# 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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### Finite Unions of Equinumerous Sets

Is the following statement provable in ZF? Assume that $A_1$ and $A_2$ are two infinite sets. If $A_1$ is equinumerous with $B_1$ and $A_2$ is equinumerous with $B_2$, then $A_1 \cup A_2$ is ...
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### Does every proof that abelian groups are amenable rely on the axiom of choice?

Does every proof that abelian groups are amenable rely on the axiom of choice? So far, any proof I've seen that all, say countable discrete, abelian groups are amenable requires some sort of argument ...
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### Example of surjective function without right-inverse (without AoC)

As I understand from this question, without the axiom of choice, we can have surjective functions without right-inverse. Is that correct? Is there any such example of a surjective function where we ...
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### Without the axiom of choice, is there always a partition refining a collection of sets that is the same size?

Let $\Omega$ be a set and let $\mathcal{C} \subset \mathcal{P}(\Omega)$ be some collection of subsets that covers $\Omega$, so $\bigcup_{C \in \mathcal{C}} = \Omega$. We would like to find a partition ...
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### Connected subset of $\mathbb{R}^2$ with disconnected sections

I call a subset of $\mathbb{R}^2$ $0$-dimensional if it has a (countable) basis made of clopen sets wrt its relative topology. For example, the subset $\mathbb{Q}\times\{0\}$ is $0$-dimensional. Now ...
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### Suppose $a$ is small and $F: a \rightarrow b$ is a surjection. Is $b$ small?

Define a class as small if there is no injection from $V$ into the class. Suppose then that $a$ is small and let $F: a \rightarrow b$ be a surjection. Is $b$ small? If we assume Choice, the answer is ...
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### Proving that a set is dedekind-infinite if and only if it has a countably infinite subset

I am interested in proving the following statement: A set is Dedekind-infinite if and only if it has a countably infinite subset. Here is my attempt: $(\Leftarrow)$ Suppose that $A$ has a countably ...
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### Proving without the Axiom of Choice (A.C.) that increasing real functions have countable discontinuities

I know three different but similar proofs of the statement: If $f:\mathbb{R}\to\mathbb{R}$ is an increasing function, then there are at most countably many discontinuities. But each of the proofs ...
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### Can't we choose any particular element from $^{A}B$ without the axiom of choice?

In "Elements of Set Theory" by Herbert B. Enderton, pg. 52 For sets $A$ and $B$ we can form the collection of functions $F$ from $A$ into $B$. Call the set of all such functions $^{A}B$: ^...
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### Linear iterative functional equation always has a solution

The following is, I assume, a known fact---however, the only proof I know uses the axiom of choice (then a theorem about well-ordered sets as well as transfinite recursion). I was wondering if anyone ...
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### Is it possible: Construct real numbers without Axiom of Choice? [duplicate]

I'm trying to construct real numbers from the Peano axiom. Then one question arises, which begins with that I have not used the Axiom of Choice so far. In the process of producing integers from ...
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### The Axiom of Choice and a definition of addition in a quotient space of a vector space

I am thinking about the Axiom of Choice and I am trying to understand the Axiom with some but a little progress. Some time ago I could not understand why the obvious "proof" of the Axiom of ...
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### How do we understand the axiom of Choice? [duplicate]

I was pondering about the infamous statement given by Russell to understand choice, "To choose one sock from an infinitely many pairs of socks one requires axiom of choice, for shoes the axiom is ...
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### Characterization of basis in terms of universal property: axiom of choice

I wonder if the proof of the following statement requires the axiom of choice: (Characterization of basis in terms of universal property) Let $V$ be a vector space, and let $S$ be a non-empty subset ...
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### Choice principles implied by antichain principle in ZF - Foundation

It is known that in ZF, the axiom of choice and the antichain principle (every partially ordered set has a maximal antichain) are equivalent. However, in ZF-Foundation, while AC still implies the ...
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### Confusion about the necessity of the axiom of choice for infinite sets [duplicate]

Given a family $(X_i)_{i \in I}$ of non-empty sets, why cannot one introduce a function $f : I \to \bigcup_{i \in I} X_i$ as follows : for all $i \in I$, there exists $x \in X_i$ — let $f(i) := x$, ...
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