Editing Forcing (mathematics) (section)

== Intuition ==
Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least <math>\aleph_2</math> of them), identified with [[subset]]s of the set <math>\mathbb{N}</math> of natural numbers, that were not there in the old universe, and thereby violate the [[continuum hypothesis]].

In order to intuitively justify such an expansion, it is best to think of the "old universe" as a [[Structure (mathematical logic)|model]] <math>M</math> of the set theory, which is itself a set in the "real universe" <math>V</math>. By the [[Löwenheim–Skolem theorem]], <math>M</math> can be chosen to be a "bare bones" model that is [[Skolem's paradox|externally countable]], which guarantees that there will be many subsets (in <math>V</math>) of <math>\mathbb{N}</math> that are not in <math>M</math>. Specifically, there is an [[Ordinal number|ordinal]] <math>\aleph_2^M</math> that "plays the role of the [[Cardinal number|cardinal]] <math>\aleph_2</math>" in <math>M</math>, but is actually countable in <math>V</math>. Working in <math>V</math>, it should be easy to find one distinct subset of <math>\mathbb{N}</math> per each element of <math>\aleph_2^M</math>. (For simplicity, this family of subsets can be characterized with a single subset <math>X \subseteq \aleph_2^M \times \mathbb{N}</math>.)

However, in some sense, it may be desirable to "construct the expanded model <math>M[X]</math> within <math>M</math>". This would help ensure that <math>M[X]</math> "resembles" <math>M</math> in certain aspects, such as <math>\aleph_2^{M[X]}</math> being the same as <math>\aleph_2^M</math> (more generally, that ''cardinal collapse'' does not occur), and allow fine control over the properties of <math>M[X]</math>. More precisely, every member of <math>M[X]</math> should be given a  (non-unique) ''name'' in <math>M</math>. The name can be thought as an expression in terms of <math>X</math>, just like in a [[simple field extension]] <math>L = K(\theta)</math> every element of <math>L</math> can be expressed in terms of <math>\theta</math>. A major component of forcing is manipulating those names within <math>M</math>, so sometimes it may help to directly think of <math>M</math> as "the universe", knowing that the theory of forcing guarantees that <math>M[X]</math> will correspond to an actual model.

A subtle point of forcing is that, if <math>X</math> is taken to be an ''arbitrary'' "missing subset" of some set in <math>M</math>, then the <math>M[X]</math> constructed "within <math>M</math>" may not even be a model. This is because <math>X</math> may encode "special" information about <math>M</math> that is invisible within <math>M</math> (e.g. the [[countability]] of <math>M</math>), and thus prove the existence of sets that are "too complex for <math>M</math> to describe".{{sfn|Cohen|2008|page=111}}
{{refn|As a concrete example, note that <math>\alpha_0</math>, the [[order type]] of all ordinals in <math>M</math>, is a countable ordinal (in <math>V</math>) that is not in <math>M</math>. If <math>X</math> is taken to be a [[well-ordering]] of <math>\mathbb{N}</math> (as a [[Relation (mathematics)|relation]] over <math>\mathbb{N}</math>, i.e. a subset of <math>\mathbb{N} \times \mathbb{N}</math>), then any <math>\mathsf{ZFC}</math> universe containing <math>X</math> must also contain <math>\alpha_0</math> (thanks to the [[axiom of replacement]]).{{sfn|Cohen|2008|page=111}} (Such a universe would also not resemble <math>M</math> in the sense that it would collapse ''all'' infinite cardinals of <math>M</math>.)}}

Forcing avoids such problems by requiring the newly introduced set <math>X</math> to be a '''generic set''' relative to <math>M</math>.{{sfn|Cohen|2008|page=111}} Some statements are "forced" to hold for any generic <math>X</math>: For example, a generic <math>X</math> is "forced" to be infinite. Furthermore, any property (describable in <math>M</math>) of a generic set is "forced" to hold under some '''forcing condition'''. The concept of "forcing" can be defined within <math>M</math>, and it gives <math>M</math> enough reasoning power to prove that <math>M[X]</math> is indeed a model that satisfies the desired properties.

Cohen's original technique, now called [[ramified forcing]], is slightly different from the '''unramified forcing''' expounded here. Forcing is also equivalent to the method of [[Boolean-valued model]]s, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.{{sfn|Shoenfield|1971}}

=== The role of the model ===
In order for the above approach to work smoothly, <math>M</math> must in fact be a [[Standard model (set theory)|standard transitive model]] in <math>V</math>, so that membership and other elementary notions can be handled intuitively in both <math>M</math> and <math>V</math>. A standard transitive model can be obtained from any standard model through the [[Mostowski collapse lemma]], but the existence of any standard model of <math>\mathsf{ZFC}</math> (or any variant thereof) is in itself a stronger assumption than the consistency of <math>\mathsf{ZFC}</math>.

To get around this issue, a standard technique is to let <math>M</math> be a standard transitive model of an arbitrary finite subset of <math>\mathsf{ZFC}</math> (any axiomatization of <math>\mathsf{ZFC}</math> has at least one [[axiom schema]], and thus an infinite number of axioms), the existence of which is guaranteed by the [[Reflection principle#In ZFC|reflection principle]]. As the goal of a forcing argument is to prove [[consistency]] results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.