TITLE: Solovay's Model and Choice
QUESTION [5 upvotes]: Reference;
Foundation for analysis without axiom of choice?
Please let me know if I'm misunderstanding something and I hope you explain this with relatively easy words. I am eager to learn, but I have no idea what forcing is like, so i couldn't understand this properly from past posts.

Solovay's model says that "In ZF, there exists a model such that all subsets of $\mathbb{R}$ is measurable." (I believe I'm misunderstanding this.. Since if it is true, then it must be true in ZFC)

Why are Solovay's model and Dependent Choice related? Is this true in ZF+DC, but not true in ZFC? How is that possible?

In the link, Asaf says, $\mathbb{R} = \bigcup_{i\in I} A_i$, where $A_i \subset \mathbb{R}$ and are mutually disjoint, nonempty and $|I|>|\mathbb{R}|$.

Really??
You can see, $|\mathbb{R}|=|\bigcup_{i\in I} A_i|=\sum_{i\in I} |A_i|≧\sum_{i\in I} 1=|I|$.
(I'm not sure if $≧$ part uses AC)
Thus $|I|≦|\mathbb{R}|$.
How come $|I|>|\mathbb{R}|$?

REPLY [2 votes]: I believe there might be a misinterpretation here.  Solovay's model is a model of ZF in which the following hold:

The axiom of dependent choice;
Every set of reals in Lebesgue measurable; and
Every set of reals has the Baire property.

Due to the usual construction of a Vitali set from the Axiom of Choice it follows in particular that the full Axiom of Choice fails in this model.  Since the Axiom of Choice fails, most of the intuition about cardinal arithmetic falls by the wayside.
(You should also note that the construction of Solovay's model does not begin with ZF or even a model of ZF+DC, but instead with a model of "ZFC + $\exists$ inaccessible cardinal".)