9
votes
0answers
183 views

Paradoxical models of $\sf ZF$ without choice [closed]

There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold ...
5
votes
1answer
137 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
2
votes
1answer
134 views

Statements equivalent to the Axiom of Choice

The Axiom of Choice reads: The product of a collection of non-empty sets is non-empty. As you know well, this axiom is equivalent to many other statements. A few examples (probably the most ...
4
votes
2answers
101 views

Zorn's Lemma and Injective Modules

In my study of injective modules over commutative rings, i noticed that Zorn's Lemma is often employed in the proofs. Here are three examples: 1) Baer's Criterion 2) the characterization of injective ...
5
votes
2answers
203 views

Astonishing and innocent results with the axiom of choice

The product of nonempty sets is nonempty. I am fascinated that such a simple and seemingly intuitive statement can lead to rather astonishing results such as the Banach-Tarski paradox or the solution ...
3
votes
3answers
225 views

Zorn's Lemma $\equiv$ Axiom of Choice

I'm confused a little bit about this, I've been told many times that Zorn's lemma is equivalent to the axiom of choice. Is it an axiom or is it lemma, I mean is there a proof of Zorn's lemma or we ...
1
vote
2answers
389 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
4
votes
1answer
893 views

Is trying to prove a theorem without Axiom of Choice useless?

Suppose there is a well-known theorem whose usual proof uses Axiom of Choice. Is trying to prove it without Axiom of Choice useless? What merits can such a proof have?
15
votes
4answers
871 views

Why is the axiom of choice separated from the other axioms?

I don't know much about set theory or foundational mathematics, this question arose just out of curiosity. As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms ...
16
votes
6answers
646 views

Implication and Interpretation of Banach Tarski

As I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being called ...