Let $LM$ denote "all subsets of $\Bbb{R}$ are Lebesgue measurable", and

$WCH$ (weak continuum hypothesis) denote "every uncountable subset of $\Bbb{R}$ can be be put into 1-1 correspondence with $\Bbb{R}$".

[**Warning:** in other contexts, weak CH means something totally different , i.e., it sometimes means $2^{\aleph_{0}} < 2^{\aleph_{1}}$].

We know that $LM$ and $WCH$ both hold in Solovay models. By forcing a (Ramsey) ultrafilter over a Solovay model one can also arrange $WCH+\neg LM$ (due to joint work of Di Prisco and Todorčević, who showed that the perfect set property holds in the generic extension).

This prompts my question ($DC$ below is dependent choice).

Question:Is it known, relative to appropriate large cardinal axioms, whether there is a model of $ZF+LM+DC+\neg WCH$?

My question arose from an FOM-question of Tim Chow, and my answer to it; see also Chow's response.

10more comments