The construction of the Vitali set does use the axiom of choice. Is there any set which is better imaginable and not borel?
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2See here https://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice – Severin Schraven Jan 30 '21 at 10:33
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For examples of "better imaginable" non-borel sets, you might look here. Even if you don't like the examples there, it links to many more examples! As an example to leave here, it is well known (for a certain definition of "well known") that continuous images of borel sets often aren't borel! This is a famous mistake made by Lebesgue, and the relevant sets are now called analytic if you want to look further into this. – HallaSurvivor Jan 30 '21 at 10:39