What is random real forcing?

Question

Hi all,

Can someone please explain the idea and the main steps in a random real forcing? - what makes it (the new real) different from adding a Cohen real? - is there a good reference for it?

Thanks.

Answer 1

This is relatively easy to answer. One of main differences between Cohen forcing and random real forcing is that random real forcing does not add an unbounded real.
That is, every function $f:\omega\to\omega$ in the forcing extension is bounded by function $g:\omega\to\omega$ in the ground model, in the sense that for all $n\in\omega$, $f(n)\leq g(n)$.

Let me sketch you a proof of this fact. Let $p$ be a set of positive measure $\varepsilon$ and assume that $p$ forces $\dot f$ to be a name for a function from $\omega$ to $\omega$. Find a maximal antichain $p_n$, $n\in\omega$, below $p$ such that each $p_n$ decides $\dot f(0)$, i.e., for some $m\in\omega$ forces $\dot f(0)$ to be $m$. Now finitely many of the $p_n$ will union up to a subset of $p$ that has a measure close to the measure of $p$. Call this union $q_0$. Now $q_0$ does not decide $\dot f(0)$, but it gives you an upper bound, the maximum of the values forced on $\dot f(0)$ by the finitely many $p_n$ that union up to $q_0$.
Iterating this procedure we get a decreasing sequence $q_i$ of conditions such that $q_i$ forces upper bounds on all $\dot f(k)$, $k\leq i$. If each $q_{i+1}$ is a sufficiently large subset of $q_i$ in terms of measure, then the intersection $q$ of the $q_i$ will be of positive measure and gives us a ground model function that is forced by $q$ to be an upper bound for $\dot f$. Combining this with the usual density argument tells you that every new function is bounded by a ground model function.

This property is known as $\omega^\omega$-boundingness and can be formulated in terms of a distributivity property of the corresponding Boolean algebra (the completion of the forcing notion, in this case the measure algebra on the reals). Jech elaborates on that (i.e., proves the equivalence of these two properties) in his book.

Since Cohen forcing adds an unbounded real, the $\omega^\omega$-boundingness of random real forcing shows that it does not add Cohen reals, or algebraically speaking, that the measure algebra has no countable atomless regular subalgebra. Also, it can be shown the Cohen forcing does not add a random real.

A good source for these questions is Bartoszynski and Judah: Set Theory of the Real Line.