# 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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### Applications of the axiom of choice to non-existence proofs

I've been thinking about Monsky's Theorem that it is impossible to partition a square into an odd number of triangles of equal area. The proof depends on a theorem of Chevalley to extend the $2$-adic ...
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### $\mathsf{AC}_{\omega_1}$ and $L(\mathbb R)$

Assume that the Axiom of Choice holds in $V$, the von Neumann universe. As Andreas Blass explained, $\mathsf{DC}$ holds in $L(\mathbb R)$. Is this the case for $\mathsf{AC}_{\omega_1}$ too? That ...
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### Why is AC needed to assert the existence of some infinite sets but not others?

In my undergraduate review for topology, we use axiom of choice to prove a result about the surjectivity of projection mappings. I have also read through a number of posts about AC on the site, and ...
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### Čech functions and the axiom of choice [migrated]

A Čech closure function on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$,...
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### On some Hahn-Banach equivalents

This question is about some equivalents of the Hahn-Banach theorem in $\textsf{ZF}$ set theory. As far as I know, the definitive reference for this sort of thing is Howard & Rubin's Consequences ...
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### Are finite projective modules locally free without axiom of choice?

Let $A$ be a commutative ring. An $A$-module $M$ is called a finite projective module if there exists a module $N$ and a natural number $n\in\mathbb N$ such that $M\oplus N\cong A^n$, and an $A$-...
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### If $x$ is a set of cardinality $\kappa$ and $\lambda<\kappa$, do we need the axiom of choice to prove that there is a subset of $x$ of size $\lambda$?

Suppose we have a set $x$ of cardinality $\kappa$. If $\lambda$ is a cardinal lower then $k$, normally we can say that there exists a subset $y$ of $x$ of size $\lambda$. To prove this - I believe - ...
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### Why does axiom of choice not imply the set of real numbers is countable?

The axiom of choice implies all sets can be well ordered. If that is true, you can well order the set of real numbers and the set of the integers. Now, why can one not just pair the set of real ...
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### Separating a point and a compact subset in a Hausdorff space without Choice

Let $K$ be a compact subset of a Hausdorff space $X$, and $x \not \in K$. Then there are disjoint open sets $U,V$ with $K \subseteq U$ and $x \in V$. The proof of this result that I have seen uses ...
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### What Choice Axiom allows you to choose from a set-sized collection of Classes?

The standard axiom of choice states that every set of non-empty sets has a choice function. And the axiom of global choice states that every class of non-empty sets has a choice function. But I’m ...
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### How to show that $f:\mathbb{R}\to \mathbb{R}$ defined as $f(x)=x$ can be decomposed as a sum of two periodic functions?

I read somewhere that the proof depends on AC. But how about showing it explicitly i.e finding an explicit example that satisfies this? I would like to see both proofs, the one that depends on AC ...
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### Is there a specific term for a function that exists solely as a set of ordered pairs and cannot be described by a mathematical formula?

I've been reading about the Axiom of Choice, and my current understanding is that we can assert a choice function exists even in cases where it may be impossible to construct a deterministic formula ...
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### Let A be an infinite set. Show that there exists a proper subset B of A such that |A| = |B|. Does your proof use the Axiom of Choice? [duplicate]

Let A be an infinite set. Show that there exists a proper subset B of A such that |A| = |B|. Does your proof use the Axiom of Choice? Any ideas on how to prove this?
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### Question on a step of “A Simple Proof of Zorn's Lemma” by Lewin

I'm reading Jonathan Lewin's "A Simple Proof of Zorn's Lemma" and cannot see the justification for one statement (which I've labeled Lemma 3 below). I'll summarize the proof below (adding a few lemma ...
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### (Long) Detailed Proof of Kőnig's Lemma (Explicit, Down to Axiom of Choice)

Kőnig's Lemma states that in an infinite, locally finite, connected graph $G$, there exists an infinite one-way path (a ray). The proof of it in my graph theory book (Introduction to Graph Theory, 4th ...
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### Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

Suppose I have an equation in some reflexive separable Banach space $X$: $$Au=f$$ for given data $f$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
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### Relationship between different versions of AoC in the context of a specific set (in ZF)

We can show that various versions of AoC, restricted to a specific set, are equivalent. For example, the following are equivalent for any set, $X$ (in ZF): [$A_X$] $X$ is well-orderable [$B_X$] ...
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### Explicit isomorphism between $L^\infty[0,1]$ and its hyperplanes

The Banach space $L^\infty[0,1]$ is isomorphic to all its hyperplanes (closed linear subspaces of codimension one). One way of seeing this is the following: all hyperplanes of a Banach space are ...
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### Understanding where my naive attempt to prove Countable Choice out of Finite Choice fails

I am aware of this and this topic, but I would like to receive a clarification concerning the foregoing naive attempt to prove Countable Choice, to see if I understood properly the question. Assume ...
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### Is the axiom of choice necessary for “a set is open iff it is a neighborhood of all its points”? [duplicate]

Note: by neighborhood of $x$ we mean a set $N$ s.t. there exists open $U$ s.t. $x \in U \subseteq N$. We only care about the "if" as the "only if" is trivial. The usual proof goes as follows. ...
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