TITLE: A better way to explain forcing?
QUESTION [67 upvotes]: Let me begin by formulating a concrete (if not 100% precise) question, and then I'll explain what my real agenda is.
Two key facts about forcing are (1) the definability of forcing; i.e., the existence of a notion $\Vdash^\star$ (to use Kunen's notation) such that $p\Vdash \phi$ if and only if $(p \Vdash^\star \phi)^M$, and (2) the truth lemma; i.e., anything true in $M[G]$ is forced by some $p\in G$.
I am wondering if there is a way to "axiomatize" these facts by saying what properties forcing must have, without actually introducing a poset or saying that $G$ is a generic filter or that forcing is a statement about all generic filters, etc.  And when I say that forcing "must have" these properties, I mean that by using these axioms, we can go ahead and prove that $M[G]$ satisfies ZFC, and only worry later about how to construct something that satisfies the axioms.

Now for my hidden agenda.  As some readers know, I have written A beginner's guide to forcing where I try to give a motivated exposition of forcing.  But I am not entirely satisfied with it, and I have recently been having some interesting email conversations with Scott Aaronson that have prompted me to revisit this topic.
I am (and I think Scott is) fairly comfortable with the exposition up to the point where one recognizes that it would be nice if one could add some function $F : \aleph_2^M \times \aleph_0 \to \lbrace 0,1\rbrace$ to a countable transitive model $M$ to get a bigger countable transitive model $M[F]$.  It's also easy to grasp, by analogy from algebra, that one also needs to add further sets "generated by $F$."  And with some more thought, one can see that adding arbitrary sets to $M$ can create contradictions, and that even if you pick an $F$ that is "safe," it's not immediately clear how to add a set that (for example) plays the role of the power set of $F$, since the "true" powerset of $F$ (in $\mathbf{V}$) is clearly the wrong thing to add.  It's even vaguely plausible that one might want to introduce "names" of some sort to label the things you want to add, and to keep track of the relations between them, before you commit to saying exactly what these names are names of. But then there seems to be a big conceptual leap to saying, "Okay, so now instead of $F$ itself, let's focus on the poset $P$ of finite partial functions, and a generic filter $G$.  And here's a funny recursive definition of $P$-names."  Who ordered all that?
In Cohen's own account of the discovery of forcing, he wrote:

There are certainly moments in any mathematical discovery when the resolution of a problem takes place at such a subconscious level that, in retrospect, it seems impossible to dissect it and explain its origin. Rather, the entire idea presents itself at once, often perhaps in a vague form, but gradually becomes more precise.

So a 100% motivated exposition may be a tad ambitious.  However, it occurs to me that the following strategy might be fruitful.  Take one of the subtler axioms, such as Comprehension or Powerset.  We can "cheat" by looking at the textbook proof that $M[G]$ satisfies the axiom.  This proof is actually fairly short and intuitive if you are willing to take for granted certain things, such as the meaningfulness of this funny $\Vdash$ symbol and its two key properties (definability and the truth lemma).  The question I have is whether we can actually produce a rigorous proof that proceeds "backwards": We don't give the usual definitions of a generic filter or of $\Vdash$ or even of $M[G]$, but just give the bare minimum that is needed to make sense of the proof that $M[G]$ satisfies ZFC.  Then we "backsolve" to figure out that we need to introduce a poset and a generic filter in order to construct something that satisfies the axioms.
If this can be made to work, then I think it would greatly help "ordinary mathematicians" grasp the proof.  In ordinary mathematics, expanding a structure $M$ to a larger structure $M[G]$ never requires anything as elaborate as the forcing machinery, so it feels like you're getting blindsided by some deus ex machina.  Of course the reason is that the axioms of ZFC are so darn complicated.  So it would be nice if one could explain what's going on by first looking at what is needed to prove that $M[G]$ satisfies ZFC, and use that to motivate the introduction of a poset, etc.
By the way, I suspect that in practice, many people learn this stuff somewhat "backwards" already.  Certainly, on my first pass through Kunen's book, I skipped the ugly technical proof of the definability of forcing and went directly to the proof that $M[G]$ satisfies ZFC.  So the question is whether one can push this backwards approach even further, and postpone even the introduction of the poset until after one sees why a poset is needed.

REPLY [30 votes]: This is an expansion of David Roberts's comment.  It may not be the sort of answer you thought you were looking for, but I think it is appropriate, among other reasons because it directly addresses your question

if there is a way to "axiomatize" these facts by saying what properties forcing must have.

In fact, modern mathematics has developed a powerful and general language for "axiomatizing properties that objects must have": the use of universal properties in category theory.  In particular, universal properties give a precise and flexible way to say what it means to "freely" or "generically" add something to a structure.
For example, suppose we have a ring $R$ and we want to "generically" add a new element.  The language of universal properties says that this should be a ring $R[x]$ equipped with a homomorphism $c:R\to R[x]$ and an element $x\in R[x]$ with the following universal property: for any ring $S$ equipped with a homomorphism $f:R\to S$ and an element $s\in S$, there exists a unique homomorphism $h:R[x]\to S$ such that $h\circ c = f$ and $h(x) = s$.
Note that this says nothing about how $R[x]$ might be constructed, or even whether it exists: it's only about how it behaves.  But this behavior is sufficient to characterize $R[x]$ up to unique isomorphism, if it exists.  And indeed it does exist, but to show this we have to give a construction: in this case we can of course use the ring of formal polynomials $a_n x^n + \cdots + a_1 x + a_0$.
From this perspective, if we want to add a function $F : \aleph_2\times \aleph_0 \to 2$ to a model $M$ of ZFC to obtain a new model $M[F]$, the correct thing to do would be to find a notion of "homomorphism of models" such that $M[F]$ can be characterized by a similar universal property: there would be a homomorphism $c:M\to M[F]$ and an $F : \aleph_2\times \aleph_0 \to 2$ in $M[F]$, such that for any model $N$ equipped with a homomorphism $f:M\to N$ and a $G : \aleph_2\times \aleph_0 \to 2$ in $N$, there is a unique homomorphism $h:M[F]\to N$ such that $h\circ c = f$ and $h(F) = G$.
The problem is that the usual phrasing of ZFC, in terms of a collection of things called "sets" with a membership relation $\in$ satisfying a list of axioms in the language of one-sorted first-order logic, is not conducive to defining such a notion of homomorphism.  However, there is an equivalent formulation of ZFC, first given by Lawvere in 1964, that works much better for this purpose.  (Amusingly, 1964 is exactly halfway between 1908, when Zermelo first proposed his list of axioms for set theory, and the current year 2020.)  In Lawvere's formulation, there is a collection of things called "sets" (although they behave differently than the "sets" in the usual presentation of ZFC) and also a separate collection of things called "functions", which together form a category (i.e. functions have sets as domain and codomain, and can be composed), and satisfy a list of axioms written in the language of category theory.  (A recent short introduction to Lawvere's theory is this article by Tom Leinster.)
Lawvere's theory is usually called "ETCS+R" (the "Elementary Theory of the Category of Sets with Replacement"), but I want to emphasize that it is really an entirely equivalent formulation of ZFC.  That is, there is a bijection between models of ZFC, up to isomorphism, and models of ETCS+R, up to equivalence of categories.  In one direction this is exceedingly simple: given a model of ZFC, the sets and functions therein as usually defined form a model of ETCS+R.  Constructing an inverse bijection is more complicated, but the basic idea is the Mostowski collapse lemma: well-founded extensional relations can be defined in ETCS+R, and the relations of this sort in any model of ETCS+R form a model of ZFC.
Since a model of ETCS+R is a structured category, there is a straightforward notion of morphism between models: a functor that preserves all the specified structure.  However, this notion of morphism has two defects.
The first is that the resulting category of models of ETCS+R is ill-behaved.  In particular, the sort of "free constructions" we are interested in do not exist in it!  However, this is a problem of a sort that is familiar in modern structural mathematics: when a category is ill-behaved, often it is because we have imposed too many "niceness" restrictions on its objects, and we can recover a better-behaved category by including more "ill-behaved" objects.  For instance, the category of manifolds does not have all limits and colimits, but it sits inside various categories of more general "smooth spaces" that do.  The same thing happens here: by dropping two of the axioms of ETCS+R we obtain the notion of an elementary topos, and the category of elementary toposes, with functors that preserve all their structure (called "logical functors"), is much better-behaved.  In particular, we can "freely adjoin a new object/morphism" to an elementary topos.
(I am eliding here the issue of the replacement/collection axiom, which is trickier to treat correctly for general elementary toposes.  But since my main point is that this direction is a blind alley for the purposes of forcing anyway, it doesn't matter.)
The second problem, however, is that these free constructions of elementary toposes do not have very explicit descriptions.  This is important because our goal is not merely to freely adjoin an $F:\aleph_2\times \aleph_0 \to 2$, but to show that the existence of such an $F$ is consistent, and for this purpose we need to know that when we freely adjoin such an $F$ the result is nontrivial.  Thus, in addition to characterizing $M[F]$ by a universal property, we need some concrete construction of it that we can inspect to deduce its nontriviality.
This problem is solved by imposing a different niceness condition on the objects of our category and changing the notion of morphism.  A Grothendieck topos is an elementary topos that, as a category, is complete and cocomplete and has a small generating set.  But, as shown by Giraud's famous theorem, it can equivalently be defined as a cocomplete category with finite limits and a small generating set where the finite limits and small colimits colimits interact nicely.  This suggests a different notion of morphism between Grothendieck toposes: a functor preserving finite limits and small colimits.  Let's call such a functor a Giraud homomorphism (it's the same as a "geometric morphism", but pointing in the opposite direction).
The category of Grothendieck toposes and Giraud homomorphisms is well-behaved, and in particular we can freely adjoin all sorts of structures to a Grothendieck topos -- specifically, any structure definable in terms of finite limits and arbitrary colimits (called "a model of a geometric theory").  (To be precise, this is a 2-category rather than a category, and the universal properties are up to isomorphism, but this is a detail, and unsurprising given the modern understanding of abstract mathematics.)  Moreover, the topos $M[G]$ obtained by freely adjoining a model $G$ of some geometric theory to a Grothendieck topos $M$ -- called the classifying topos of the theory of $G$ -- has an explicit description in terms of $M$-valued "sheaves" on the syntax of the theory of $G$.  This description allows us to check, in any particular case, that it is nontrivial.  But for other purposes, it suffices to know the universal property of $M[G]$.  In this sense, the universal property of a classifying topos is an answer to your question:

when I say that forcing "must have" these properties, I mean that by using these axioms, we can go ahead and prove that $M[G]$ satisfies ZFC, and only worry later about how to construct something that satisfies the axioms.

Only one thing is missing: not every Grothendieck topos is a model of ETCS+R, hence $M[G]$ may not itself directly yield a model of ZFC.  We solve this in three steps.  First, since ZFC satisfies classical logic rather than intuitionistic logic (the natural logic of categories), we force $M[G]$ to become Boolean.  Second, by restricting to "propositional" geometric theories we ensure that the result also satisfies the axiom of choice.  Finally, we pass to the "internal logic" of the topos, which is to say that we allow "truth values" lying in its subobject classifier rather than in the global poset of truth values $2$.  We thereby get an "internal" model of ETCS+R, and hence also an "internal" model of ZFC.
So where does the complicated machinery in the usual presentation of forcing come from?  Mostly, it comes from "beta-reducing" this abstract picture, writing out explicitly the meaning of "well-founded extensional relation internal to Boolean sheaves on the syntax of a propositional geometric theory".  The syntax of a propositional geometric theory yields, as its Lindenbaum algebra, a poset.  The Boolean sheaves on that poset are, roughly, those that satisfy the usual "denseness" condition in forcing.  The "internal logic" valued in the subobject classifier corresponds to the forcing relation over the poset.  And the construction of well-founded extensional relations translates to the recursive construction of "names".
(Side note: this yields the "Boolean-valued models" presentation of forcing.  The other version, where we take $M$ to be countable inside some larger model of ZFC and $G$ to be an actual generic filter living in that larger model, is, at least to first approximation, an unnecessary complication.  By comparison (and in jesting reference to Asaf's answer), if we want to adjoin a new transcendental to the field $\mathbb{Q}$, we can simply construct the field of rational functions $\mathbb{Q}(x)$.  From the perspective of modern structural mathematics, all we care about are the intrinsic properties of $\mathbb{Q}(x)$; it's irrelevant whether it happens to be embeddable in some given larger field like $\mathbb{R}$ by setting $x=\pi$.)
The final point is that it's not necessary to do this beta-reduction.  As usual in mathematics, we get a clearer conceptual picture, and have less work to do, when working at an appropriate level of abstraction.  We prove the equivalence of ZFC and ETCS+R once, abstractly.  Similarly, we show that we have an "internal" model of ETCS+R in any Grothendieck topos.  These proofs are easier to write and understand in category-theoretic language, using the intrinsic characterization of Grothendieck toposes rather than anything to do with sites or sheaves.  With that done, the work of forcing for a specific geometric theory is reduced to understanding the relevant properties of its category of Boolean sheaves, which are simple algebraic structures.