Random forcing request

Question

I want to begin study random forcing and understand it completely, since a intuitively as formally point of view. What text or paper do you recommend me?

Thanks a lot

Answer 1

I'll turn my comments into an answer and try to provide some intuition. I assume you're familiar with Cohen forcing (and I suggest you learn about that first otherwise), since in some sense that can be made precise random forcing is to Cohen forcing as measure zero subsets of $\Bbb R$ are to meager sets of $\Bbb R$.

A good reference is Kanamori's "the higher infinite", one of the standard set theory textbooks. In particular in chapter 11 Kanamori introduces random forcing, as the poset whose conditions are subsets of the reals of positive Lebesgue measure, ordered by inclusion and Borel codes.

The idea behind Borel codes is that after a forcing adding real numbers if you look, for example, at $[0,1]^V$ in $V[G]$ it will usually be a messy set, much more complicated that $[0,1]^{V[G]}$, since it's missing a lot of the reals in $[0,1]$ from the point of view of $V[G]$, but we'd like a way to have a definition of $[0,1]$ such that $V$ interprets it as $[0,1]^V$ and $V[G]$ as $[0,1]^{V[G]}$. The answer to this issue are Borel codes, they are a way to code Borel subsets of $\Bbb R$ as elements of $\omega^\omega$, in a robust way, meaning that if $c$ and $d$ are Borel codes and $A_c$ and $A_d$ their interpretations as Borel sets, then a bunch of basic assertions, such as $A_c=\varnothing$, $A_d=\Bbb R\setminus A_c$, $A_c\subseteq A_d$ and so on are absolute between $V$ and $V[G]$. It's possible to define Borel codes for all Borel subsets of the reals, and this is indeed what Solovay did in his 1970 paper, but since every measurable $X\subseteq\Bbb R$ has a $G_\delta$ superset with the same measure of $X$, it's enough to code open, closed and $G_\delta$, which is what Kanamori does.

In chapter 11 Kanamori goes on to show the standard results about this forcing: it is c.c.c., the generic filter determines a new real, called a random real, and the filter can be recovered from this real, there's a characterization of the random real in terms of Borel codes and finally Solovay's construction of a model of $\mathsf{ZF}+\mathsf{DC}$ in which all subsets of $\Bbb R$ are measurable, have the perfect set property and have the Baire property.

Later on, in chapter 17, Kanamori shows how to add $\kappa$ many random reals to a ground model to prove another result of Solovay, namely that if $\kappa$ is measurable and you force by adding $\kappa$ many random reals, then in $V[G]$, $\mathfrak c=\kappa$ and $\mathfrak c$ is real-valued measurable. If you want to see another application of this forcing, Friedman's paper "a consistent Fubini-Tonelli theorem for nonmeasurable functions" uses it to prove that a strong version of Fubini's theorem is consistent with $\mathsf{ZFC}$.