TITLE: Parts of Set Theory immune to independence
QUESTION [14 upvotes]: The motivation for asking this question is a passage (3.2) in an article by Greg Hjorth where he said that "...it is also an attractive feature of the theory of Borel cardinalities and of the theory of $L(\mathbb{R})$ cardinality that these are largely immunized against independence." 
My question is: What other parts of Set Theory are so immunised? Assuming something extra is okay(say Determinacy), but whatever it is, should settle 'most' of the questions.
The wording above is not too clear, but I'm not sure how I can make a stronger statement(suggestions about this would be great). I guess that if you ask high-level enough questions(definability hierarchy wise), independence will come in sooner or later. However, I'm not really asking this from the point of view of absoluteness, but from the point of view of what large class of questions can be settled by what small set of tools.
A bonus question is: Why is the part about Borel cardinalities true? I guess this might have some simple absoluteness explanation, but the 'largely' tells me there is more to it than I think.
Thanks in advance.
P.S. I'm not sure if I've tagged this appropriately, perhaps a 'soft-question' or 'big-list' one would be okay (although I wonder how big the list would be!). I hope people will retag it if they find it suitable.

REPLY [3 votes]: Some of Shelah's work has shown that parts of set theory that seemed to be dominated by independence results (for example, cardinal arithmetic) actually have a lot of nontrivial structure that is provable in ZFC.  I don't know enough about the area to say any more, but I can point to Shelah's survey paper "Cardinal arithmetic for skeptics."