Last time I took you on a road trip to infinity. We zipped past a bunch of countable ordinals
and stopped for gas at the first one after all these. It’s called Heuristically, you can imagine it like this:
More rigorously, it’s the smallest ordinal obeying the equation
But I’m sure you have a question. What comes after ?
Well, duh! It’s
and then eventually we get to
and after a long time
and then eventually
and then eventually….
Oh, I see! You wanted to know the first really interesting ordinal after
Well, this is a matter of taste, but you might be interested in This is the first ordinal after that satisfies this equation:
How do we actually reach this ordinal? Well, just as was the limit of this sequence:
is the limit of this:
You may wonder what I mean by the ‘limit’ of an increasing sequence of ordinals. I just mean the smallest ordinal greater than or equal to every ordinal in that sequence. Such a thing is guaranteed to exist, since if we treat ordinals as well-ordered sets, we can just take the union of all the sets in that sequence.
Here’s a picture of taken from David Madore’s interactive webpage:
In what sense is the first "really interesting" ordinal after ?
For one thing, it’s first that can’t be built out of and using finitely many additions, multiplications and exponentiations. In other words, if we use Cantor normal form to describe ordinals (as explained last time), and allow expressions involving as well as and we get a notation for all ordinals up to
What’s the next really interesting ordinal after ? As you might expect, it’s called This is the next solution of
and it’s defined to be the limit of this sequence:
Maybe now you get the pattern. In general, is the
th solution of We can define this, if we’re smart, for any ordinal
So, we can keep driving on through fields of ever larger ordinals:
which is the first ordinal bigger than
Let’s stop and take a look!
Nice! Okay, back in the car…
As you can see, this gets boring after a while: it’s suspiciously similar to the beginning of our trip through the ordinals. The same ordinals are now showing up as subscripts in this epsilon notation. But we’re moving much faster now, since I’m skipping over much bigger gaps, not bothering to mention all sorts of ordinals like
Anyway… while we’re zipping along, I might as well finish telling you the story I started last time. My friend David Sternlieb and I were driving across South Dakota on Route 80. We kept seeing signs for the South Dakota Tractor Museum. When we finally got there, we were driving pretty darn fast, out of boredom—about 85 miles an hour. And guess what happened then!
Oh — wait a minute—this one is sort of interesting:
Then come some more like that:
until we reach this:
As we keep speeding up, we see:
So, anyway: by the time we got that tractor museum, we were driving really fast. And, all we saw as we whizzed by was a bunch of rusty tractors out in a field! It was over in a split second! It was a real anticlimax — just like this anecdote, in fact.
But that’s just the way it is when you’re driving through these ordinals! Every ordinal, no matter how large, looks pretty pathetic and small compared to the ones ahead — so you keep speeding up, looking for something ‘really new and different’. But when you find one, it turns out to be part of a larger pattern, and soon that gets boring too.
For example, when we reach the limit of this sequence:
our notation fizzles out again, since this is the first solution of
We could make up a new name for this ordinal, like I don’t think this name is very common, though I’ve seen it. We could call it the Tractor Museum of Countable Ordinals.
Now we can play the whole game again, defining the zeta number to be the th solution of
sort of like how we defined the epsilons. This kind of equation, where something equals some function of itself, is called a fixed point equation.
But since we’ll have to play this game infinitely often, we might as well be more systematic about it!
The Veblen hierarchy
As you can see, we keep running into new, qualitatively different types of ordinals. First we ran into the powers of omega. Then we ran into the epsilons, and then the zetas. It’s gonna keep happening! For each type of ordinal, our notation fizzles out when we reach the first ‘fixed point’— when the xth ordinal of this type is actually equal to x.
So, instead of making up infinitely many Greek letters for different types of ordinals let’s index them… by ordinals! For each ordinal we’ll have a type of ordinal. We’ll let be the th ordinal of type
We can use the fixed point equation to define in terms of In other words, we start off by defining
and then define
to be the th solution of
where we start counting at so the first solution is called the ‘zeroth’.
We can even make sense of when itself is infinite! Suppose is a limit of smaller ordinals. Then we define to be the limit of as approaches I’ll make this more precise next time.
We get infinitely many different types of ordinals, called the Veblen hierarchy. So, concretely, the Veblen hierarchy starts with the powers of
and then it goes on to the ‘epsilons’:
and then it goes on to what I called the ‘zetas’:
But that’s just the start!
The Feferman–Schütte ordinal
Boosting the subscript in increases the result much more than boosting so let’s focus on that and just let The Veblen hierarchy contains ordinals like this:
and then ordinals like this:
and then ordinals like this:
and then this:
where of course I’m skipping huge infinite stretches of ‘boring’ ones. But note that
In short, we can plug the phi function into itself—and we get the biggest effect if we plug it into the subscript!
So, if we’re in a rush to reach some really big countable ordinals, we can try these:
But the limit of these is an ordinal that has
This is called the Feferman–Schütte ordinal and denoted
In fact, the Feferman–Schütte ordinal is the smallest solution of
Since this equation is self-referential, we can’t describe Feferman–Schütte ordinal using the Veblen hierarchy—at least, not without using the Feferman–Schütte ordinal!
Indeed, some mathematicians have made a big deal about this ordinal, claiming it’s
the smallest ordinal that cannot be described without self-reference.
This takes some explaining, and it’s somewhat controversial. After all, there’s a sense in which every fixed point equation is self-referential. But there’s a certain precise sense in which the Feferman–Schütte ordinal is different from previous ones.
Anyway, you have admit that this is a very cute description of the Fefferman–Schuette ordinal: “the smallest ordinal that cannot be described without self-reference.” Does it use self-reference? It had better—otherwise we have a contradiction!
It’s a little scary, like this picture:
More importantly for us, the Veblen hierarchy fizzles out when we hit the Feferman–Schuette ordinal. Let me say what I mean by that.
Veblen normal form
Veblen normal form relies on this result:
Theorem. Any ordinal can be written uniquely as
where is a natural number, each term is less than or equal to the previous one, and for all
Note that we can also use this theorem to write out the ordinals and , and so on, recursively. So, it gives us a notation for ordinals.
However, this notation is only useful when all the ordinals are less than the ordinal that we’re trying to describe. Otherwise we need to already have a notation for to express in Veblen normal form!
So, the power of this notation eventually fizzles out. And the place where it does is Feferman–Schütte ordinal. Every ordinal less than this can be expressed in terms of , addition, and the function using just finitely many symbols!
As I hope you see, the power of the human mind to see a pattern and formalize it gives the quest for large countable ordinals a strange quality. As soon as we see a systematic way to generate a sequence of larger and larger ordinals, we know this sequence has a limit that’s larger then all of those! And this opens the door to even larger ones….
So, this whole journey feels a bit like trying to outrace our car’s own shadow as we drive away from the sunset: the faster we drive, the faster it shoots ahead of us. We’ll never win.
On the other hand, we’ll only lose if we get tired.
So it’s interesting to hear what happens next. We don’t have to give up. The usual symbol for the Feferman–Schütte ordinal should be a clue. It’s called And that’s because it’s just the start of a new series of even bigger countable ordinals!
I’m dying to tell you about those. But this is enough for today.