TITLE: An informal description of forcing.
QUESTION [36 upvotes]: Is it possible for someone to give a technical but nevertheless non 'jargonish' description of forcing in say less than 200 words. If that is impossible could someone give a descriptive outline of how we get to forcing. Like with the Poincare conjecture proof you would say, first we had formalization of topology, then we came up with certain results such as Ricci flows, and then further research into that led to the solution of some other problem, even though each of these stages are quite technical in nature. Is such a description possible for forcing? All the descriptions seem to be either short and too technical, or as long as an entire book.
I understand what the continuum hypothesis is. And the diagonal argument. How could forcing be motivated from that point?

REPLY [4 votes]: I made the following statement in a Reddit discussion (in /r/math) about the Cohen model for not-CH:
"The point of forcing is that you can describe this process using a 'forcing language' definable in the ground model, so that the ground model can talk in this language about what is happening in the extension. So, consistency of ZFC (in the ground model, where you define the forcing language to talk about the extension) implies consistency of ZFC+not-CH (in the extension, what the forcing language talks about). This is why it is all stated as relative consistency."
To put a bit more detail to this, in the context of @PedroPimenta's answer, the poset $P$ in the ground model is used to establish "relative truth values" for statements in the extension -- the forcing relation "$p\Vdash\phi$" is defined syntactically (i.e. inductively on the length of $\phi$), so there is something to prove before you can say that "$(\exists p\in G)p\Vdash \phi$ iff $\phi$ is valid in $M[G]$."  Now the generic set $G$ is, first of all, a filter in $P$, so statements forced by $p\in G$ are "mutually consistent" (i.e. you didn't introduce any contradictions in the extension), and second, $G$ is "generic" meaning it intersects all dense subsets of $P$, so the forcing language says "as much as possible" about truth in the extension.
From here, as @PedroPimenta suggests, it is down to a clever choice for $P$ and $G$.  For the Cohen model, $P$ consists of finite partial functions from $\omega_2$ to $2^\omega$, ordered by reverse inclusion.  The rest of the argument, such as showing that $P$ has the countable chain condition (to avoid cardinal collapses in the extension), is almost purely combinatorial.  My gripe with the Boolean algebra approach to forcing is that it hides these combinatorial arguments under another layer of structure.  For this reason, I recommend the following reference:
Set Theory: An Introduction to Independence Proofs
A particularly nice article by Baumgartner here highlights the combinatorial nature of forcing:
Independence Proofs and Combinatorics
Shoenfield's original article on the "unramified" approach to forcing remains quite readable:
Unramified Forcing