Forcing (mathematics) - Random Reals

Random Reals

In the Borel sets ( Bor(I), ⊆, I ) example, the generic filter converges to a real number r, called a random real. A name for the decimal expansion of r (in the sense of the canonical set of decimal intervals that converge to r) can be given by letting r = { ( Eˇ, E ) : E =, 0 ≤ k < 10n }. This is, in some sense, just a subname of G.

To recover G from r, one takes those Borel subsets of I that "contain" r. Since the forcing poset is in V, but r is not in V, this containment is actually impossible. But there is a natural sense in which the interval in V "contains" a random real whose decimal expansion begins 0.5. This is formalized by the notion of "Borel code".

Every Borel set can, nonuniquely, be built up, starting from intervals with rational endpoints and applying the operations of complement and countable unions, a countable number of times. The record of such a construction is called a Borel code. Given a Borel set B in V, one recovers a Borel code, and then applies the same construction sequence in V, getting a Borel set B*. One can prove that one gets the same set independent of the construction of B, and that basic properties are preserved. For example, if B⊆C, then B*⊆C*. If B has measure zero, then B* has measure zero.

So given r, a random real, one can show that G = { B (in V) : r∈B* (in V) }. Because of the mutual interdefinability between r and G, one generally writes V for V.

A different interpretation of reals in V was provided by Dana Scott. Rational numbers in V have names that correspond to countably many distinct rational values assigned to a maximal antichain of Borel sets, in other words, a certain rational-valued function on I = . Real numbers in V then correspond to Dedekind cuts of such functions, that is, measurable functions.