Large Countable Ordinals (Part 2)

4 July, 2016

Last time I took you on a road trip to infinity. We zipped past a bunch of countable ordinals

$\omega , \; \omega^\omega,\; \omega^{\omega^\omega}, \;\omega^{\omega^{\omega^\omega}}, \dots$

and stopped for gas at the first one after all these. It’s called $\epsilon_0.$ Heuristically, you can imagine it like this:

$\epsilon_0 = \omega^{\omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}}$

More rigorously, it’s the smallest ordinal $x$ obeying the equation

$x = \omega^x$

Beyond εo

But I’m sure you have a question. What comes after $\epsilon_0$?

Well, duh! It’s

$\epsilon_0 + 1$

Then comes

$\epsilon_0 + 2$

and then eventually we get to

$\epsilon_0 + \omega$

and then

$\epsilon_0 + \omega^2 ,\dots, \epsilon_0 + \omega^3,\dots \epsilon_0 + \omega^4,\dots$

and after a long time

$\epsilon_0 + \epsilon_0 = \epsilon_0 2$

and then eventually

$\epsilon_0^2$

and then eventually….

Oh, I see! You wanted to know the first really interesting ordinal after $\epsilon_0.$

Well, this is a matter of taste, but you might be interested in $\epsilon_1.$ This is the first ordinal after $\epsilon_0$ that satisfies this equation:

$x = \omega^x$

How do we actually reach this ordinal? Well, just as $\epsilon_0$ was the limit of this sequence:

$\omega , \; \omega^\omega,\; \omega^{\omega^\omega}, \;\omega^{\omega^{\omega^\omega}}, \dots$

$\epsilon_1$ is the limit of this:

$\epsilon_0 + 1, \; \omega^{\epsilon_0 + 1}, \; \omega^{\omega^{\epsilon_0 + 1}}, \; \omega^{\omega^{\omega^{\epsilon_0 + 1}}},\dots$

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 $\epsilon_1,$ taken from David Madore’s interactive webpage:

In what sense is $\epsilon_1$ the first "really interesting" ordinal after $\epsilon_0$?

For one thing, it’s first that can’t be built out of $1, \omega$ and $\epsilon_0$ 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 $\epsilon_0$ as well as $1$ and $\omega,$ we get a notation for all ordinals up to $\epsilon_1.$

What’s the next really interesting ordinal after $\epsilon_1$? As you might expect, it’s called $\epsilon_2.$ This is the next solution of

$x = \omega^{x}$

and it’s defined to be the limit of this sequence:

$\epsilon_1 + 1, \; \omega^{\epsilon_1 + 1}, \;\omega^{\omega^{\epsilon_1 + 1}}, \; \omega^{\omega^{\omega^{\epsilon_1 + 1}}},\dots$

Maybe now you get the pattern. In general, $\epsilon_{\alpha}$ is the
$\alpha$th solution of $x = \omega^{x}.$ We can define this, if we’re smart, for any ordinal $\alpha.$

So, we can keep driving on through fields of ever larger ordinals:

$\epsilon_2,\dots, \epsilon_{3},\dots, \epsilon_{4}, \dots$

and eventually

$\epsilon_{\omega}$

which is the first ordinal bigger than $\epsilon_0, \epsilon_1, \epsilon_2, \dots$

Let’s stop and take a look!

Nice! Okay, back in the car…

$\epsilon_{\omega+1},\dots, \epsilon_{\omega+2},\dots, \epsilon_{\omega+1},\dots$

and then

$\epsilon_{\omega^2},\dots , \epsilon_{\omega^3},\dots, \epsilon_{\omega^4},\dots$

and then

$\epsilon_{\omega^{\omega}},\dots, \epsilon_{\omega^{\omega^{\omega}}},\dots$

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

$\epsilon_{\omega^{\omega}} + \epsilon_{\omega 248} + \omega^{\omega^{\omega + 17}} + 1$

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:

$\displaystyle{ \epsilon_{\epsilon_0} }$

Then come some more like that:

$\epsilon_{\epsilon_1},\dots, \epsilon_{\epsilon_2},\dots \epsilon_{\epsilon_3},\dots$

until we reach this:

$\epsilon_{\epsilon_{\omega}}$

and then

$\epsilon_{\epsilon_{\omega^{\omega}}},\dots, \epsilon_{\epsilon_{\omega^{\omega^{\omega}}}},\dots$

As we keep speeding up, we see:

$\epsilon_{\epsilon_{\epsilon_0}},\dots \epsilon_{\epsilon_{\epsilon_{\epsilon_0}}},\dots \epsilon_{\epsilon_{\epsilon_{\epsilon_{\epsilon_0}}}},\dots$

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:

$\epsilon_0, \epsilon_{\epsilon_0}, \epsilon_{\epsilon_{\epsilon_0}}, \epsilon_{\epsilon_{\epsilon_{\epsilon_0}}}, \epsilon_{\epsilon_{\epsilon_{\epsilon_{\epsilon_0}}}},\dots$

our notation fizzles out again, since this is the first solution of

$x = \epsilon_{x}$

We could make up a new name for this ordinal, like $\zeta_0.$ 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 $\zeta_{\alpha}$ to be the $\alpha$th solution of

$x = \epsilon_x$

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 $\gamma$ we’ll have a type of ordinal. We’ll let $\phi_\gamma(\alpha)$ be the $\alpha$th ordinal of type $\gamma.$

We can use the fixed point equation to define $\phi_{\gamma+1}$ in terms of $\phi_{\gamma}.$ In other words, we start off by defining

$\phi_0(\alpha) = \omega^{\alpha}$

and then define

$\phi_{\gamma+1}(\alpha)$

to be the $\alpha$th solution of

$x = \phi_{\gamma}(x)$

where we start counting at $\alpha = 0,$ so the first solution is called the ‘zeroth’.

We can even make sense of $\phi_\gamma(\alpha)$ when $\gamma$ itself is infinite! Suppose $\gamma$ is a limit of smaller ordinals. Then we define $\phi_\gamma(x)$ to be the limit of $\phi_\beta(x)$ as $\beta$ approaches $\gamma.$ 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 $\omega:$

$\phi_0(\alpha) = \omega^\alpha$

and then it goes on to the ‘epsilons’:

$\phi_1(\alpha) = \epsilon_\alpha$

and then it goes on to what I called the ‘zetas’:

$\phi_2(\alpha) = \zeta_\alpha$

But that’s just the start!

The Feferman–Schütte ordinal

Boosting the subscript $\gamma$ in $\phi_\gamma(\alpha)$ increases the result much more than boosting $\alpha,$ so let’s focus on that and just let $\alpha = 0.$ The Veblen hierarchy contains ordinals like this:

$\phi_{\omega}(0), \; \phi_{\omega+1}(0), \; \phi_{\omega+2}(0), \dots$

and then ordinals like this:

$\phi_{\omega^2}(0), \; \phi_{\omega^3}(0), \; \phi_{\omega^4}(0), \dots$

and then ordinals like this:

$\phi_{\omega^\omega}(0), \; \phi_{\omega^{\omega^\omega}}(0), \; \phi_{\omega^{\omega^{\omega^{\omega}}}}(0), \dots$

and then this:

$\phi_{\epsilon_0}(0), \phi_{\epsilon_{\epsilon_0}}(0), \phi_{\epsilon_{\epsilon_{\epsilon_0}}}(0), \dots$

where of course I’m skipping huge infinite stretches of ‘boring’ ones. But note that

$\phi_{\omega}(0) = \phi_{\phi_0(0)}(0)$

and

$\phi_{\epsilon_0}(0) = \phi_{\phi_1(0)}(0)$

and

$\phi_{\zeta_0}(0) = \phi_{\phi_2(0)}(0)$

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:

$\phi_0(0), \; \phi_{\phi_0(0)}(0) , \; \phi_{\phi_{\phi_0(0)}(0)}(0), \dots$

But the limit of these is an ordinal $x$ that has

$x = \phi_x(0)$

This is called the Feferman–Schütte ordinal and denoted $\Gamma_0.$

In fact, the Feferman–Schütte ordinal is the smallest solution of

$x = \phi_x(0)$

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!

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

The Veblen hierarchy gives a notation for ordinals called the Veblen normal form. You can think of this as a high-powered version of Cantor normal form, which we discussed last time.

Veblen normal form relies on this result:

Theorem. Any ordinal $\beta$ can be written uniquely as

$\beta = \phi_{\gamma_1}(\alpha_1) + \dots + \phi_{\gamma_{k}}(\alpha_k)$

where $k$ is a natural number, each term is less than or equal to the previous one, and $\alpha_i < \phi_{\gamma_i}(\alpha_i)$ for all $i.$

Note that we can also use this theorem to write out the ordinals $\beta_i$ and $\gamma_i$, and so on, recursively. So, it gives us a notation for ordinals.

However, this notation is only useful when all the ordinals $\alpha_i, \gamma_i$ are less than the ordinal $\beta$ that we’re trying to describe. Otherwise we need to already have a notation for $\beta$ to express $\alpha$ 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 $0$, addition, and the function $\phi,$ using just finitely many symbols!

The moral

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 $\Gamma_0.$ 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.

Large Countable Ordinals (Part 1)

29 June, 2016

I love the infinite.

It may not exist in the physical world, but we can set up rules to think about it in consistent ways, and then it’s a helpful concept. The reason is that infinity is often easier to think about than very large finite numbers.

Finding rules to work with the infinite is one of the great triumphs of mathematics. Cantor’s realization that there are different sizes of infinity is truly wondrous—and by now, it’s part of the everyday bread and butter of mathematics.

Trying to create a notation for these different infinities is very challenging. It’s not a fair challenge, because there are more infinities than expressions we can write down in any given alphabet! But if we seek a notation for countable ordinals, the challenge becomes more fair.

It’s still incredibly frustrating. No matter what notation we use it fizzles out too soon… making us wish we’d invented a more general notation. But this process of ‘fizzling out’ is fascinating to me. There’s something profound about it. So, I would like to tell you about this.

Today I’ll start with a warmup. Cantor invented a notation for ordinals that works great for ordinals less than a certain ordinal called ε0. Next time I’ll go further, and bring in the ‘single-variable Veblen hierarchy’! This lets us describe all ordinals below a big guy called the ‘Feferman–Schütte ordinal’.

In the post after that I’ll bring in the ‘multi-variable Veblen hierarchy’, which gets us all the ordinals below the ‘small Veblen ordinal’. We’ll even touch on the ‘large Veblen ordinal’, which requires a version of the Veblen hierarchy with infinitely many variables. But all this is really just the beginning of a longer story. That’s how infinity works: the story never ends!

To describe countable ordinals beyond the large Veblen ordinal, most people switch to an entirely different set of ideas, called ‘ordinal collapsing functions’. I may tell you about those someday. Not soon, but someday. My interest in the infinite doesn’t seem to be waning. It’s a decadent hobby, but hey: some middle-aged men buy fancy red sports cars and drive them really fast. Studying notions of infinity is cooler, and it’s environmentally friendly.

I can even imagine writing a book about the infinite. Maybe these posts will become part of that book. But one step at a time…

Cardinals versus ordinals

Cantor invented two different kinds of infinities: cardinals and ordinals. Cardinals say how big sets are. Two sets can be put into 1-1 correspondence iff they have the same number of elements—where this kind of ‘number’ is a cardinal. You may have heard about cardinals like aleph-nought (the number of integers), 2 to power aleph-nought (the number of real numbers), and so on. You may have even heard rumors of much bigger cardinals, like ‘inaccessible cardinals’ or ‘super-huge cardinals’. All this is tremendously fun, and I recommend starting here:

• Frank R. Drake, Set Theory, an Introduction to Large Cardinals, North-Holland, 1974.

There are other books that go much further, but as a beginner, I found this to be the most fun.

But I don’t want to talk about cardinals! I want to talk about ordinals.

Ordinals say how big ‘well-ordered’ sets are. A set is well-ordered if it comes with a relation ≤ obeying the usual rules:

Transitivity: if x ≤ y and y ≤ z then x ≤ z

Reflexivity: x ≤ x

Antisymmetry: if x ≤ y and y ≤ x then x = y

and one more rule: every nonempty subset has a smallest element!

For example, the empty set

$\{\}$

is well-ordered in a trivial sort of way, and the corresponding ordinal is called

$0$

Similarly, any set with just one element, like this:

$\{0\}$

is well-ordered in a trivial sort of way, and the corresponding ordinal is called

$1$

Similarly, any set with two elements, like this:

$\{0,1\}$

becomes well-ordered as soon as we decree which element is bigger; the obvious choice is to say 0 < 1. The corresponding ordinal is called

$2$

Similarly, any set with three elements, like this:

$\{0,1,2\}$

becomes well-ordered as soon as we linearly order it; the obvious choice here is to say 0 < 1 < 2. The corresponding ordinal is called

$3$

Perhaps you’re getting the pattern — you’ve probably seen these particular ordinals before, maybe sometime in grade school. They’re called finite ordinals, or "natural numbers".

But there’s a cute trick they probably didn’t teach you then: we can define each ordinal to be the set of all ordinals less than it:

$0 = \{\}$ (since no ordinal is less than 0)
$1 = \{0\}$ (since only 0 is less than 1)
$2 = \{0,1\}$ (since 0 and 1 are less than 2)
$3 = \{0,1,2\}$ (since 0, 1 and 2 are less than 3)

and so on. It’s nice because now each ordinal is a well-ordered set of the size that ordinal stands for. And, we can define one ordinal to be "less than or equal" to another precisely when its a subset of the other.

Infinite ordinals

What comes after all the finite ordinals? Well, the set of all finite ordinals is itself well-ordered:

$\{0,1,2,3,\dots \}$

So, there’s an ordinal corresponding to this — and it’s the first infinite ordinal. It’s usually called $\omega,$ pronounced ‘omega’. Using the cute trick I mentioned, we can actually define

$\omega = \{0,1,2,3,\dots\}$

What comes after this? Well, it turns out there’s a well-ordered set

$\{0,1,2,3,\dots,\omega\}$

containing the finite ordinals together with $\omega,$ with the obvious notion of "less than": $\omega$ is bigger than the rest. Corresponding to this set there’s an ordinal called

$\omega+1$

As usual, we can simply define

$\omega+1 = \{0,1,2,3,\dots,\omega\}$

At this point you could be confused if you know about cardinals, so let me throw in a word of reassurance. The sets $\omega$ and $\omega+1$ have the same cardinality: they are both countable. In other words, you can find a 1-1 and onto function between these sets. But $\omega$ and $\omega+1$ are different as ordinals, since you can’t find a 1-1 and onto function between them that preserves the ordering. This is easy to see, since $\omega+1$ has a biggest element while $\omega$ does not.

Indeed, all the ordinals in this series of posts will be countable! So for the infinite ones, you can imagine that all I’m doing is taking your favorite countable set and well-ordering it in ever more sneaky ways.

Okay, so we got to $\omega + 1.$ What comes next? Well, not surprisingly, it’s

$\omega+2 = \{0,1,2,3,\dots,\omega,\omega+1\}$

Then comes

$\omega+3, \omega+4, \omega+5,\dots$

and so on. You get the idea.

I haven’t really defined ordinal addition in general. I’m trying to keep things fun, not like a textbook. But you can read about it here:

The main surprise is that ordinal addition is not commutative. We’ve seen that $\omega + 1 \ne \omega,$ since

$\omega + 1 = \{1, 2, 3, \dots, \omega \}$

is an infinite list of things… and then one more thing that comes after all those!. But $1 + \omega = \omega,$ because one thing followed by a list of infinitely many more is just a list of infinitely many things.

With ordinals, it’s not just about quantity: the order matters!

ω+ω and beyond

Okay, so we’ve seen these ordinals:

$1, 2, 3, \dots, \omega, \omega + 1, \omega + 2, \omega+3, \dots$

What next?

Well, the ordinal after all these is called $\omega+\omega.$ People often call it "omega times 2" or $\omega 2$ for short. So,

$\omega 2 = \{0,1,2,3,\dots,\omega,\omega+1,\omega+2,\omega+3,\dots.\}$

It would be fun to have a book with $\omega$ pages, each page half as thick as the previous page. You can tell a nice long story with an $\omega$-sized book. I think you can imagine this. And if you put one such book next to another, that’s a nice picture of $\omega 2.$

It’s worth noting that $\omega 2$ is not the same as $2 \omega.$ We have

$\omega 2 = \omega + \omega$

while

$2 \omega = 2 + 2 + 2 + \cdots$

where we add $\omega$ of these terms. But

$2 + 2 + 2 + \cdots = (1 + 1) + (1 + 1) + (1 + 1) \dots = \omega$

so

$2 \omega = \omega$

This is not a proof, because I haven’t given you the official definition of how to multiply ordinals. You can find it here:

• Wikipedia, Ordinal arithmetic: multiplication.

Using this you can prove that what I’m saying is true. Nonetheless, I hope you see why what I’m saying might make sense. Like ordinal addition, ordinal multiplication is not commutative! If you don’t like this, you should study cardinals instead.

What next? Well, then comes

$\omega 2 + 1, \omega 2 + 2,\dots$

and so on. But you probably have the hang of this already, so we can skip right ahead to $\omega 3.$

In fact, you’re probably ready to skip right ahead to $\omega 4,$ and $\omega 5,$ and so on.

In fact, I bet now you’re ready to skip all the way to "omega times omega", or $\omega^2$ for short:

$\omega^2 = \{0,1,2\dots\omega,\omega+1,\omega+2,\dots ,\omega 2,\omega 2+1,\omega 2+2,\dots\}$

Suppose you had an encyclopedia with $\omega$ volumes, each one being a book with $\omega$ pages. If each book is twice as thin as one before, you’ll have $\omega^2$ pages — and it can still fit in one bookshelf! Here’s the idea:

What comes next? Well, we have

$\omega^2+1, \omega^2+2, \dots$

and so on, and after all these come

$\omega^2+\omega, \omega^2+\omega+1, \omega^2+\omega+2, \dots$

and so on — and eventually

$\omega^2 + \omega^2 = \omega^2 2$

and then a bunch more, and then

$\omega^2 3$

and then a bunch more, and then

$\omega^2 4$

and then a bunch more, and more, and eventually

$\omega^2 \omega = \omega^3$

You can probably imagine a bookcase containing $\omega$ encyclopedias, each with $\omega$ volumes, each with $\omega$ pages, for a total of $\omega^3$ pages. That’s $\omega^3.$

ωω

I’ve been skipping more and more steps to keep you from getting bored. I know you have plenty to do and can’t spend an infinite amount of time reading this, even if the subject is infinity.

So if you don’t mind me just mentioning some of the high points, there are guys like $\omega^4$ and $\omega^5$ and so on, and after all these comes

$\omega^\omega$

Let’s try to we imagine this! First, imagine a book with $\omega$ pages. Then imagine an encyclopedia of books like this, with $\omega$ volumes. Then imagine a bookcase containing $\omega$ encyclopedias like this. Then imagine a room containing $\omega$ bookcases like this. Then imagine a floor with library with $\omega$ rooms like this. Then imagine a library with $\omega$ floors like this. Then imagine a city with $\omega$ libraries like this. And so on, ad infinitum.

You have to be a bit careful here, or you’ll be imagining an uncountable number of pages. To name a particular page in this universe, you have to say something like this:

the 23rd page of the 107th book of the 20th encyclopedia in the 7th bookcase in 0th room on the 1000th floor of the 973rd library in the 6th city on the 0th continent on the 0th planet in the 0th solar system in the…

But it’s crucial that after some finite point you keep saying “the 0th”. Without that restriction, there would be uncountably many pages! This is just one of the rules for how ordinal exponentiation works. For the details, read:

• Wikipedia, Ordinal arithmetic: exponentiation.

As they say,

But for infinite exponents, the definition may not be obvious.

Here’s a picture of $\omega^\omega,$ taken from David Madore’s wonderful interactive webpage:

On his page, if you click on any of the labels for an initial portion of an ordinal, like $\omega, \omega^2, \omega^3$ or $\omega^4$ here, the picture will expand to show that portion!

And here’s another picture, where each turn of the clock’s hand takes you to a higher power of $\omega$:

Ordinals up to ε0

Okay, so we’ve reached $\omega^\omega.$ Now what?

Well, then comes $\omega^\omega + 1,$ and so on, but I’m sure that’s boring by now. And then come ordinals like

$\omega^\omega 2,\dots, \omega^\omega 3, \dots, \omega^\omega 4, \dots$

$\omega^\omega \omega = \omega^{\omega + 1}$

Then eventually come ordinals like

$\omega^\omega \omega^2 , \dots, \omega^\omega \omega^3, \dots, \omega^\omega \omega^4, \dots$

and so on, leading up to

$\omega^\omega \omega^\omega = \omega^{\omega + \omega} = \omega^{\omega 2}$

This actually reminds me of something that happened driving across South Dakota one summer with a friend of mine. We were in college, so we had the summer off, so we drive across the country. We drove across South Dakota all the way from the eastern border to the west on Interstate 90.

This state is huge — about 600 kilometers across, and most of it is really flat, so the drive was really boring. We kept seeing signs for a bunch of tourist attractions on the western edge of the state, like the Badlands and Mt. Rushmore — a mountain that they carved to look like faces of presidents, just to give people some reason to keep driving.

Anyway, I’ll tell you the rest of the story later — I see some more ordinals coming up:

$\omega^{\omega 3},\dots \omega^{\omega 4},\dots \omega^{\omega 5},\dots$

We’re really whizzing along now just to keep from getting bored — just like my friend and I did in South Dakota. You might fondly imagine that we had fun trading stories and jokes, like they do in road movies. But we were driving all the way from Princeton to my friend Chip’s cabin in California. By the time we got to South Dakota, we were all out of stories and jokes.

Hey, look! It’s

$\omega^{\omega \omega}= \omega^{\omega^2}$

That was cool. Then comes

$\omega^{\omega^3}, \dots \omega^{\omega^4}, \dots \omega^{\omega^5}, \dots$

and so on.

Anyway, back to my story. For the first half of our half of our trip across the state, we kept seeing signs for something called the South Dakota Tractor Museum.

Oh, wait, here’s an interesting ordinal:

$\omega^{\omega^\omega}$

Let’s stop and take look:

That was cool. Okay, let’s keep driving. Here comes

$\omega^{\omega^\omega} + 1, \omega^{\omega^\omega} + 2, \dots$

and then

$\omega^{\omega^\omega} + \omega, \dots, \omega^{\omega^\omega} + \omega 2, \dots, \omega^{\omega^\omega} + \omega 3, \dots$

and then

$\omega^{\omega^\omega} + \omega^2, \dots, \omega^{\omega^\omega} + \omega^3, \dots$

and eventually

$\omega^{\omega^\omega} + \omega^\omega$

and eventually

$\omega^{\omega^\omega} + \omega^{\omega^\omega} = \omega^{\omega^\omega} 2$

and then

$\omega^{\omega^\omega} 3, \dots, \omega^{\omega^\omega} 4, \dots, \omega^{\omega^\omega} 5, \dots$

and eventually

$\omega^{\omega^\omega} \omega = \omega^{\omega^\omega + 1}$

After a while we reach

$\omega^{\omega^\omega + 2}, \dots, {\omega^\omega + 3}, \dots {\omega^\omega + 4}, \dots$

and then

$\omega^{\omega^\omega + \omega}, \dots, \omega^{\omega^\omega + \omega 2}, \dots, \omega^{\omega^\omega + \omega 3}, \dots$

and then

$\omega^{\omega^\omega + \omega^2}, \dots, \omega^{\omega^\omega + \omega^3}, \dots, \omega^{\omega^\omega + \omega^4}, \dots$

and then

$\omega^{\omega^\omega + \omega^\omega} = \omega^{\omega^\omega 2}$

and then

$\omega^{\omega^\omega 3}, \dots, \omega^{\omega^\omega 4} , \dots$

and then

$\omega^{\omega^\omega \omega} = \omega^{\omega^{\omega + 1}}$

and eventually

$\omega^{\omega^{\omega + 2}}, \dots, \omega^{\omega^{\omega + 3}}, \dots, \omega^{\omega^{\omega + 4}}, \dots$

This is pretty boring; we’re already going infinitely fast, but we’re still just picking up speed, and it’ll take a while before we reach something interesting.

Anyway, we started getting really curious about this South Dakota Tractor Museum — it sounded sort of funny. It took 250 kilometers of driving before we passed it. We wouldn’t normally care about a tractor museum, but there was really nothing else to think about while we were driving. The only thing to see were fields of grain, and these signs, which kept building up the suspense, saying things like

ONLY 100 MILES TO THE SOUTH DAKOTA TRACTOR MUSEUM!

We’re zipping along really fast now:

$\omega^{\omega^{\omega^\omega}}, \dots, \omega^{\omega^{\omega^{\omega^\omega}}},\dots , \omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}},\dots$

What comes after all these?

At this point we need to stop for gas. Our notation for ordinals just ran out!

The ordinals don’t stop; it’s just our notation that fizzled out. The set of all ordinals listed up to now — including all the ones we zipped past — is a well-ordered set called

$\epsilon_0$

or "epsilon-nought". This has the amazing property that

$\epsilon_0 = \omega^{\epsilon_0}$

And it’s the smallest ordinal with this property! It looks like this:

It’s an amazing fact that every countable ordinal is isomorphic, as an well-ordered set, to some subset of the real line. David Madore took advantage of this to make his pictures.

Cantor normal form

I’ll tell you the rest of my road story later. For now let me conclude with a bit of math.

There’s a nice notation for all ordinals less than $\epsilon_0,$ called ‘Cantor normal form’. We’ve been seeing lots of examples. Here is a typical ordinal in Cantor normal form:

$\omega^{\omega^{\omega^{\omega+\omega+1}}} \; + \; \omega^{\omega^\omega+\omega^\omega} \; + \; \omega^\omega \;+\; \omega + \omega + 1 + 1 + 1$

The idea is that you write it out using just + and exponentials and 1 and $\omega.$

Here is the theorem that justifies Cantor normal form:

Theorem. Every ordinal $\alpha$ can be uniquely written as

$\alpha = \omega^{\beta_1} c_1 + \omega^{\beta_2}c_2 + \cdots + \omega^{\beta_k}c_k$

where $k$ is a natural number, $c_1, c_2, \ldots, c_k$ are positive integers, and $\beta_1 > \beta_2 > \cdots > \beta_k \geq 0$ are ordinals.

It’s like writing ordinals in base $\omega.$

Note that every ordinal can be written this way! So why did I say that Cantor normal form is nice notation for ordinals less than $\epsilon_0$? Here’s the problem: the Cantor normal form of $\epsilon_0$ is

$\epsilon_0 = \omega^{\epsilon_0}$

So, when we hit $\epsilon_0,$ the exponents $\beta_1 ,\beta_2, \dots, \beta_k$ can be as big as the ordinal $\alpha$ we’re trying to describe! So, while the Cantor normal form still exists for ordinals $\geq \epsilon_0,$ it doesn’t give a good notation for them unless we already have some notation for ordinals this big!

This is what I mean by a notation ‘fizzling out’. We’ll keep seeing this problem in the posts to come.

But for an ordinal $\alpha$ less than $\epsilon_0,$ something nice happens. In this case, when we write

$\alpha = \omega^{\beta_1} c_1 + \omega^{\beta_2}c_2 + \cdots + \omega^{\beta_k}c_k$

all the exponents $\beta_1, \beta_2, \dots, \beta_k$ are less than $\alpha.$ So we can go ahead and write them in Cantor normal form, and so on… and because ordinals are well-ordered, this process ends after finitely many steps.

So, Cantor normal form gives a nice way to write any ordinal less than $\epsilon_0$ using finitely many symbols! If we abbreviate $\omega^0$ as $1,$ and write multiplication by positive integers in terms of addition, we get expressions like this:

$\omega^{\omega^{\omega^{\omega^{1 + 1} +\omega+1}}} \; + \; \omega^{\omega^\omega+\omega^\omega} \; + \; \omega^{\omega+1+1} \;+\; \omega + 1$

They look like trees. Even better, you can write a computer program that does ordinal arithmetic for ordinals of this form: you can add, multiply, and exponentiate them, and tell when one is less than another.

So, there’s really no reason to be scared of $\epsilon_0.$ Remember, each ordinal is just the set of all smaller ordinals. So you can think of $\epsilon_0$ as the set of tree-shaped expressions like the one above, with a particular rule for saying when one is less than another. It’s a perfectly reasonable entity. For some real excitement, we’ll need to move on to larger ordinals. We’ll do that next time.

For more, see:

• Wikipedia, Cantor normal form.

Exponential Zero

25 June, 2016

guest post by David A. Tanzer

Here is a mathematical riddle.  Consider the function below, which is undefined for negative values, sends zero to one, and sends positive values to zero.   Can you come up with a nice compact formula for this function, which uses only the basic arithmetic operations, such as addition, division and powers?  You can’t use any special functions, including things like sign and step functions, which are by definition discontinuous.

In college, I ran around showing people the graph, asking them to guess the formula.  I even tried it out on some professors there, U. Penn.  My algebra prof, who was kind of intimidating, looked at it, got puzzled, and then got irritated. When I showed him the answer, he barked out: Is this exam over??!  Then I tried it out during office hours on E. Calabi, who was teaching undergraduate differential geometry.  With a twinkle in his eye, he said, why that’s zero to the x!

The graph of 0x is not without controversy.   It is reasonable that for positive x, we have that 0x is zero.  Then 0-x = 1/0x = 1/0, so the function is undefined for negative values.  But what about 00?  This question is bound to come up in the course of one’s general mathematical education, and has been the source of long, ruminative arguments.

There are three contenders for 00:  undefined, 0, and 1.  Let’s try to define it in a way that is most consistent with the general laws of exponents — in particular, that for all a, x and y, ax+y = ax ay, and a-x = 1/ax. Let’s stick to these rules, even when a, x and y are all zero.

Then 00 equals its own square, because 00 = 00 + 0 = 00 00. And it equals its reciprocal, because 00 = 0-0 = 1/00. By these criteria, 00 equals 1.

That is the justification for the above graph — and for the striking discontinuity that it contains.

Here is an intuition for the discontinuity. Consider the family of exponential curves bx, with b as the parameter.  When b = 1, you get the constant function 1.  When b is more than 1, you get an increasing exponential, and when it is between 0 and 1, you get a decreasing exponential.  The intersection of all of these graphs is the “pivot” point x = 0, y = 1.  That is the “dot” of discontinuity.

What happens to bx, as b decreases to zero?  To the right of the origin, the curve progressively flattens down to zero.  To the left it rises up towards infinity more and more steeply.  But it always crosses through the point x = 0, y = 1, which remains in the limiting curve.  In heuristic terms, the value y = 1 is the discontinuous transit from infinitesimal values to infinite values.

There are reasons, however, why 00 could be treated as indeterminate, and left undefined.  These were indicated by the good professor.

Dr. Calabi had a truly inspiring teaching style, back in the day. He spoke of Italian paintings, and showed a kind of geometric laser vision.  In the classroom, he showed us the idea of torsion using his arms to fly around the room like an airplane.  There’s even a manifold named after him, the Calabi-Yau manifold.

He went on to talk about the underpinnings of this quirky function.  First he drew attention to the function f(x,y) = xy, over the complex domain, and attempted to sketch its level sets.  He focused on the behavior of the function when x and y are close to zero.   Then he stated that every one of the level sets $L(z) = \{(x,y)|x^y = z\}$ comes arbitrarily close to (0,0).

This means that xy has a wild singularity at the origin: every complex number z is the limit of xy along some path to zero.  Indeed, to reach z, just take a path in L(z) that approaches (0,0).

To see why the level sets all approach the origin, take logs, to get ln(xy) = y ln(x) = ln(z).  That gives y = ln(z) / ln(x), which is a parametric formula for L(z).  As x goes to zero, ln(x) goes to negative infinity, so y goes to zero.  These are paths (x, ln(z)/ln(x)), completely within L(z), which approach the origin.

In making these statements, we need to keep in mind that xy is multi-valued.  That’s because xy = e y ln(x), and ln(x) is multi-valued. That is because ln(x) is the inverse of the complex exponential, which is many-to-one: adding any integer multiple of $2 \pi i$ to z leaves ez unchanged.  And that follows from the definition of the exponential, which sends a + bi to the complex number with magnitude a and phase b.

Footnote:  to visualize these operations, represent the complex numbers by the real plane.  Addition is given by vector addition.  Multiplication gives the vector with magnitude equal to the product of the magnitudes, and phase equal to the sum of the phases.   The positive real numbers have phase zero, and the positive imaginary numbers are at 90 degrees vertical, with phase $\pi / 2$.

For a specific (x,y), how many values does xy have?  Well, ln(x) has a countable number of values, all differing by integer multiples of $2 \pi i$.  This generally induces a countable number of values for xy.  But if y is rational, they  collapse down to a finite set.  When y = 1/n, for example, the values of y ln(x) are spaced apart by $2 \pi i / n$, and when these get pumped back through the exponential function, we find only n distinct values for x 1/n — they are the nth roots of x.

So, to speak of the limit of xy along a path, and of the partition of $\mathbb{C}^2$ into level sets, we need to work within a branch of xy.   Each branch induces a different partition of $\mathbb{C}^2$.  But for every one of these partitions, it holds true that all of the level sets approach the origin.  That follows from the formula for the level set L(z), which is y = ln(z) / ln(x).  As x goes to zero, every branch of ln(x) goes to negative infinity.  (Exercise:  why?)  So y also goes to zero.  The branch affects the shape of the paths to the origin, but not their existence.

Here is a qualitative description of how the level sets fit together:  they are like spokes around the origin, where each spoke is a curve in one complex dimension.  These curves are 1-D complex manifolds, which are equivalent to  two-dimensional surfaces in $\mathbb{R}^4$.  The partition comprises a two-parameter family of these surfaces, indexed by the complex value of xy.

What can be said about the geometry and topology of this “wheel of manifolds”?  We know they don’t intersect.  But are they “nicely” layered, or twisted and entangled?  As we zoom in on the origin, does the picture look smooth, or does it have a chaotic appearance, with infinite fine detail?  Suggestive of chaos is the fact that the gradient

$\nabla x^y = (y x^{y-1}, \ln(y) x^y) = (y/x, \ln(y)) x^y$

is also “wildly singular” at the origin.

These questions can be explored with plotting software.  Here, the artist would have the challenge of having only two dimensions to work with, when the “wheel” is really a structure in four-dimensional space.  So some interesting cross-sections would have to be chosen.

Exercises:

• Speak about the function bx, where b is negative, and x is real.
• What is $0^\pi$, and why?
• What is $0^i$?

Moral: something that seems odd, or like a joke that might annoy your algebra prof, could be more significant than you think.  So tell these riddles to your professors, while they are still around.

Azimuth News (Part 5)

11 June, 2016

I’ve been rather quiet about Azimuth projects lately, because I’ve been too busy actually working on them. Here’s some of what’s happening:

Jason Erbele is finishing his thesis, entitled Categories in Control: Applied PROPs. He successfully gave his thesis defense on Wednesday June 8th, but he needs to polish it up some more. Building on the material in our paper “Categories in control”, he’s defined a category where the morphisms are signal flow diagrams. But interestingly, not all the diagrams you can draw are actually considered useful in control theory! So he’s also found a subcategory where the morphisms are the ‘good’ signal flow diagrams, the ones control theorists like. For these he studies familiar concepts like controllability and observability. When his thesis is done I’ll announce it here.

Brendan Fong is also finishing his thesis, called The Algebra of Open and Interconnected Systems. Brendan has already created a powerful formalism for studying open systems: the decorated cospan formalism. We’ve applied it to two examples: electrical circuits and Markov processes. Lately he’s been developing the formalism further, and this will appear in his thesis. Again, I’ll talk about it when he’s done!

Blake Pollard and I are writing a paper called “A compositional framework for open chemical reaction networks”. Here we take our work on Markov processes and throw in two new ingredients: dynamics and nonlinearity. Of course Markov processes have a dynamics, but in our previous paper when we ‘black-boxed’ them to study their external behaviour, we got a relation between flows and populations in equilibrium. Now we explain how to handle nonequilibrium situations as well.

Brandon Coya, Franciscus Rebro and I are writing a paper that might be called “The algebra of networks”. I’m not completely sure of the title, nor who the authors will be: Brendan Fong may also be a coauthor. But the paper explores the technology of PROPs as a tool for describing networks. As an application, we’ll give a new shorter proof of the functoriality of black-boxing for electrical circuits. This new proof also applies to nonlinear circuits. I’m really excited about how the theory of PROPs, first introduced in algebraic topology, is catching fire with all the new applications to network theory.

I expect all these projects to be done by the end of the summer. Near the end of June I’ll go to the Centre for Quantum Technologies, in Singapore. This will be my last summer there. My main job will be to finish up the two papers that I’m supposed to be writing.

There’s another paper that’s already done:

Kenny Courser has written a paper “A bicategory of decorated cospans“, pushing Brendan’s framework from categories to bicategories. I’ll explain this very soon here on this blog! One goal is to understand things like the coarse-graining of open systems: that is, the process of replacing a detailed description by a less detailed description. Since we treat open systems as morphisms, coarse-graining is something that goes from one morphism to another, so it’s naturally treated as a 2-morphism in a bicategory.

So, I’ve got a lot of new ideas to explain here, and I’ll start soon! I also want to get deeper into systems biology.

In the fall I’ve got a couple of short trips lined up:

• Monday November 14 – Friday November 18, 2016 – I’ve been invited by Yoav Kallus to visit the Santa Fe Institute. From the 16th to 18th I’ll attend a workshop on Statistical Physics, Information Processing and Biology.

• Monday December 5 – Friday December 9 – I’ve been invited to Berkeley for a workshop on Compositionality at the Simons Institute for the Theory of Computing, organized by Samson Abramsky, Lucien Hardy, and Michael Mislove. ‘Compositionality’ is a name for how you describe the behavior of a big complicated system in terms of the behaviors of its parts, so this is closely connected to my dream of studying open systems by treating them as morphisms that can be composed to form bigger open systems.

Here’s the announcement:

The compositional description of complex objects is a fundamental feature of the logical structure of computation. The use of logical languages in database theory and in algorithmic and finite model theory provides a basic level of compositionality, but establishing systematic relationships between compositional descriptions and complexity remains elusive. Compositional models of probabilistic systems and languages have been developed, but inferring probabilistic properties of systems in a compositional fashion is an important challenge. In quantum computation, the phenomenon of entanglement poses a challenge at a fundamental level to the scope of compositional descriptions. At the same time, compositionally has been proposed as a fundamental principle for the development of physical theories. This workshop will focus on the common structures and methods centered on compositionality that run through all these areas.

I’ll say more about both these workshops when they take place.

Programming with Data Flow Graphs

5 June, 2016

Network theory is catching on—in a very practical way!

Google recently started a new open source library called TensorFlow. It’s for software built using data flow graphs. These are graphs where the edges represent tensors—that is, multidimensional arrays of numbers—and the nodes represent operations on tensors. Thus, they are reminiscent of the spin networks used in quantum gravity and gauge theory, or the tensor networks used in renormalization theory. However, I bet the operations involved are nonlinear! If so, they’re more general.

TensorFlow™ is an open source software library for numerical computation using data flow graphs. Nodes in the graph represent mathematical operations, while the graph edges represent the multidimensional data arrays (tensors) communicated between them. The flexible architecture allows you to deploy computation to one or more CPUs or GPUs in a desktop, server, or mobile device with a single API. TensorFlow was originally developed by researchers and engineers working on the Google Brain Team within Google’s Machine Intelligence research organization for the purposes of conducting machine learning and deep neural networks research, but the system is general enough to be applicable in a wide variety of other domains as well.

What is a Data Flow Graph?

Data flow graphs describe mathematical computation with a directed graph of nodes & edges. Nodes typically implement mathematical operations, but can also represent endpoints to feed in data, push out results, or read/write persistent variables. Edges describe the input/output relationships between nodes. These data edges carry dynamically-sized multidimensional data arrays, or tensors. The flow of tensors through the graph is where TensorFlow gets its name. Nodes are assigned to computational devices and execute asynchronously and in parallel once all the tensors on their incoming edges becomes available.

TensorFlow Features

Deep Flexibility. TensorFlow isn’t a rigid neural networks library. If you can express your computation as a data flow graph, you can use TensorFlow. You construct the graph, and you write the inner loop that drives computation. We provide helpful tools to assemble subgraphs common in neural networks, but users can write their own higher-level libraries on top of TensorFlow. Defining handy new compositions of operators is as easy as writing a Python function and costs you nothing in performance. And if you don’t see the low-level data operator you need, write a bit of C++ to add a new one.

True Portability. TensorFlow runs on CPUs or GPUs, and on desktop, server, or mobile computing platforms. Want to play around with a machine learning idea on your laptop without need of any special hardware? TensorFlow has you covered. Ready to scale-up and train that model faster on GPUs with no code changes? TensorFlow has you covered. Want to deploy that trained model on mobile as part of your product? TensorFlow has you covered. Changed your mind and want to run the model as a service in the cloud? Containerize with Docker and TensorFlow just works.

Connect Research and Production. Gone are the days when moving a machine learning idea from research to product require a major rewrite. At Google, research scientists experiment with new algorithms in TensorFlow, and product teams use TensorFlow to train and serve models live to real customers. Using TensorFlow allows industrial researchers to push ideas to products faster, and allows academic researchers to share code more directly and with greater scientific reproducibility.

Auto-Differentiation. Gradient based machine learning algorithms will benefit from TensorFlow’s automatic differentiation capabilities. As a TensorFlow user, you define the computational architecture of your predictive model, combine that with your objective function, and just add data — TensorFlow handles computing the derivatives for you. Computing the derivative of some values w.r.t. other values in the model just extends your graph, so you can always see exactly what’s going on.

Language Options. TensorFlow comes with an easy to use Python interface and a no-nonsense C++ interface to build and execute your computational graphs. Write stand-alone TensorFlow Python or C++ programs, or try things out in an interactive TensorFlow iPython notebook where you can keep notes, code, and visualizations logically grouped. This is just the start though — we’re hoping to entice you to contribute SWIG interfaces to your favorite language — be it Go, Java, Lua, JavaScript, or R.

Maximize Performance. Want to use every ounce of muscle in that workstation with 32 CPU cores and 4 GPU cards? With first-class support for threads, queues, and asynchronous computation, TensorFlow allows you to make the most of your available hardware. Freely assign compute elements of your TensorFlow graph to different devices, and let TensorFlow handle the copies.

Who Can Use TensorFlow?

TensorFlow is for everyone. It’s for students, researchers, hobbyists, hackers, engineers, developers, inventors and innovators and is being open sourced under the Apache 2.0 open source license.

TensorFlow is not complete; it is intended to be built upon and extended. We have made an initial release of the source code, and continue to work actively to make it better. We hope to build an active open source community that drives the future of this library, both by providing feedback and by actively contributing to the source code.

Why Did Google Open Source This?

If TensorFlow is so great, why open source it rather than keep it proprietary? The answer is simpler than you might think: We believe that machine learning is a key ingredient to the innovative products and technologies of the future. Research in this area is global and growing fast, but lacks standard tools. By sharing what we believe to be one of the best machine learning toolboxes in the world, we hope to create an open standard for exchanging research ideas and putting machine learning in products. Google engineers really do use TensorFlow in user-facing products and services, and our research group intends to share TensorFlow implementations along side many of our research publications.

For more details, try this:

Very Long Proofs

28 May, 2016

In the 1980s, the mathematician Ronald Graham asked if it’s possible to color each positive integer either red or blue, so that no triple of integers $a,b,c$ obeying Pythagoras’ famous equation:

$a^2 + b^2 = c^2$

all have the same color. He offered a prize of \$100.

Now it’s been solved! The answer is no. You can do it for numbers up to 7824, and a solution is shown in this picture. But you can’t do it for numbers up to 7825.

To prove this, you could try all the ways of coloring these numbers and show that nothing works. Unfortunately that would require trying

3 628 407 622 680 653 855 043 364 707 128 616 108 257 615 873 380 491 654 672 530 751 098 578 199 115 261 452 571 373 352 277 580 182 512 704 196 704 700 964 418 214 007 274 963 650 268 320 833 348 358 055 727 804 748 748 967 798 143 944 388 089 113 386 055 677 702 185 975 201 206 538 492 976 737 189 116 792 750 750 283 863 541 981 894 609 646 155 018 176 099 812 920 819 928 564 304 241 881 419 294 737 371 051 103 347 331 571 936 595 489 437 811 657 956 513 586 177 418 898 046 973 204 724 260 409 472 142 274 035 658 308 994 441 030 207 341 876 595 402 406 132 471 499 889 421 272 469 466 743 202 089 120 267 254 720 539 682 163 304 267 299 158 378 822 985 523 936 240 090 542 261 895 398 063 218 866 065 556 920 106 107 895 261 677 168 544 299 103 259 221 237 129 781 775 846 127 529 160 382 322 984 799 874 720 389 723 262 131 960 763 480 055 015 082 441 821 085 319 372 482 391 253 730 679 304 024 117 656 777 104 250 811 316 994 036 885 016 048 251 200 639 797 871 184 847 323 365 327 890 924 193 402 500 160 273 667 451 747 479 728 733 677 070 215 164 678 820 411 258 921 014 893 185 210 250 670 250 411 512 184 115 164 962 089 724 089 514 186 480 233 860 912 060 039 568 930 065 326 456 428 286 693 446 250 498 886 166 303 662 106 974 996 363 841 314 102 740 092 468 317 856 149 533 746 611 128 406 657 663 556 901 416 145 644 927 496 655 933 158 468 143 482 484 006 372 447 906 612 292 829 541 260 496 970 290 197 465 492 579 693 769 880 105 128 657 628 937 735 039 288 299 048 235 836 690 797 324 513 502 829 134 531 163 352 342 497 313 541 253 617 660 116 325 236 428 177 219 201 276 485 618 928 152 536 082 354 773 892 775 152 956 930 865 700 141 446 169 861 011 718 781 238 307 958 494 122 828 500 438 409 758 341 331 326 359 243 206 743 136 842 911 727 359 310 997 123 441 791 745 020 539 221 575 643 687 646 417 117 456 946 996 365 628 976 457 655 208 423 130 822 936 961 822 716 117 367 694 165 267 852 307 626 092 080 279 836 122 376 918 659 101 107 919 099 514 855 113 769 846 184 593 342 248 535 927 407 152 514 690 465 246 338 232 121 308 958 440 135 194 441 048 499 639 516 303 692 332 532 864 631 075 547 542 841 539 848 320 583 307 785 982 596 093 517 564 724 398 774 449 380 877 817 714 717 298 596 139 689 573 570 820 356 836 562 548 742 103 826 628 952 649 445 195 215 299 968 571 218 175 989 131 452 226 726 280 771 962 970 811 426 993 797 429 280 745 007 389 078 784 134 703 325 573 686 508 850 839 302 112 856 558 329 106 490 855 990 906 295 808 952 377 118 908 425 653 871 786 066 073 831 252 442 345 238 678 271 662 351 535 236 004 206 289 778 489 301 259 384 752 840 495 042 455 478 916 057 156 112 873 606 371 350 264 102 687 648 074 992 121 706 972 612 854 704 154 657 041 404 145 923 642 777 084 367 960 280 878 796 437 947 008 894 044 010 821 287 362 106 232 574 741 311 032 906 880 293 520 619 953 280 544 651 789 897 413 312 253 724 012 410 831 696 803 510 617 000 147 747 294 278 502 175 823 823 024 255 652 077 422 574 922 776 413 427 073 317 197 412 284 579 070 292 042 084 295 513 948 442 461 828 389 757 279 712 121 164 692 705 105 851 647 684 562 196 098 398 773 163 469 604 125 793 092 370 432

possibilities. But recently, three mathematicians cleverly figured out how to eliminate most of the options. That left fewer than a trillion to check!

So they spent 2 days on a supercomputer, running 800 processors in parallel, and checked all the options. None worked. They verified their solution on another computer.

This is one of the world’s biggest proofs: it’s 200 terabytes long! That’s about equal to all the digitized text held by the US Library of Congress. There’s also a 68-gigabyte digital signature—sort of a proof that a proof exists—if you want to skim it.

It’s interesting that these 200 terabytes were used to solve a yes-or-no question, whose answer takes a single bit to state: no.

I’m not sure breaking the world’s record for the longest proof is something to be proud of. Mathematicians prize short, insightful proofs. I bet a shorter proof of this result will eventually be found.

Still, it’s fun that we can do such things. Here’s a story about the proof:

• Evelyn Lamb, Two-hundred-terabyte maths proof is largest ever, Nature, May 26, 2016.

and here’s the actual paper:

• Marijn J. H. Heule, Oliver Kullmann and Victor W. Marek, Solving and verifying the Boolean Pythagorean triples problem via cube-and-conquer.

The ‘cube-and-conquer’ paradigm is a “hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers”. The actual benefit of this huge proof is developing such methods for solving big problems! In the comments to my G+ post, Roberto Bayardo explained:

CDCL == “conflict-directed clause learning”: when you hit a dead end in backtrack search, this is the process of recording a (hopefully small) clause that prevents you from making the same mistake again…a type of memorization, essentially.

Look-ahead: in backtrack search, you repeat the process of picking an unassigned variable and then picking an assignment for that variable until you reach a “dead end” (upon deriving a contradiction). Look-ahead involves doing some amount of processing on the remaining formula after each assignment in order to simplify it. This includes identifying variables for which one of its assignments can be quickly eliminated. “Unit propagation” is a type of look-ahead, though I suspect in this case they mean something quite a bit more sophisticated.﻿

Arnaud Spiwack has a series of blog posts introducting SAT solvers here:

A much longer proof

By the way, despite the title of the Nature article, in the comments to my G+ post about this, Michael Nielsen pointed out a much longer proof by Chris Jefferson, who wrote:

Darn, I had no idea one could get into the media with this kind of stuff.

I had a much larger “proof”, where we didn’t bother storing all the details, in which we enumerated 718,981,858,383,872 semigroups, towards counting the semigroups of size 10.

Uncompressed, it would have been about 63,000 terabytes just for the semigroups, and about a thousand times that to store the “proof”, which is just the tree of the search.

Of course, it would have compressed extremely well, but also I’m not sure it would have had any value, you could rebuild the search tree much faster than you could read it from a disc, and if anyone re-verified our calculation I would prefer they did it by a slightly different search, which would give us much better guarantees of correctness.

His team found a total of 12,418,001,077,381,302,684 semigroups of size 10. They only had to find 718,981,858,383,872 by a brute force search, which is 0.006% of the total:

• Andreas Distler, Chris Jefferson, Tom Kelsey, and Lars Kottho, The semigroups of order 10, in Principles and Practice of Constraint Programming, Springer Lecture Notes in Computer Science 7514, Springer, Berlin, pp. 883–899.

Insanely long proofs

All the proofs mentioned so far are downright laconic compared to those discussed here:

• John Baez, Insanely long proofs, 19 October 2012.

For example, if you read this post you’ll learn about a fairly short theorem whose shortest proof using Peano arithmetic contains at least

$\displaystyle{ 10^{10^{1000}} }$

symbols. This is so many that if you tried to write down the number of symbols in this proof—not the symbols themselves, but just the number of symbols—in ordinary decimal notation, you couldn’t do it if you wrote one digit on each proton, neutron and electron in the observable Universe!

The Busy Beaver Game

21 May, 2016

This month, a bunch of ‘logic hackers’ have been seeking to determine the precise boundary between the knowable and the unknowable. The challenge has been around for a long time. But only now have people taken it up with the kind of world-wide teamwork that the internet enables.

A Turing machine is a simple model of a computer. Imagine a machine that has some finite number of states, say N states. It’s attached to a tape, an infinitely long tape with lots of squares, with either a 0 or 1 written on each square. At each step the machine reads the number where it is. Then, based on its state and what it reads, it either halts, or it writes a number, changes to a new state, and moves either left or right.

The tape starts out with only 0’s on it. The machine starts in a particular ‘start’ state. It halts if it winds up in a special ‘halt’ state.

The Busy Beaver Game is to find the Turing machine with N states that runs as long as possible and then halts.

The number BB(N) is the number of steps that the winning machine takes before it halts.

In 1961, Tibor Radó introduced the Busy Beaver Game and proved that the sequence BB(N) is uncomputable. It grows faster than any computable function!

A few values of BB(N) can be computed, but there’s no way to figure out all of them.

As we increase N, the number of Turing machines we need to check increases faster than exponentially: it’s

$(4(n+1))^{2n}$

Of course, many could be ruled out as potential winners by simple arguments. But the real problem is this: it becomes ever more complicated to determine which Turing machines with N states never halt, and which merely take a huge time to halt.

Indeed, matter what axiom system you use for math, as long as it has finitely many axioms and is consistent, you can never use it to correctly determine BB(N) for more than some finite number of cases.

So what do people know about BB(N)?

For starters, BB(0) = 0. At this point I should admit that people don’t count the halt state as one of our N states. This is just a convention. So, when we consider BB(0), we’re considering machines that only have a halt state. They instantly halt.

Next, BB(1) = 1.

Next, BB(2) = 6.

Next, BB(3) = 21. This was proved in 1965 by Tibor Radó and Shen Lin.

Next, BB(4) = 107. This was proved in 1983 by Allan Brady.

Next, BB(5). Nobody knows what BB(5) equals!

The current 5-state busy beaver champion was discovered by Heiner Marxen and Jürgen Buntrock in 1989. It takes 47,176,870 steps before it halts. So, we know

BB(5) ≥ 47,176,870.

People have looked at all the other 5-state Turing machines to see if any does better. But there are 43 machines that do very complicated things that nobody understands. It’s believed they never halt, but nobody has been able to prove this yet.

We may have hit the wall of ignorance here… but we don’t know.

That’s the spooky thing: the precise boundary between the knowable and the unknowable is unknown. It may even be unknowable… but I’m not sure we know that.

Next, BB(6). In 1996, Marxen and Buntrock showed it’s at least 8,690,333,381,690,951. In June 2010, Pavel Kropitz proved that

$\displaystyle{ \mathrm{BB}(6) \ge 7.412 \cdot 10^{36,534} }$

You may wonder how he proved this. Simple! He found a 6-state machine that runs for

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steps and then halts!

Of course, I’m just kidding when I say this was simple. The machine is easy enough to describe, but proving it takes exactly this long to run takes real work! You can read about such proofs here:

• Pascal Michel, The Busy Beaver Competition: a historical survey.

I don’t understand them very well. All I can say at this point is that many of the record-holding machines known so far are similar to the famous Collatz conjecture. The idea there is that you can start with any positive integer and keep doing two things:

• if it’s even, divide it by 2;

• if it’s odd, triple it and add 1.

The conjecture is that this process will always eventually reach the number 1. Here’s a graph of how many steps it takes, as a function of the number you start with:

Nice pattern! But this image shows how it works for numbers up to 10 million, and you’ll see it doesn’t usually take very long for them to reach 1. Usually less than 600 steps is enough!

So, to get a Turing machine that takes a long time to halt, you have to take this kind of behavior and make it much more long and drawn-out. Conversely, to analyze one of the potential winners of the Busy Beaver Game, people must take that long and drawn-out behavior and figure out a way to predict much more quickly when it will halt.

Next, BB(7). In 2014, someone who goes by the name Wythagoras showed that

$\displaystyle{ \textrm{BB}(7) > 10^{10^{10^{10^{10^7}}}} }$

It’s fun to prove lower bounds on BB(N). For example, in 1964 Milton Green constructed a sequence of Turing machines that implies

$\textrm{BB}(2N) \ge 3 \uparrow^{N-2} 3$

Here I’m using Knuth’s up-arrow notation, which is a recursively defined generalization of exponentiation, so for example

$\textrm{BB}(10) \ge 3 \uparrow^{3} 3 = 3 \uparrow^2 3^{3^3} = 3^{3^{3^{3^{\cdot^{\cdot^\cdot}}}}}$

where there are $3^{3^3}$ threes in that tower.

But it’s also fun to seek the smallest N for which we can prove BB(N) is unknowable! And that’s what people are making lots of progress on right now.

Sometime in April 2016, Adam Yedidia and Scott Aaronson showed that BB(7910) cannot be determined using the widely accepted axioms for math called ZFC: that is, Zermelo—Fraenkel set theory together with the axiom of choice. It’s a great story, and you can read it here:

• Scott Aaronson, The 8000th Busy Beaver number eludes ZF set theory: new paper by Adam Yedidia and me, Shtetl-Optimized, 3 May 2016.

• Adam Yedidia and Scott Aaronson, A relatively small Turing machine whose behavior is independent of set theory, 13 May 2016.

Briefly, Yedidia created a new programming language, called Laconic, which lets you write programs that compile down to small Turing machines. They took an arithmetic statement created by Harvey Friedman that’s equivalent to the consistency of the usual axioms of ZFC together with a large cardinal axiom called the ‘stationary Ramsey property’, or SRP. And they created a Turing machine with 7910 states that seeks a proof of this arithmetic statement using the axioms of ZFC.

Since ZFC can’t prove its own consistency, much less its consistency when supplemented with SRP, their machine will only halt if ZFC+SRP is inconsistent.

Since most set theorists believe ZFC+SRP is consistent, this machine probably doesn’t halt. But we can’t prove this using ZFC.

In short: if the usual axioms of set theory are consistent, we can never use them to determine the value of BB(7910).

The basic idea is nothing new: what’s new is the explicit and rather low value of the number 7910. Poetically speaking, we know the unknowable starts here… if not sooner.

However, this discovery set off a wave of improvements! On the Metamath newsgroup, Mario Carneiro and others started ‘logic hacking’, looking for smaller and smaller Turing machines that would only halt if ZF—that is, Zermelo–Fraenkel set theory, without the axiom of choice—is inconsistent.

By just May 15th, Stefan O’Rear seems to have brought the number down to 1919. He found a Turing machine with just 1919 states that searches for an inconsistency in the ZF axioms. Interestingly, this turned out to work better than using Harvey Friedman’s clever trick.

Thus, if O’Rear’s work is correct, we can only determine BB(1919) if we can determine whether ZF set theory is consistent. However, we cannot do this using ZF set theory—unless we find an inconsistency in ZF set theory.

For details, see:

• Stefan O’Rear, A Turing machine Metamath verifier, 15 May 2016.

I haven’t checked his work, but it’s available on GitHub.

What’s the point of all this? At present, it’s mainly just a game. However, it should have some interesting implications. It should, for example, help us better locate the ‘complexity barrier’.

I explained that idea here:

• John Baez, The complexity barrier, Azimuth, 28 October 2011.

Briefly, while there’s no limit on how much information a string of bits—or any finite structure—can have, there’s a limit on how much information we can prove it has!

This amount of information is pretty low, perhaps a few kilobytes. And I believe the new work on logic hacking can be used to estimate it more accurately!