Let $n$ be a strictly positive natural integer. Let us consider the topological group $(\mathbb{R}^n,+)$ with its usual structure.

In ZF, can we deduce some form of the axiom of choice from the statement "there exist a discontinuous group endomorphism of $(\mathbb{R}^n,+)$", and if so, how?


Well, this depends on what do you mean by "a form of choice". If you mean a statement of the form "Every family with property ... admits a choice function", then the answer, to my best of knowledge, is unknown to this date.

It is very important to point out that any such implication will be very local. That is, even if such a choice principle is provable it is likely one which only applies to particular sets of real numbers. It is easily arrangeable for any global choice principle (i.e. one which does not require all the sets we choose from to be subsets of a particular set) to fail, but the real numbers to be well-orderedable, and so all the $\sf ZFC$ theorems hold for it.

In the diagrams appearing in both Herrlich's The Axiom of Choice and Moore's Zermelo's Axiom of Choice there is really just the one obvious conclusion:

If there exists such endomorphism, then there exists a non-measurable set. I believe that we can also conclude the existence of a set of real without the Baire property, by the same argument.

To my knowledge, we don't even know if this assumption implies the existence of a Hamel basis (although my guess would be that it doesn't).

For sake of completeness, I should also point out that the statement itself is unprovable from $\sf ZF$, and in fact even when augmented by $\sf DC$, we still can't prove that there is a discontinuous endomorphism. The reason is that it is consistent (with $\sf ZF+DC$) that every set of reals has the Baire property, and from this follows that every homomorphism between Polish groups is continuous.

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    $\begingroup$ Asaf, I think you misread nombre's question; he was asking whether his statement implies some form of choice. $\endgroup$ – Paul McKenney Jan 3 '14 at 15:18
  • $\begingroup$ Jeepers. I must be more tired than I thought I was. Well, unlikely. I've been up and running for 24 hours now. Let me edit. $\endgroup$ – Asaf Karagila Jan 3 '14 at 15:19
  • $\begingroup$ @Paul: Thank you for pointing that out! $\endgroup$ – Asaf Karagila Jan 3 '14 at 15:33
  • $\begingroup$ Thanks a lot! I had no idea this was an open problem and I was driving myself crazy not figuring how to prove or disprove it. $\endgroup$ – nombre Jan 3 '14 at 17:08
  • $\begingroup$ @nombre: What did you try to prove? $\endgroup$ – Asaf Karagila Jan 3 '14 at 18:31

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