This is a follow-up to my previous question. In that question, I asked whether "All additive functions are linear" imply every set of reals is measurable. The answer is no. So, a natural follow-up is, does "All additive functions are linear" imply every set of reals has the Baire property? Just to be clear, the background theory is ZF set theory without the axiom of choice.
2026-04-08 02:59:55.1775617195
Does "All additive functions are linear" imply every set of reals has the Baire property?
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No. It follows from "Every set of reals is Lebesgue measurable", which is known by the work of Shelah to be independent from the Baire proprety. This was shown by Shelah to be consistent in his paper
It's almost the same proofs as those that show that the Baire property implies continuity also work for Lebesgue measurability, and I am guessing they will also work for a number of other properties which may very well be independent as well.