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.

learn more… | top users | synonyms

11
votes
2answers
924 views

Equivalent statements of the Axiom of Choice

As a little project for myself this winter break, I'm trying to go through as much of Enderton's Elements of Set Theory as I can. I hit a snag trying to show two forms of the Axiom of Choice are ...
3
votes
1answer
568 views

Dimension of the sequence space and its dual, depending on status of (AC) and (CH)

Let's consider the sequence space $E =\mathbb R^{\mathbb N}$. If I believe in Choice, I have an isomorphism $E \simeq \mathbb R^{(\mathfrak c)}$ for some cardinal $\mathfrak c$. I further have some ...
15
votes
2answers
2k views

Axiom of Choice Examples

In the wikipedia article, two examples are given which use/ do not use the axiom of choice. They are: Given an infinite pair of socks, one needs AC to pick one sock out of each pair. Given an ...
25
votes
5answers
907 views

Axiom of choice - to use or not to use

I was wondering if there are examples of results in mathematics that were first proven using axiom of choice and later someone found a proof of the result without using the axiom of choice.
7
votes
9answers
905 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
17
votes
3answers
687 views

Where do we need the axiom of choice in Riemannian geometry?

A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the ...
1
vote
1answer
253 views

Specific equivalent to the Axiom of Choice involving the empty set

I'm trying to remember a particular theorem of ZF but unfortunately my memory is quite incomplete. The theorem is of the form (some set operation) is either (expected answer) or the empty set. If ...
14
votes
2answers
946 views

Transfinite Induction and the Axiom of Choice

My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set? I have read ...
64
votes
8answers
18k 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 ...