The axiom-of-choice tag has no wiki summary.

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### Cardinality: Why is there no “ℵ½”?

A wikipedia page/paragraph on ℵ₁ states:
"The definition of ℵ₁ implies (in
ZF, Zermelo-Fraenkel set theory
without the axiom of choice) that no
cardinal number is between ℵ₀ and
ℵ₁."
"If the axiom ...

**8**

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**8**answers

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### Choice vs. countable choice

This question arose after reading the answers (and the comments to the answers) to Why worry about the axiom of choice?.
First things first. In my intuitive conception of the hierarchy of sets, the ...

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**15**answers

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### Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...

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**0**answers

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### Explicit element of $(\ell^{\infty})^* - \ell^1$? [duplicate]

Possible Duplicate:
What’s an example of a space that needs the Hahn-Banach Theorem?
It is well known that the dual of $\ell^{\infty}$ properly contains $\ell^1$ (over $\mathbb{N}$, ...

**8**

votes

**1**answer

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### Why does this sum depend on the Axiom of Choice?

On page 168 of Mathematical Fallacies and Paradoxes, it states that the fact that the series
$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots $
has a sum depends on the Axiom of Choice. Where ...

**2**

votes

**1**answer

2k views

### What is a universal function?

This question stems from Dick Lipton's recent blog post on the Axiom of Choice. I asked there but got no takers. I promise I'm not an inept Googler, but I couldn't find a satisfactory answer. I ...

**6**

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**5**answers

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### Subset of the plane that intersects every line exactly twice

In a comment to this question, Tim Gowers remarked that using the axiom of choice, once can show that there exists a subset of the plane that intersects every line exactly twice (although it has yet ...

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votes

**2**answers

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### Is it still impossible to partition the plane into Jordan curves without choice?

It is an easy exercise to show that the Euclidean plane cannot be partitioned into round circles (note however that it is possible to do so for $\mathbb{R}^3$). It seems almost obvious that it is not ...

**12**

votes

**2**answers

732 views

### Ultrafilters vs Well-orderings

This question was actually asked by John Stillwell in a comment to an answer to this question. I thought I would advertise it as a separate question since no one has yet answered and I am also ...

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**15**answers

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### Most 'unintuitive' application of the Axiom of Choice?

It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even ...

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**2**answers

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### Choice function for Borel sets?

Let's say we want to define a choice function for certain particular subsets $S \subset2^{\mathbb{R}}$, i.e. we want a function $c:S \rightarrow \mathbb{R}$ such that $c(X)\in X$ for every $X\in S$. ...

**2**

votes

**5**answers

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### Decidability of the Axiom of Choice

Everything that I read regarding Set Theory states that the Axiom of Choice is independent and undecidable within the context of Zermelo-Frankel Set Theory. My question is this: Is there any ...

**1**

vote

**2**answers

627 views

### Axiom of Choice and Order Types

A beginner's question:
We know: "Since order-equivalence is an equivalence relation, it partitions the class of all sets into equivalence classes." (from Wikipedia)
This holds since every set can be ...

**57**

votes

**3**answers

5k views

### Does every non-empty set admit a group structure (in ZF)?

It is easy to see that in ZFC, any non-empty set $S$ admits a group structure: for finite $S$ identify $S$ with a cyclic group, and for infinite $S$, the set of finite subsets of $S$ with the binary ...

**2**

votes

**4**answers

710 views

### Set theories that do require the existence of urelements?

I am looking for an axiomatic set theory that not only admits the existence of urelements/atoms (via two-sortedness or an additional unary predicate) but requires it, e.g. by an axiom like "for each ...

**20**

votes

**2**answers

634 views

### How much choice is needed to show that formally real fields can be ordered?

Background: a field is formally real if -1 is not a sum of squares of elements in that field. An ordering on a field is a linear ordering which is (in exactly the sense that you would guess if you ...

**2**

votes

**6**answers

976 views

### Splitting lemma under assumption of the axiom of choice

The splitting lemma says:
Given a short exact sequence with maps $q$ and $r$:
$0 \rightarrow A \overset{q}{\rightarrow} B \overset{r}{\rightarrow} C \rightarrow 0$
then the following are ...

**11**

votes

**5**answers

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### Where are some interesting places where the axiom of choice crops up in category theory?

The two that come to mind are splitting epics in Set and taking the Skel of a category. Surely there are lots of other interesting (and maybe upsetting) places where this comes up.