For more than 2000 years, geometers and mathematicians tried in vain to prove the parallel postulate from the other axioms of Euclidean geometry. These other axioms said^[1]:

1. From any point to any point, one can draw a straight line
2. One can extend a finite straight line continuously in a straight line.
3. One can describe a circle by a center and distance (radius).
4. All right angles are equal to one another.

These axioms are beautiful in their simplicity. They clarify precisely what we feel like geometry is trying to describe intuitively. The parallel postulate read as follows:

5. If a straight line intersects two straight lines, making the interior angles on one side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.

“Blegh! Gross! We hate it!” said the mathematicians.

This doesn’t feel like it should be a fundamental notion of geometry. Sure, perhaps we could try and write it simpler, but it still isn’t analogous to capturing my basic instinct of what a line is, or what a circle is. If you showed me an example of three straight lines on a piece of paper, I could see that they do exhibit this property — but it doesn’t feel basic. Yet we needed it to help us build the rest of geometry, along with the four simple axioms.

Mathematicians’ attempts to prove that the parallel postulate could be proven from the other axioms were in vain because in fact, it does not follow from the other axioms. You can have a perfectly reasonable geometry where this axiom does not hold. Either because all lines eventually meet (spherical geometry), or because there are many “parallel lines” which never meet through a given point (hyperbolic geometry).

These different notions of geometry would today be known as models. They offer different models of the first four axioms of geometry. In at least one of these models, the parallel postulate is also true (Euclidean geometry), and yet there are other models, as shown in the image above, where the parallel postulate is not true.

In fact, there are models for all consistent mathematical theories — that is, so long as your theory isn’t self-contradictory, there’s a model for it^[2]. Including the theory in which we do all of our modern mathematics: Set Theory (so long as set theory is consistent).

I. What is the Constructible Universe?

The Constructible Universe of sets, often referred to as L, is one of these models of set theory. It is — in some ways — the simplest model of set theory. It was defined first by Gödel in 1938. It is often the model that lay-people have in mind when they think about “the Universe of sets.”

It was built by Gödel in stages, as follows:

1. At the first stage of building L, we say that “L nought,” is just the empty set. There is nothing in our universe yet.
2. At each subsequent stage, we add all the sets that can be defined from the stage before. So at the second stage “L one,” we can define the empty set. Therefore “L one” is just the set that contains the empty set.
3. We keep doing this forever. At some points, we’ll need to “reach an infinity.” At this point, we take every set we have defined so far, and bundle them all up into a new set, which we call “L some-infinity.” And then we keep going.

So, for example, once we reach the first “L infinity,” we bundle everything up. We then use the second step in our recipe to get to “L infinity plus one”, “L infinity plus two,” and so on until we have to reach “L infinity times two,” where we apply our third step again, bundling everything up.

(I should say, that although there are such entities as “infinity plus one,” “infinity plus two,” they are not bigger than infinity, they just come after infinity in order, but not in size — don’t worry about it too much^[3]).

We do this forever. There is no stopping point. There are always larger and larger infinities to get to. If you think you’ve reached the largest infinity, set theory will stop you, point ahead at the next infinity, and show you how much further you have to go (you still have to go so much further than you’ve already gone)!^[4]

If we step back, and consider the class^[5] of all sets you will eventually build, you will be able to show that all of the 9 axioms of set theory^[6] are satisfied within this structure. That is, if you believe that this structure can be built, then you believe that set theory must be consistent. Further, despite us making no use of the axiom of choice to build this structure, it does satisfy the axiom of choice (yet another arrow in the Axiom of Choice’s quiver).

The reason that Gödel built this structure was to prove that the Continuum hypothesis (the first of Hilbert’s 23 problems for mathematicians in the 20th century) is consistent with set theory. The continuum hypothesis says that there is no “intermediate size” between the size of the whole numbers, and the size of the real numbers.

L believes that the continuum hypothesis is true, and so long as set theory is consistent, L must also exist. So it must be possible to have a version of set theory that believes the continuum hypothesis — much like it is possible to have a version of geometry that believes the parallel postulate.

So why not accept the continuum hypothesis is true, say that L is set theory, and all that remains for mathematicians is to analyze L. Set theory by itself is not able to prove that there’s anything outside of L. It looks rather tempting…

II. Forcing us to look again

In 1963, depending on who you ask, Paul Cohen either: destroyed the dream that we might one day resolve the continuum hypothesis; or he resolved the continuum hypothesis once and for all. He demonstrated that there is a model of set theory where the continuum hypothesis fails: managing to create the equivalent of “spherical geometry” for set theory.

The method he used to do this was called forcing. Explaining forcing in a rigorous manner here would be impossible, but let me attempt to briefly outline the idea in two not-so-easy steps!

First, we begin with the L we built. It is a fact that if there is some model of a theory, no matter how big, then there is a model of that theory which contains only countably many elements.

Let’s pause here and realize that something weird is happening. There is a countable structure that believes everything that L believes, including that there are uncountable sets — yet every set in this structure is countable: the structure itself is countable! This is known as Skolem’s paradox, and it is strange. Essentially, the explanation of the paradox is to realize that “uncountable” just means “not countable” — there is not a function that counts all the elements in the set. From our perspective, looking at this model, we can see that of course the sets it contains are actually countable. However, this model cannot see the function that counts the elements in the “uncountable” set. So from within the model this set is uncountable — are you starting to see why Cantor went crazy yet?

Okay, now on to the second and final step. This countable model that’s been created — which believes and proves everything that L believes and proves — has a copy of the “real numbers^[7]” within it. Within L, there is no size of infinity that lies between the real numbers and the whole numbers, so this countable model must think the same thing. However, from the outside we can see that this countable model can see barely any of the real numbers. From the outside, the “real numbers” of this model are countable! We know that there are more real real numbers than that.

So we think of a clever trick we can play on this model. Under cover of darkness, we sneak in and add a new collection of real numbers — we force it to see more real numbers^[8], being careful not to break any of the axioms of set theory that the model believes in. We add only countably many of them, but it’s enough that this model now believes that the number of real numbers it can see is not the first uncountable infinity, there are the second uncountable infinity many real numbers! There are too many real numbers in this model — enough that it thinks it can squeeze the first uncountable infinity between the number of whole numbers and the number of real numbers. So this model still satisfies the axioms of set theory, and yet it doesn’t agree with the continuum hypothesis!

So the continuum hypothesis is neither provable nor unprovable from the axioms of set theory. It is a statement much like the parallel postulate, there are cases where it is true and cases where it is false.

But the consequences of Cohen’s method went far beyond the continuum hypothesis^[9], it opened up a multiverse of different possible interpretations of the axioms of set theory. The same argument that was used to refute the continuum hypothesis was able to be modified to make the size of the reals as big as you would like! It could be modified again, and again, and again to show that a plethora of statements were neither provable nor disprovable from the axioms. It was, in a word, a revolution.

So set theorists started to believe that there is no one model of set theory. That set theory is destined to forever be another branch of mathematics like geometry — where you can believe what you want to believe and analyze whichever model interests you the most. Set theorists began to let a thousand flowers bloom, and even argue that this was the correct philosophical interpretation of the results that had occurred up to this point. This is the pluralist/multiverse view of set theory. They were arguing that it was the correct one — or at least the one we ought to adopt.

III. The Return of L

Yet those who wished for there to be one true Universe of Sets never quite went away. There were always those who thought that one day we could reconcile our desire to have a strong set theory with our intuitions for how sets should behave: “Wouldn’t it be nice if the continuum hypothesis were true?” they would argue, “Can’t we agree on the one True Universe?” These acolytes would argue that we have all agreed on a single version of the whole numbers for number theory, and a single version of the real numbers for analysis, why can we not do the same thing for set theory?

To this end, Hugh Woodin has proposed a field of research aimed at showing that there exists a structure — Ultimate L — that functions very similarly to L, and yet is much stronger than it. Those who believe in a true Universe of Sets think that there is strong evidence for very large numbers. Numbers so large that you cannot prove that they exist from set theory alone. However, if we could construct a structure like L, that is also able to contain these large numbers, all of a sudden, this Ultimate L looks a lot more appealing compared to the multiverse of sets we’ve been able to garner from forcing. If we can all agree on it, we can finally decide the truth of statements like the continuum hypothesis. Unfortunately, as yet, the project developing Ultimate L remains conjectural, and even if it’s shown to work, the committed pluralist can just add it to their collection of multiverses!

IV. Conclusion

Ultimately, I must admit that I come down on the pluralist side of the argument. As a society we have proceeded over time to reject the “unique truth” that Euclidean geometry once thought it provided, in favor of a richer view of geometry. We have rejected the “unique truth” of the real numbers, with work developing analysis in the hyperreals^[10]. Algebraists have never even felt the need to limit themselves to finding the “One True Group” — it’s not clear what such a thing would even mean, and certainly it is clear why it’s not desirable to seek it.

Is it not richer to live in a Universe teeming with possibilities, where around every set-theoretical corner we don’t know what we will find? Where in every new model of set theory we explore, there may be some interesting structure waiting to surprise and astound us? I certainly think so — but this is not to say that giving up this one True Universe when it comes to mathematics in general is not a sacrifice, it certainly is. It’s just a sacrifice worth making.

1. ^{^}

   Axioms adapted from Thomas Heath’s translation.

2. ^{^}

   Link, if you’re curious

3. ^{^}

   Just read some of this wikipedia page if you can’t help but worry — I promise it’s fine.

4. ^{^}

   You may wish to listen to some music while you build:

5. ^{^}

   What we have built is too big to be a set. Far too big! It is a class which is what set theorists call an object that they can describe, but that if it were a set would lead us to a contradiction.

6. ^{^}

   Some people say there are only eight. It’s much like planets, those people are no fun! (Yes I know two of them are technically axiom schema, don’t be a pedant).

7. ^{^}

   Of course, they are not the true real numbers — or are they? Everything that is true about the actual real numbers is true about this countable set of “real numbers,” from within the model.

8. ^{^}

   We really do have to be very careful not to change anything else about the model, in particular, so that the model doesn’t start thinking that the uncountable infinities change sizes, as that could mess up the whole trick.

9. ^{^}

   There is a reason Cohen is the only mathematician in the field of mathematical logic to ever win the Fields medal— although Gödel certainly would have won were he eligible for it after 1950.

10. ^{^}

    Although there are other reasons why it is not so nice to work with the hyperreals — relating to a property called “categoricity.”