tag in tandem with (set-theory).

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### Countable Choice from Finite Sets

Consider the following 4 statements: Axiom of countable choice Axiom of countable choice from finite sets Axiom of countable choice from Dedekind finite sets Existence of a choice function for any ...
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### Axiom of Choice in characterizing openness in subspace

Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$. $E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$. The forwards direction usually ...
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### Why is the Axiom of Choice Necessary in ZFC

Within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice $(ZFC)$, when we considered the method of constructing the set of natural numbers, we regarded it as the smallest inductive ...
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### Are the sets of the form $a=\{a\}$ different?

Suppose we adopt all the ZF axioms except the axiom of foundation. I suppose even adding the AC would not harm my question. Somewhere on the internet, I read that in this case, the axiom of ...
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### Is every set an image of a totally ordered set?

It is known that the statement "Every set admits a total order" is independent of ZF. See here, for example. However, can it be proven in ZF that for every set $Y$, there exists a totally ...
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### How does this proof that for every infinite set $A$ there exists an injection $f : \mathbb{N} \rightarrow A$ rely on the axiom of choice?

I'm currently taking a course in proof-writing, and the following question came up on a problem set: Prove that for every infinite set $A$, there is a one-to-one function $f : \mathbb{N} \rightarrow A$...
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### What is the most general form of the distributive law for $\cup$ and $\cap$?

The following forms of one of the distributive laws increase in generality as we move down the list: \begin{align} (A\cup B)\cap C &= (A\cap C)\cup(B\cap C)\\\\ &\,\Uparrow\\\\ (\bigcup_{a\in ...
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### Can the separability of a space depend on the axiom of choice?

Does there exist a topological space $X$ such that in $\mathsf{ZFC}$, $X$ is separable, but such that it is consistent with $\mathsf{ZF}$ that $X$ is not separable? The motivation behind this question ...
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### No 3 vectors independent over $\mathbb{Z}$ in $\mathbb{Z}^2$, without AoC

Q: Are there three $\mathbb{Z}^2$ vectors independent over $\mathbb{Z}$ ? Context: This problem arise naturally when I'm characterizing possible sub-"latice" in $\mathbb{Z}^2$. Formally let ...
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### How Can I Finish off this Proof on Axiom of Choice?

Question Prove that the following is equivalent to the Axiom of Choice: Every surjective map has a right inverse. Attempt I already showed that if the Axiom of Choice holds, then every surjective map ...
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### How to prove consistency with choice for large cardinal extensions?

How can we know if an extension of $\sf ZF$ by some large cardinal property that results in a consistency strength beyond $0^{\#}$ is compatible with choice or not? I mean the easiest way to know if ...
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### Without choice, what can be the (finite) automorphism groups of $\mathbb F_2$-vector spaces?

My motivation for this question is similar to the one in this question. However, that question only asks about the possibility of an infinite $\mathbb F_2$-vector space having trivial automorphism ...
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### How much choice is needed to prove that $|G| \nless |G/ \sim|$ for any equivalence relation $\sim$ on set $G$?

$G/ \sim$ is the set of $\sim$-equivalence classes in $G$ and $|G/ \sim|$ is the cardinality of $G/ \sim$. $|A| \leq |B|$ means that there is an injective function from $A$ to $B$. $|A| < |B|$ ...
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### If $f$ is surjective, it has a right inverse

I've been struggling to understand how the surjection of a function $f : X \rightarrow Y$ implies that it has a right inverse. My questions basically reside on the application of the axiom of choice ...
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### How to prove that CC($\mathbb R$) is true in permutation models [closed]

I've familiarised myself with models of set theory and am beginning to understand the basics, but am still very far away from being a proper model theorist. I currently live under the impression that ...
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### Finite Content Not Continuous on the Empty Set

I have come across a lemma which states: Let $\mu$ be a finite content on an algebra $\mathcal{A}$. $\mu$ is a pre-measure if and only if $\mu$ is continuous on the empty set, i.e., for every sequence ...