In his answer to a Math Stack Exchange question of Katlus, Asaf Karagila wrote the following:
"It is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an unmeasurable set of real numbers, so in Solovay's model we have that there are no sets of real numbers which have size $\aleph_1$. In fact, in Solovay's model every uncountable set of reals is of size continuum, and in some sense the continuum hypothesis holds."
Three questions regarding this statement:
i) Since "it is a theorem that from $ZF+DC+$"$\aleph_1$$\le$$|$$\mathbb R$$|$" we can prove that there is an unmeasurable set of real numbers" could someone provide an explicit construction of this set in the aforementioned system?
ii) Are there models of $ZF$$+$"Every set of reals is Lebesgue measurable"$+$$\lnot$$CH$?
iii) In the models mentioned in ii), does $DC$ fail?