Editing Forcing (mathematics) (section)

== Forcing conditions and forcing posets ==
Each forcing condition can be regarded as a ''finite'' piece of information regarding the object <math>X</math> adjoined to the model. There are many different ways of providing information about an object, which give rise to different '''forcing notions'''. A general approach to formalizing forcing notions is to regard forcing conditions as abstract objects with a [[poset]] structure.

A '''forcing poset''' is an ordered triple, <math> (\mathbb{P},\leq,\mathbf{1}) </math>, where <math> \leq </math> is a [[preorder]] on <math> \mathbb{P} </math>, and <math> \mathbf{1} </math> is the largest element. Members of <math> \mathbb{P} </math> are the '''forcing conditions''' (or just '''conditions'''). The order relation <math> p \leq q </math> means "<math> p </math> is '''stronger''' than <math> q </math>". (Intuitively, the "smaller" condition provides "more" information, just as the smaller interval <math> [3.1415926,3.1415927] </math> provides more information about the number [[Pi|{{pi}}]] than the interval <math> [3.1,3.2] </math> does.) Furthermore, the preorder <math> \leq </math> must be [[Atom (order theory)|atomless]], meaning that it must satisfy the '''splitting condition''':

*For each <math> p \in \mathbb{P} </math>, there are <math> q,r \in \mathbb{P} </math> such that <math> q,r \leq p </math>, with no <math> s \in \mathbb{P} </math> such that <math> s \leq q,r </math>.

In other words, it must be possible to strengthen any forcing condition <math>p</math> in at least two incompatible directions. Intuitively, this is because <math>p</math> is only a finite piece of information, whereas an infinite piece of information is needed to determine <math>X</math>.

There are various conventions in use. Some authors require <math> \leq </math> to also be [[antisymmetric relation|antisymmetric]], so that the relation is a [[partial order]]. Some use the term [[partial order]] anyway, conflicting with standard terminology, while some use the term [[preorder]]. The largest element can be dispensed with. The reverse ordering is also used, most notably by [[Saharon Shelah]] and his co-authors.

=== Examples ===

Let <math>S</math> be any infinite set (such as <math>\mathbb{N}</math>), and let the generic object in question be a new subset <math>X \subseteq S</math>. In Cohen's original formulation of forcing, each forcing condition is a ''finite'' set of sentences, either of the form <math>a \in X</math> or <math>a \notin X</math>, that are self-consistent (i.e. <math>a \in X</math> ''and'' <math>a \notin X</math> for the same value of <math>a</math> do not appear in the same condition). This forcing notion is usually called '''Cohen forcing'''.

The forcing poset for Cohen forcing can be formally written as <math> (\operatorname{Fin}(S,2),\supseteq,0) </math>, the finite partial functions from <math> S </math> to <math> 2 ~ \stackrel{\text{df}}{=} ~ \{ 0,1 \} </math> under ''reverse'' inclusion. Cohen forcing satisfies the splitting condition because given any condition <math>p</math>, one can always find an element <math>a \in S</math> not mentioned in <math>p</math>, and add either the sentence <math>a \in X</math> or <math>a \notin X</math> to <math>p</math> to get two new forcing conditions, incompatible with each other.

Another instructive example of a forcing poset is <math> (\operatorname{Bor}(I),\subseteq,I) </math>, where <math> I = [0,1] </math> and <math> \operatorname{Bor}(I) </math> is the collection of [[Borel subset]]s of <math> I </math> having non-zero [[Lebesgue measure]]. The generic object associated with this forcing poset is a '''random real number''' <math>r \in [0, 1]</math>. It can be shown that <math>r</math> falls in every Borel subset of <math>[0, 1]</math> with measure 1, provided that the Borel subset is "described" in the original unexpanded universe (this can be formalized with the concept of ''Borel codes''). Each forcing condition can be regarded as a random event with probability equal to its measure. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.