The tag has no wiki summary.

learn more… | top users | synonyms

14
votes
11answers
3k views

Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
13
votes
4answers
1k views

Nilradicals without Zorn's lemma

It's well known that the nilradical of a commutative ring with identity $A$ is the intersection of all the prime ideals of $A$. Every proof I found (e.g. in the classical "Commutative Algebra" by ...
3
votes
2answers
1k views

Axiom of Computable Choice versus Axiom of Choice

What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions? I guess I ...
11
votes
2answers
2k views

AC in group isomorphism between R and R^2

Using the axiom of choice, one can show that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups. In particular, they are both vector spaces over $\mathbb{Q}$ and AC gives bases of ...
6
votes
3answers
3k views

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
votes
8answers
2k views

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 ...
92
votes
15answers
17k views

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 ...
5
votes
0answers
1k views

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
1answer
931 views

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
1answer
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
votes
5answers
2k views

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 ...
3
votes
2answers
742 views

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
2answers
736 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 ...
45
votes
15answers
11k views

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 ...
6
votes
2answers
473 views

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
5answers
1k views

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
2answers
630 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 ...
64
votes
3answers
6k 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
4answers
724 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
2answers
645 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
6answers
980 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
5answers
891 views

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.