Editing Forcing (mathematics) (section)

== Random reals ==

{{main article|Random algebra}}

Random forcing can be defined as forcing over the set <math>P</math> of all compact subsets of <math>[0,1]</math> of positive measure ordered by relation <math>\subseteq</math> (smaller set in context of inclusion is smaller set in ordering and represents condition with more information). There are two types of important dense sets:

# For any positive integer <math>n</math> the set <math display="block">D_n= \left \{p\in P : \operatorname{diam}(p)<\frac 1n \right \}</math> is dense, where <math>\operatorname{diam}(p)</math> is diameter of the set <math>p</math>.
# For any Borel subset <math>B \subseteq [0,1]</math> of measure 1, the set <math display="block">D_B=\{p\in P : p\subseteq B\}</math> is dense.

For any filter <math>G</math> and for any finitely many elements <math>p_1,\ldots,p_n\in G</math> there is <math>q\in G</math> such that holds <math>q\leq p_1,\ldots,p_n</math>. In case of this ordering, this means that any filter is set of compact sets with finite intersection property. For this reason, intersection of all elements of any filter is nonempty. If <math>G</math> is a filter intersecting the dense set <math>D_n</math> for any positive integer <math>n</math>, then the filter <math>G</math> contains conditions of arbitrarily small positive diameter. Therefore, the intersection of all conditions from <math>G</math> has diameter 0. But the only nonempty sets of diameter 0 are singletons. So there is exactly one real number <math>r_G</math> such that <math>r_G\in\bigcap G</math>.

Let <math>B\subseteq[0,1]</math> be any Borel set of measure 1. If <math>G</math> intersects <math>D_B</math>, then <math>r_G\in B</math>.

However, a generic filter over a countable transitive model <math>V</math> is not in <math>V</math>. The real <math>r_G</math> defined by <math>G</math> is provably not an element of <math>V</math>. The problem is that if <math>p\in P</math>, then <math>V\models</math> "<math>p</math> is compact", but from the viewpoint of some larger universe <math>U\supset V</math>, <math>p</math> can be non-compact and the intersection of all conditions from the generic filter <math>G</math> is actually empty. For this reason, we consider the set <math>C=\{\bar p : p \in G \}</math> of topological [[Closure (topology)|closures]] of conditions from G (i.e., <math> \bar p = p \cup \{\inf(p) \} \cup \{\sup(p) \} </math>). Because of <math>\bar p\supseteq p</math> and the finite intersection property of <math>G</math>, the set <math>C</math> also has the finite intersection property. Elements of the set <math>C</math> are bounded closed sets as closures of bounded sets.{{clarify|date=July 2018}} Therefore, <math>C</math>
is a set of compact sets{{clarify|date=July 2018}} with the finite intersection property and thus has nonempty intersection. Since <math>\operatorname{diam}(\bar p) = \operatorname{diam}(p)</math> and the ground model <math>V</math> inherits a metric from the universe <math>U</math>, the set <math>C</math> has elements of arbitrarily small diameter. Finally, there is exactly one real that belongs to all members of the set <math>C</math>. The generic filter <math>G</math> can be reconstructed from <math>r_G</math> as <math>G=\{p\in P : r_G\in\bar p\}</math>.

If <math>a</math> is name of <math>r_G</math>,{{clarify|date=July 2018}} and for <math>B\in V</math> holds <math>V\models</math>"<math>B</math> is Borel set of measure 1", then holds

:<math>V[G]\models \left (p\Vdash_{\mathbb{P}}a\in\check{B} \right )</math>

for some <math>p\in G</math>. There is name <math>a</math> such that for any generic filter <math>G</math> holds

:<math>\operatorname{val}(a,G)\in\bigcup_{p\in G}\bar p.</math>

Then

:<math>V[G]\models \left (p\Vdash_{\mathbb{P}}a\in\check{B} \right )</math>

holds for any condition <math>p</math>.

Every Borel set can, non-uniquely, 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 <math>B</math> in <math>V</math>, one recovers a Borel code, and then applies the same construction sequence in <math>V[G]</math>, getting a Borel set <math>B^*</math>. It can be proven that one gets the same set independent of the construction of <math>B </math>, and that basic properties are preserved. For example, if <math>B \subseteq C</math>, then <math>B^* \subseteq C^*</math>. If <math>B</math> has measure zero, then <math>B^*</math> has measure zero. This mapping <math>B\mapsto B^*</math> is injective.

For any set <math>B\subseteq[0,1]</math> such that <math>B\in V</math> and <math>V\models</math>"<math>B</math> is a Borel set of measure 1" holds <math>r\in B^*</math>.

This means that <math>r</math> is "infinite random sequence of 0s and 1s" from the viewpoint of <math>V</math>, which means that it satisfies all statistical tests from the ground model <math>V</math>.

So given <math>r</math>, a random real, one can show that

:<math> G = \left \{ B ~ (\text{in } V) \mid r \in B^* ~ (\text{in } V[G]) \right \}. </math>

Because of the mutual inter-definability between <math>r</math> and <math> G </math>, one generally writes <math>V[r]</math> for <math>V[G]</math>.

A different interpretation of reals in <math>V[G]</math> was provided by [[Dana Scott]]. Rational numbers in <math> V[G] </math> 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 <math>I = [0,1] </math>. Real numbers in <math>V[G]</math> then correspond to [[Dedekind cut]]s of such functions, that is, [[measurable function]]s.