Rudimentary results in iterated forcing.

Question

What are some relatively basic and fun results using iterated forcing that I can include in my undergraduate thesis in the section I'm going to title "Rudimentary applications." It's the section following the section titled "Fundamental Theorems of Iterated Forcing."

Eventually, in some other section, I'm going to include the result by Dow, Tall, and Weiss on the consistency of the Normal Moore Space conjecture, which is obtained by an iteration of cohen forcing of supercompact length.

Answer 1

The proof of the consistency of ZFC+$\neg$CH+MA uses an iterated forcing argument; I wouldn't necessarily call it "rudimentary," but it's definitely simpler than the consistency of the normal Moore space conjecture.

A simpler result, which is however of less general interest in my opinion, is the consistency relative to ZFC of ZFC+"There are exactly three degrees of constructibility of real numbers." The proof is a two-step iteration of Sacks forcing over a model of ZFC+V=L: the first step produces a unique nonzero constructibility degree of a real, and the second step produces a unique nonzero constructibility degree of a real above the first one. This is a nice case where we see vividly the difference between the product and the support, since the product $\mathbb{S}\times\mathbb{S}$ of two Sacks forcings over $L$ yields four degrees of constructibility of reals: the trivial one, the left generic, the right generic, and their join. (A nice coda to this is the observation that the crucial property is that the Sacks forcing of a generic extension is not the same as the Sacks forcing of the ground model.)

Finally, there are cardinal characteristic separations. These are simpler than the MA case and more inherently interesting in my opinion than the "three degrees" result. There are many of them, and I'll leave it to you to pick a favorite.

Answer 2

1. The grandfather, O.G., result by iteration: Martin's Axiom.

   You can include also its more sophisticated offspring, $\sf PFA$ and $\sf MM$ or other forcing axioms, but then you need to start with a supercompact cardinal.

2. Another very interesting result would be the Mitchell proof that assuming a weakly compact cardinal, it is consistent that $\aleph_2$ has the tree property. This is done with a mixed-support of Levy collapses and Cohen reals.

3. And of course, if we are talking about nontrivial supports, then Easton's theorem that assuming $\sf GCH$, any function on regular cardinal satisfying: $F(\kappa)\leq F(\lambda)$ for $\kappa\leq\lambda$, and $\operatorname{cf}(F(\kappa))>\kappa$ is a possible continuum function for regular cardinals.

   (Of course, Easton's original proof is by using a product, but you can also use an iteration. And a product is a kind of iteration, after all...)