Examples of countable and uncountable Bernstein sets

Question

I'm looking for concrete examples of a countable Bernstein set and an uncountable Bernstein set. I haven't been able to find or construct any specific examples so far.

I'm fairly certain that Bernstein sets are not Lebesgue-measureable — John White, Aug 14 '20 at 20:36
Do you know the usual construction by transfinite induction? It's not exactly explicit, but it is the best you can get by Brian's answer — Alessandro Codenotti, Aug 15 '20 at 07:13
In this answer I went through the construction mentioned by Alessandro Codenotti. — Brian M. Scott, Aug 15 '20 at 16:34

score 5 · Accepted Answer · answered Aug 14 '20 at 20:53

Bernstein sets are non-measurable, so there are no concrete examples: assuming the existence of an inaccessible cardinal, it is consistent with $\mathsf{ZF}$ that all subsets of the reals are measurable.

All Borel sets are measurable, so there cannot be a countable Bernstein set.

Examples of countable and uncountable Bernstein sets

1 Answers1