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### Unprovable statements in ZF [duplicate]

Possible Duplicate: Advantage of accepting the axiom of choice Advantage of accepting non-measurable sets As you all know, Banach-Tarski paradox is solely a consequence of Axiom of Choice, and ...
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### What are some unintuitive consequences if we assume that axiom of choice is wrong? [duplicate]

Accetping axiom of choice gives rise to all sort of paradoxes. What if we assume the contrary?
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### 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'...
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### Does every set have a group structure?

I know that there is no vector space having precisely $6$ elements. Does every set have a group structure?
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### 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.
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### 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 ...
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### Axiom of choice and automorphisms of vector spaces

A classical exercise in group theory is "Show that if a group has a trivial automorphism group, then it is of order $1$ or $2$." I think that the straightforward solution uses that a exponent two ...
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### Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
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### Algebraic closure for $\mathbb{Q}$ or $\mathbb{F}_p$ without Choice?

I know the usual proof of the existence of an algebraic closure for any field using Zorn's Lemma. The answer to this previous question makes it clear that in general, some nonconstructive axiom (not ...
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### 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?
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### 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...
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### Axiom of choice in set theory

Just as the title stated, what is the main point of axiom of choice? I do not quite understand what is written in the axiom. The axiom that I know is: Given any collection of non-empty sets, there ...
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### A question about cardinal arithmetics without the Axiom of Choice

Is multiplication of infinite cardinals defined in ZF without Choice?
I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
### What does a well ordering of $\mathbb{R}$ look like? [duplicate]
Possible Duplicate: Is there a known well ordering of the reals? I am having a hard time wrapping my head around what a well-ordering of $\mathbb{R}$ looks like. I have seen the presentation of a ...