TITLE: Why does the Solovay-Tennenbaum theorem work?
QUESTION [6 upvotes]: I have decided to learn iterations at long last, and I am reading through Jech's Set Theory for now. The standard example after explaining what is an iteration with finite support is the following theorem (quoted from Jech, Theorem 16.13):

Theorem (Solovay-Tennenbaum). Assume GCH and let $\kappa$ be a regular cardinal greater than $\aleph_1$. There is a c.c.c. notion of forcing $P$ such that the generic extension $V[G]$ by $P$ satisfies Martin's Axiom and $2^{\aleph_0}=\kappa$.

I believe that I understood the proof mechanically, but even though I feel that I understand it in a more profound way - I have this feeling that I really didn't get the point. It feels like magic. 
What I managed to understand is that we have enumerations of all isomorphism classes of small posets in each stage, and generically we reach them all. Alas, like any new forcing argument, this too feels like some sort of witchcraft.
I also understand completely why $2^{\aleph_0}=\kappa$, and I have to admit that this is one clever argument, but I can't really see the full intuition behind it. I just see that it works, and I can reason why it works, by MA there is a generic for any $<\kappa$ dense sets, but this means that we have to have at least $\kappa$ dense subsets in the universe so $2^{\aleph_0}\geq\kappa$, where the other side of the inequality follows from the fact that $|P|\leq\kappa$ and $P$ is c.c.c.

Can anyone shed some intuition behind the proof? Some insights about why it works?

REPLY [9 votes]: Let me add a couple of comments to Joel's and Martin's answers.  They both pointed out that you get $\kappa$ reals for a much simpler reason than having forced MA, namely that at $\kappa$ stages you chose to add a Cohen real. You might ask what happens if you never choose to add a real.  Indeed, that's the original Solovay-Tennenbaum situation, where each step was forcing with a Suslin tree, which doesn't add reals.  Nevertheless, you get new reals anyway.  Cohen reals appear at the limit stages of cofinality $\omega$ in any finite-support iteration.  
I think of MA as a compatibility result.  There are lots of things that would naturally be done by ccc forcing --- killing a Suslin tree, adding a single measure-zero set that covers all the ground model's (coded) measure-zero sets (amoeba forcing), adding Cohen reals, etc.  The main point of MA is that you can do all these things together, and mixing them together won't produce disasters.  Notice that, in other circumstances, mixing innocuous-looking forcings can produce disasters, expecially collapsing cardinals. For example, if you specialize an Aronszajn tree and I add a path through it, then between us we've collapsed $\aleph_1$.  Similarly if you shoot a club through a stationary co-stationary set and I shoot a club through its complement.  The key technical fact underlying MA is that a finite-support iteration of arbitrary ccc forcings is still ccc.  So you can mix these forcings at will, without any risk of collapsing cardinals.