Linked Questions

1
vote
3answers
188 views

What does Axiom of Choice mean [duplicate]

So this a fundamental assumption in mathematics. Can someone explain informally what it actually is please. My guess is that its when we say in proofs that "Let $x \in X$". But I am not sure.
102
votes
4answers
9k views

What are the Axiom of Choice and Axiom of Determinacy?

Would someone please explain: What does the Axiom of Choice mean, intuitively? What does the Axiom of Determinancy mean, intuitively, and how does it contradict the Axiom of Choice? as simple words ...
81
votes
2answers
18k views

Is there a known well ordering of the reals?

So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My ...
29
votes
6answers
5k views

Axiom of Choice and finite sets

So I am relatively familiar with the Axiom of Choice and a few of its equivalent forms (Zorn's Lemma, Surjective implies right invertible, etc.) but I have never actually taken a set theory course. I ...
45
votes
2answers
4k views

Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
15
votes
6answers
2k views

Aren't constructive math proofs more “sound”?

Since constructive mathematics allows us to avoid things like Russell's Paradox, then why don't they replace traditional proofs? How do we know the "regular" kind of mathematics are free of paradox ...
12
votes
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 ...
6
votes
5answers
285 views

Does 'let x be a member of S…' require axiom of choice? [duplicate]

A lot of proofs start like this 'Let x be some member of the set S, then...' (If S are the natural numbers, this is especially common) Do all of those proofs ...
8
votes
1answer
1k views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...
0
votes
2answers
561 views

The Banach-Tarski paradox and the notion of measure

Was reading through this question and the answer given by @triple_sec lists some mindboggling results that are implied by the axiom of choice. Specifically: [Geometry] Banach–Tarski paradox. (The ...
1
vote
1answer
123 views

Axiom of choice, proving a function is onto

I had some questions about the Axiom of choice. suppose I have a function f:A->B, where A and B are infinite sets, and I have to prove f is onto. So as a general strategy I pick an arbitrary element ...
3
votes
1answer
187 views

Why do we need the axiom of choice? [duplicate]

I don't mean why is it important. I mean why can't we just define the "selector function" like $S\colon \mathbb{F} \to A, $ such that $S(X) = x \in X$, without an axiom? Why can't we do that but we ...
1
vote
1answer
111 views

Can one explain when the Axiom of Choice is needed in simple terms?

Consider the sequence of sets $(A_1,A_2,A_3,\dots)$. Assume for each $A_n$, there exists bijective mapping $f_n: \{1,2,3,4\} \rightarrow A_n$ for each $n$. If I want to construct the sequence $(f_1(2)...
1
vote
1answer
129 views

Can't find the demonstration of a theorem about recursion [closed]

Here's the theorem : Let $E$ be a set, $g$ a function from $E$ into $E$, and $a$ an element of $E$. There exists a unique function $f$ defined from $\mathbb{N}$ into $E$ such that $f(0)=a$ and $f(n+...