Linked Questions

5
votes
1answer
963 views

Why is the axiom of choice controversial? [duplicate]

In other words, what are the arguments for ZF over ZFC, and what philosophical issues have people raised against including it as a standard axiom of set theory?
1
vote
1answer
57 views

Uses of Axiom of Choice [duplicate]

I am a first-year maths student but I occasionally drift away from our taught material. Some years ago I saw the ZFC axioms for the first time, but now that I am in college, and although the stuff I'...
46
votes
6answers
6k views

Foundation for analysis without axiom of choice?

Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ...
39
votes
5answers
3k views

Advantage of accepting the axiom of choice

What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)? It seems that there is no clear reason to prefer over other axioms.. Thanks for help.
28
votes
5answers
1k 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.
25
votes
3answers
2k views

Is there a *simple* example of how the axiom of choice can lead to a counterintuitive result?

This question is quite subjective, but here goes: The axiom of choice notoriously leads to many extremely counterintuitive results, like the Hausdorff, Von Neumann, and (most famously) Banach-Tarski ...
4
votes
3answers
225 views

What's the thorny issue on: “If all $S\in \ell $ are nonempty, does it follow that $\prod_{S\in \ell} S$ is nonempty? when $\ell$ is infinite?”

I'm reading Paolo Aluffi's ALGEBRA, Chapter 0. Here he proposes that there's a thorny issue: What is this thorny issue?
2
votes
1answer
1k views

Countable infinity and the axiom of choice

Many of the states of affairs about infinite cardinalities and their size, depend on the axiom of choice. What would a comprehensive list be of the properties of the cardinality of the natural numbers ...
8
votes
1answer
309 views

How did the axiom of choice become the boogeyman?

The axiom of choice has a reputation as being a nefarious actor in set theory, the source of all sorts of horrible1, nonconstructive things. For example, to quote wikipedia: The axiom of choice ...
-1
votes
1answer
82 views

Is the axiom of choice independent of the Unrestricted Comprehension Principle?

In Why is the axiom of choice separated from the other axioms?, a comment on the top answer mentions how 'Choice is special in that it isn't a special case of the problematic Unrestricted ...