# Tagged Questions

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### What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
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### Multiple context available for the AC?

I am surprised at the language used in connection with the axiom of choice. From the answer to a question a made (which turned out to be duplicate) about involvement of AC in Wilesâ€™ proof of Fermatâ€™s ...
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### If AC is false , is this statement about the halting problem true?

Assume AC is false. (AC = axiom of choice ) Let $n,m$ be positive integers. Let $f: \Bbb N \rightarrow \Bbb N$ and $f(m)=m$. Let $g(n,m)=1$ if the iterations $f(n),f(f(n)),...$ converges to $m$. ...
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### How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
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### A confusion about Axiom of Choice and existence of maximal ideals.

The proof of the theorem of statement that every ring has a maximal ideal uses zorn's lemma or the axiom of choice.Now, the defintion of ring as well as the definition of maximal ideal don't depend on ...
### Does (Infer $\phi$ from $\psi$) imply (Infer $\phi^L$ from $\psi^L$)?
I am studying set theory on my own on Drake's famous book and I'm stuck on the (finitary) prove of the relative consistency of the Axiom of Choice. Is it true that a if we were able to infer $\xi$ ...