TITLE: Models of set theory where not every set can be linearly ordered
QUESTION [5 upvotes]: Can anybody point me towards a model of set theory where not every set can be linearly ordered, and a corresponding proof. I have seen it claimed that in Fraenkels second permutation model that there is a set that cannot be linearly ordered, but cannot find a proof.
Essentially, I am asking for a proof that without choice sometimes the linear ordering principle fails.

REPLY [7 votes]: An interesting example of a different kind is any model where all sets of reals have the Baire property. In any such set the quotient of $\mathbb R$ by the Vitali equivalence relation is not linearly orderable. See here for a sketch.  
Examples of such models are Solovay's model where all sets of reals are Lebesgue measurable, or natural models of the axiom of determinacy, or Shelah's model from section 7 of

MR0768264 (86g:03082a). Shelah, Saharon. Can you take Solovay's inaccessible away? Israel J. Math. 48 (1984), no. 1, 1–47.