Asaf Karagila
I don't have much choice...

Forcing. This Has To Stop.

There are 9 comments on this post.

Most, if not all, set theorists at one point or another were asked by a fellow mathematician to explain how forcing works. And many chose to give as an opening analogy field extensions. You can talk about how the construction of an algebraic closure is a bit similar, since the generic filter is a bit like the maximal ideal you use to make this construction; or you can talk about adding a transcendental number and the things that change as you add it.

But both these analogies would be wrong. They only take you so far, and not further. And if you wish to give a proper explanation to your listener, there will be no escape from the eventual logic and set theory of it all. I stopped, or at least I'm doing my best, using these analogies. I do, however, use the analogy of "How many roots does \(x^{42}-2\) has?" as an example for everyday independence (none in \(\mathbb Q\), two in \(\mathbb R\) and many in \(\mathbb C\)). But this is to motivate a different part of the explanation: the use of models of set theory (e.g. "How can you add a real number??", well how can you add a root to a polynomial?) and the fact that we don't consider the universe per se. Of course, in a model of \(\ZFC\) we can always construct the rest of mathematics internally, but this is not the issue now. Just like we have a model of one theory, we can have a model for another.

So why do we do it? Why do we keep explaining adding generic sets using the analogy of field extensions? Well, the easy answer is that field extensions are something that most mathematicians can understand pretty easily and it shows how we can "enlarge a universe".

But here's why we should stop doing that (and why I stopped doing that):

  1. Most people think about field extensions as being subfields of the complex numbers. This allows for a particular fixed background universe from which we can draw the numbers that we add. In set theory it is easy to think that there is one universe of sets, and that we force over that universe, in which case, where did the generic set come from?

    Moreover, if \(F\) is an algebraically closed field, and we add a transcendental element to \(F\), then there is a nice closure operation after which the resulting field has the same first-order theory as \(F\) (and if \(F\) is uncountable, then there is also an isomorphism between them). Short of miraculous black magic, I don't know of any such example in the case of set theory. Once you extend a model by forcing, there's no definable process to restore the theory of the model by enlarging the model without adding ordinals (the obvious case is \(V=L\)).

  2. The analogy is not accurate, and we can do better. For example, we can consider actual generic objects. The term generic comes from topology (and to my knowledge, that is where it trickled into algebraic geometry as well; but please correct me if I'm wrong about that). We say that a point $x$ is generic if it is an element of every dense open set. Generic objects are a lot like that. We have a partial order, it has a topology, and the generic real is something which lies in the intersection of all the dense sets from the ground model. So we can talk about adding a generic, or considering a generic point. This way it's also easy to explain why it has so many properties -- each property happens on a dense open set, so the generic point must have it. Or we can talk about what actually happens. We start with a countable model of set theory. There's no shame in that. It's like considering a countable field or another countable structure. Since it's countable, it's certainly not the collection of all sets. Be sure to explain why the model is only countable from the outside and not internally; and what does it mean that "the model thinks that ...", true it's not that easy, but there's a large payoff. You're not lying anywhere.
  3. Recently I watched a video of Richard Feynman being asked by a layman to explain why magnets behave the way they do. And Feynman said that's a very good question, and proceeded for five minutes to give analogy of a curious alien which would ask all sort of questions that you and I would take for granted; and then he went to say that the reason he didn't answer the question is that all the analogies he could make, or other people make, boil down to electromagnetic forces acting on a microscopic level. So if you would compare magnets to rubber, then the next reasonable question would be how does the rubber work, and by inquiring further you'd finally reach the original question again, how does the electromagnetic force work. So Feynman didn't want to deceive the layman, or confuse them with analogies which ultimately explain nothing. And I think that's a wonderful approach when you try to teach someone something. If electromagnetic force is one of the fundamental forces, we have to take it the way it is, and we can't explain it in simpler terms. Similarly, if forcing is a technique which is in its own class in mathematics, then we can't quite explain it in terms of other techniques. Every analogy would break down and cause confusion. In that case, maybe it is the simplest solution to just start right away from forcing, logic and set theory? Motivate what you do in terms of logic. We are interested in truth values of statements, this statement is a statement of the form "There exists a set such that ...", so we want to approximate this set, so by carefully choosing a set from outside the model our approximations are actualized in the new model.
  4. One prominent set theorist once told me that an incredibly smart mathematician (from representation theory) once asked him to explain forcing to him. He began with the analogy of field extensions, and it seems to be fine, and as he continued he defined the generic extension. Now we want to examine what sort of sentences are true in that generic extension, and we have this magical theorem which tells us exactly when something is true in the generic extension. Once the sentence "a formula in the language of forcing" was uttered, the eyes became vacant and the rest was moot. And neither of the mathematician is stupid, and the set theorist involved is a wonderful teacher. So why didn't it work? What can we learn from this? My guess is that the analogy sets a particular direction, and when you break from that analogy, it becomes confusing to the listener. If you weren't prepared to hear the term "the language of forcing", then you won't be able to jump over that hurdle when you reach it. And any analogy to field extensions hides this hurdle from the listener.

It seems to me, if so, that there are plenty of reasons not to use analogies, and plenty of reasons to explain things as they are. And forcing is not a trivial idea, remember that it completely revolutionized set theory. So if your audience can't grasp it over coffee, it's not a big deal. Perhaps using broad strokes to paint a rough image of approximations is better in that case, or at least better than giving the wrong idea.

There are 9 comments on this post.

[…] The strongest man in the world cannot resist the powers of set theoretical forcing. And Asaf Karagila will make sure you won’t wrongly use the analogy of field extensions to explain forcing. […]

By Joel David Hamkins
(Jun 10 2014, 16:04)

I have often used this analogy when explaining forcing to non-logicians, and despite your remarks, I think I'll continue to do so, because it is a great way to convey the fact, which is often surprising or even eye-opening to those not familiar with it, that set theorists study different models of set theory in a way that is similar to how group theorists study groups or how geometers study geometrical spaces. The case of algebraic extensions is a familiar way to explain the concept of non-elementary extensions, since the question of whether a given polynomial has a root or not depends on in which field you are considering it. Transcendental extensions help to convey the idea of genericity, as well as the fact that all the new objects in the field extension are definable from objects in the ground field plus the new generic object. Although it is true, as you mention, that the analogy doesn't tell the whole story; but we never thought it did, and that is a misunderstanding of when to use the analogy. Rather, we use the analogy as an entryway, a way to explain a little about what forcing is like to people who might otherwise be completely mystified by it. And in my experience, it serves very well for that purpose. For those who want to learn more about forcing, of course more precise explanations will be forthcoming.

Joel, I address this very issue in the second paragraph. I agree with this analogy, used for this purpose. To exhibit how we move from one model to another. And I agree these analogies are very useful when you want to exhibit day-to-day independence problems, which mathematicians often take for granted.

My point is that this analogy cannot be used to explain the mechanics of forcing. And this is something which I have seen, heard about, and did in the past. You can't use the idea of how the algebraic closure is constructed to explain how a Cohen real is approximated by its initial segments from the ground model. This analogy breaks down fairly fast once you try and get into any form of details.

But yes, this is very true, you can use the notion of algebraic numbers to give example on how a relatively obvious question has different answers in different models. This is still philosophically difficult to grasp, to some people, because often we take the complex numbers as something which obviously exists, so the algebraic numbers come "from there"; whereas the generic filter comes from nowhere. But this can be solved quite easily by either explaining that we don't really force over the universe (well, at least if we want generic filters to actually exist) or asking your listener to accept only the existence of the rational numbers as a premise, in which case what is \(\sqrt 2\)? It's something which we can define magically, but it's not a number per se.

Of course, between the two approaches, I prefer the first one.

Oh, then perhaps we agree more than I thought from what you wrote. I have never used the analogy except as an initial way to introduce the very idea of forcing, and indeed I would find it unsatisfactory as fuller account of forcing, if the more precise details were not part of the discussion. But for this initial purpose, the analogy works extremely well, and I think the analogy is more robust than you admit. When you say "it has to stop" I had taken you to mean that we should stop using the analogy altogether. That isn't what you meant?

What I meant was that the fields analogy is unsuitable as a technical analogy (I've heard it being used, for example, that we approximate the algebraic closure by adding more and more information about roots).

This sort of analogy should be stopped. The examples of routine independence and how we jump from one model to another, that's just a natural example. I also like the commutativity axiom in group theory when talking about an axiom (this seems like a suitable example for why certain axioms are not standard in modern set theory: why aren't all groups abelian?).

So please, keep using these analogies to explain what is independence; but don't start using it to explain the technical details of forcing.

This is your blog, and so I don't want to make a fuss here, but I disagree with the main thesis of this post. I believe that the analogy is more robust than your remarks allow, and it provides an excellent way to describe the relationship between a ground model and a forcing extension for those who know nothing about it. The analogy extends to the fact that both ground field/model and extension are models of the basic underlying theory; the extension has a new object that is not present in the ground model; this new object relates to ground model objects in a concrete manner with respect to the fundamental operations; the entirety of the extension is generated from the objects of the ground model and the new generic object; there is a corresponding sense of "name" in both cases where one can name the objects of the extension (such as \(x^3 3x\)) using objects in the ground model; the ground model and extension often have different truths, usually having to do specifically with the existence of the new generic object; in both cases the extension methods are one of the fundamental model-construction methods in their respective subjects; in both cases one often understands the extension by means of analyzing the automorphism groups of the structure that was used to generate the new object; in the case of ordered fields, one has fruitful approximations to the new generic object (the cut it determines), and in this way the ground field has some access to what will be true of the new object; the larger collection of all field extensions has a structure theory not unlike that for the forcing extensions of a given model, giving rise to a category-theoretic or multiverse perspective; there are different types of field extensions (algebraic, transcendental; degree of transcendence) just as there are different types of forcing extensions; in both cases one can construct an extension with a specific new type of object by choosing the construction method appropriately (e.g. adding an algebraic root or forcing with a particular forcing notion); algebraic extensions are like forcing extensions in which the generic filter is definable; both field extensions and forcing extensions are constructed in a quotient procedure by a filter; and so on. Your objections that the analogy is not a perfect technical description of forcing seems no reason not to continue using it to describe the basic situation of forcing for people who are new to it.

First of all, let me assure you that I am quite happy with this discussion. I often hope that someone will pick up the post and start a discussion in the comment, but that rarely happens. I usually get no comments, or comments saying that the post is nice/interesting. This discussion is very refreshing, and I'm thankful that you're going with it this far.

I am willing to concede that it has more than I just present it. But I also think that all those properties are only similar in a superficial way (okay, maybe not that superficial, but not deep enough). If you spent the first ten minutes of your explanation saying that this is very much like field extensions, and then you get into some details and it stops being similar, and in fact can become very different, then you haven't made the analogy clear enough.

But now that we talk about this, it might be possible to stretch this analogy into some of the details, but not in a sufficient way. And in any case, I wouldn't use it to explain any of the details because once you dot the I's and cross the T's these two turn out to be quite different. So telling someone how much these two constructions are alike seems like a counterproductive move.

Again, I'm not saying that the analogy is not useful to describe what forcing is, in very broad strokes, or why we are interested in these extensions, or even to give some motivation for the whole shindig. I'm just saying that if you want to get into any details, then this analogy breaks down pretty quickly. So if you plan to cover any details in your explanation, then it might be a good idea to avoid this analogy beyond the very general motivations.

By François G. Dorais
(Jun 10 2014, 20:12)

I've often used a different explanation via field extensions. This one takes a lot more to set up right but it has the advantage of solving a very real problem of everyday mathematics and it requires nothing more than a basic abstract algebra course. Here is the recipe...

Start with a field \(K\) and add infinitely many indeterminates \(T = \{t_0,t_1,t_2,\ldots\}\) to form the polynomial ring \(K[T] = K[t_0,t_1,t_2,\ldots]\). There is a natural forcing here: proper, finitely generated ideals in \(K[T]\), ordered by inclusion. The union of the generic filter is a ideal \(\Gamma\) in \(K[T]\). There is no need to explain the meaning of "generic" right away but make sure everybody is convinced that \(\Gamma\) is a proper ideal before moving on so everybody is clear what a "filter" is.

At this point, simply ask: What is \(K[T]/\Gamma\)? The initial reaction is usually that \(K[T]/\Gamma\) could be anything at all. Of course, this is where "generic" pops its head: \(K[T]/\Gamma\) is the algebraic closure of \(K\)! First explain why \(\Gamma\) must be a maximal ideal, so \(K[T]/\Gamma\) is a field extension of \(K\). Then, for a more subtle use of density, explain why \(K[T]/\Gamma\) is algebraically closed. It's rare for people to ask why every element of \(K[T]/\Gamma\) is algebraic over \(K\) but it's good to keep an answer in mind.

The upshot is that this construction of the algebraic closure is thousands of times better than the usual construction by transfinite recursion with annoying bookkeeping. Of course, it is well-known to set theorists that for every construction by transfinite recursion with annoying bookkeeping there is an easier forcing construction lurking in the background...

That's an interesting idea. I didn't think about this one (nor I have heard it before).

At this rate, I'll give a talk in the student seminar and explain forcing with these analogies. :-)

[That would be a cruel twist of fate, after I called out to stop using them.]

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