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While Vitali's proof that Vitali sets are not Lebesgue measurable is easy to understand, it feels quite magical for me. Hence, this post.

When I heard of it in a lecture yesterday, I was superbly fascinated - it is such a beautiful proof! However, as mentioned, it feels like magic. Why should - intuitively - the Vitali set not be measurable? What makes it that "bad"?

I understand that translation-invariance is the main problem. I also understand that Axiom of Choice can be difficult to talk about. I too understand that the Vitali set is quite pathological.

So this is a somewhat vague question. But I still hope there are chances for deeper intuition, as I rarely believe in "magic" in mathematics but instead always in clearer intuition at a deeper level.

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    $\begingroup$ The way I choose to "accept" that the Vitali set makes sense is that someone created a non-numerable set with the points spread across the $[0,1]$ interval in the most non-trivial way possible $\endgroup$ – RGS Nov 6 '18 at 10:22
  • $\begingroup$ The magic is precisely the axiom of choice. Note that in some set-theoretic model without the axiom of choice, every subset of the reals is Lebesgue measurable. Therefore, even the mere existence (not to mention non-measurability) of Vitali set is an artefact of the axiom of choice. $\endgroup$ – user1551 Nov 6 '18 at 10:45
  • $\begingroup$ I like RGS' comment. And yeah, user1551, I know of that - so you're right, of course! Axiom of choice is quite crazy, isn't it... $\endgroup$ – Qi Zhu Nov 6 '18 at 20:54

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