Solovay's inaccessible over a weak set theory without choice
  We study the consistency strength of Lebesgue measurability for $\Sigma^1_3$ sets over Zermelo set theory ($Z$) in a completely choiceless context. We establish a result analogous to the Solovay-Shelah theorem. 
Haim Horowitz, Saharon Shelah
math.LO