Linked Questions

1
vote
0answers
88 views

Naive understanding of choice axiom [duplicate]

It's written in many resources that if we consider only finite sets, that choice axiom can be skipped and be proven from other ZF axioms. I remember I've read the following explanation. If $A$ is a ...
1
vote
1answer
47 views

Uses of Axiom of Choice [duplicate]

I am a first-year maths student but I occasionally drift away from our taught material. Some years ago I saw the ZFC axioms for the first time, but now that I am in college, and although the stuff I'...
2
votes
1answer
55 views

Does the proof of rank-nullity theorem from Lang's “Linear Algebra” involves axiom of choice? [duplicate]

Disclaimer: I never had (yet c:) a rigorous exposure to set theory (independence proofs, and similar stuff...). I was wondering if, in the following proof of the rank-nullity theorem form Lang's "...
115
votes
8answers
42k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of Choice ...
31
votes
9answers
4k views

Why is the Axiom of Choice not needed when the collection of sets is finite?

According to Wikipedia: Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object ...
12
votes
5answers
2k views

Do we need Axiom of Choice to make infinite choices from a set?

According to the answers to this question, we do not need choice to pick from a finite product of nonempty sets, even if each of the sets is infinite. The axiom of choice is required to ensure that a ...
12
votes
3answers
4k views

Infinite sets with cardinality less than the natural numbers

Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?
12
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3answers
2k views

How do I choose an element from a non-empty set?

Suppose I have a non-empty set $A$. How do I choose an element $x\in A$? More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is ...
5
votes
7answers
630 views

Existence in variants of the Axiom of Choice

Let $\{A_i: i \in I\}$ be a nonempty family of nonempty sets. Why is it allowed to prove the Axiom of Choice using the Well Ordering Principle as follows: There is a well-ordering of $\cup_{i \in I}...
6
votes
2answers
593 views

Does the axiom of choice have any use for finite sets?

It is well known that certain properties of infinite sets can only be shown using (some form of) the axiom of choice. I'm reading some introductory lectures about ZFC and I was wondering if there are ...
4
votes
2answers
1k views

$d(x,A)=0\iff $ every neighborhood of $X$ contains a point of $A$

Mendelson, Introduction to Topology, p.52 $(8)$. Let $A$ be a non-empty subset of a metric space $(X,d)$. Let $x\in X$. Prove that $d(x,A)=0$ if, and only if, every nieghborhood $V$ of $x$ contains ...
6
votes
3answers
265 views

Axiom of Choice (for example in the Snake Lemma)

If we have to make a choice, but in the end it doesn't matter what choice we made, did we really make a choice to begin with? More explicitly, somewhere in the standard diagram-chasing proof of the ...
1
vote
2answers
106 views

Can ZF prove that every surjection with finite codomain has a section?

Can ZF prove the following? Theorem. Let $X$ and $Y$ denote sets and $e : X \rightarrow Y$ denote a surjection. If $Y$ is finite, then $e$ has a section, i.e. a function $m : Y \rightarrow X$ such ...
3
votes
1answer
129 views

Finite family of infinite sets / A.C.

Let $\{A_i\mid i\in n\}$ be a finite family of infinite sets. ( That is, $A_i$ is infinite for every $i\in n$ and $n\in \mathbb{N}$) Here, we can choose representative $a_i$ from each $A_i$ and ...
1
vote
1answer
161 views

The class of well-founded sets satisfies the axiom of foundation and the axiom of choice

I'm reviewing a theorem we saw in class. The class $WF$ satisfies the axiom of foundation. Furthermore, if the axiom of choice is true, then $WF$ satisfies the axiom of choice. $WF$ here is such ...

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