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.