structural:expansion, stylistic:tone, linguistic:paraphrasing, quality:corruption
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 ℵ 2 {\displaystyle \aleph _{2}} of them), identified with subsets of the set N {\displaystyle \mathbb {N} } 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 model M {\displaystyle M} of the set theory, which is itself a set in the "real universe" V {\displaystyle V}. By the Löwenheim–Skolem theorem, M {\displaystyle M} can be chosen to be a "bare bones" model that is externally countable, which guarantees that there will be many subsets (in V {\displaystyle V}) of N {\displaystyle \mathbb {N} } that are not in M {\displaystyle M}. Specifically, there is an ordinal ℵ 2 M {\displaystyle \aleph _{2}^{M}} that "plays the role of the cardinal ℵ 2 {\displaystyle \aleph _{2}} " in M {\displaystyle M}, but is actually countable in V {\displaystyle V}. Working in V {\displaystyle V}, it should be easy to find one distinct subset of N {\displaystyle \mathbb {N} } per each element of ℵ 2 M {\displaystyle \aleph _{2}^{M}}. (For simplicity, this family of subsets can be characterized with a single subset X ⊆ ℵ 2 M × N {\displaystyle X\subseteq \aleph _{2}^{M}\times \mathbb {N} }.)

However, in some sense, it may be desirable to "construct the expanded model M {\displaystyle M} within M {\displaystyle M} ". This would help ensure that M {\displaystyle M} "resembles" M {\displaystyle M} in certain aspects, such as ℵ 2 M {\displaystyle \aleph _{2}^{M}} being the same as ℵ 2 M {\displaystyle \aleph _{2}^{M}} (more generally, that cardinal collapse does not occur), and allow fine control over the properties of M {\displaystyle M}. More precisely, every member of M {\displaystyle M} should be given a (non-unique) name in M {\displaystyle M}. The name can be thought as an expression in terms of X {\displaystyle X}, just like in a simple field extension L = K (θ) {\displaystyle L=K(\theta)} every element of L {\displaystyle L} can be expressed in terms of θ {\displaystyle \theta }. A major component of forcing is manipulating those names within M {\displaystyle M}, so sometimes it may help to directly think of M {\displaystyle M} as "the universe", knowing that the theory of forcing guarantees that M {\displaystyle M} will correspond to an actual model.

A subtle point of forcing is that, if X {\displaystyle X} is taken to be an arbitrary "missing subset" of some set in M {\displaystyle M}, then the M {\displaystyle M} constructed "within M {\displaystyle M} " may not even be a model. This is because X {\displaystyle X} may encode "special" information about M {\displaystyle M} that is invisible within M {\displaystyle M} (e.g. the countability of M {\displaystyle M}), and thus prove the existence of sets that are "too complex for M {\displaystyle M} to describe".

Forcing avoids such problems by requiring the newly introduced set X {\displaystyle X} to be a generic set relative to M {\displaystyle M}. Some statements are "forced" to hold for any generic X {\displaystyle X}: For example, a generic X {\displaystyle X} is "forced" to be infinite. Furthermore, any property (describable in M {\displaystyle M}) of a generic set is "forced" to hold under some forcing condition. The concept of "forcing" can be defined within M {\displaystyle M}, and it gives M {\displaystyle M} enough reasoning power to prove that M {\displaystyle M} 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 models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.