Also available at http://math.ucr.edu/home/baez/week236.html
July 26, 2006
This Week's Finds in Mathematical Physics (Week 236)
John Baez
This week I'd like to catch you up on some papers about
categorification and quantum mechanics.
But first, since it's summer vacation, I'd like to take you on
a little road trip - to infinity. And then, for fun, a little
detective story about the history of the icosahedron.
Cantor invented two kinds of infinities: cardinals and ordinals.
Cardinals are more familiar. They 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.
But today I want to talk about ordinals. Ordinals say how big
"well-ordered" sets are. A set is well-ordered if it's linearly
ordered and 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.
Now, what comes after all the finite ordinals? Well,
the set of all finite ordinals is itself well-ordered:
{0,1,2,3,...}
So, there's an ordinal corresponding to this - and it's the first
*infinite* ordinal. It's usually called omega. Using the cute
trick I mentioned, we can actually define
omega = {0,1,2,3,...}
Now, what comes after this? Well, it turns out there's a
well-ordered set
{0,1,2,3,...,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,...,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", but they're 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.)
Now, what comes next? Well, not surprisingly, it's
omega+2 = {0,1,2,3,...,omega,omega+1}
Then comes
omega+3, omega+4, omega+5,...
and so on. You get the idea.
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,...,omega,omega+1,omega+2,omega+3,....}
What next? Well, then comes
omega 2 + 1, omega 2 + 2,...
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 squared" for short:
omega^2 =
{0,1,2...omega,omega+1,omega+2,...,omega2,omega2+1,omega2+2,...}
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. But it would be even more fun to have
an encyclopedia with omega volumes, each being an omega-sized book,
each half as thick as the previous volume. Then you have omega^2
pages - and it can still fit in one bookshelf!
What comes next? Well, we have
omega^2+1, omega^2+2, ...
and so on, and after all these come
omega^2+omega, omega^2+omega+1, omega^2+omega+2, ...
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.
I'm 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 Week's Finds, 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.
And then 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,..., omega^omega 3, ..., omega^omega 4, ...
leading up to
omega^omega omega = omega^{omega + 1}
Then eventually come ordinals like
omega^omega omega^2 , ..., omega^omega omega^3, ...
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},... omega^{omega 4},... omega^{omega 5},...
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}, ... omega^{omega^4}, ... omega^{omega^5}, ...
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 - let's slow down and
take a look:
omega^{omega^omega}
I like that! Okay, let's keep driving:
omega^{omega^omega} + 1, omega^{omega^omega} + 2, ...
and then
omega^{omega^omega} + omega, ..., omega^{omega^omega} + omega 2, ...
and then
omega^{omega^omega} + omega^2, ..., omega^{omega^omega} + omega^3, ...
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, ..., omega^{omega ^ omega} 4, ...
and eventually
omega^{omega^omega} omega = omega^{omega^omega + 1}
and then
omega^{omega^omega + 2}, ..., omega^{omega^omega + 3}, ...
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}}, ... omega^{omega^{omega^{omega^omega}}},...
What comes after all these?
At this point we need to stop for gas. Our notation for ordinals
runs out at this point!
The ordinals don't stop; it's just our notation that gives 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.
In fact, all the ordinals smaller than epsilon_0 can be drawn as
trees. You write them in "Cantor normal form" like this:
omega^{omega^omega + omega} + omega^omega + omega + omega + 1 + 1 + 1
using just + and exponentials and 1 and omega, and then you turn
this notation into a picture of a tree. I'll leave it as a puzzle
to figure out how.
So, the set of (finite, rooted) trees becomes a well-ordered set
whose ordinal is epsilon_0. Trees are important in combinatorics
and computer science, so epsilon_0 is not really so weird after all.
Another cool thing is that Gentzen proved the consistency of the
usual axioms for arithmetic - "Peano arithmetic" - with the help
of epsilon_0. He did this by drawing proofs as trees, and using
this to give an inductive argument that there's no proof in Peano
arithmetic that 0 = 1. But, this inductive argument goes beyond
the simple kind you use to prove facts about all natural numbers.
It uses induction up to epsilon_0.
You can't formalize Gentzen's argument in Peano arithmetic: thanks
to Goedel, this system can't proof itself consistent unless it's *not*.
I used to think this made Gentzen's proof pointless, especially since
"induction up to epsilon_0" sounded like some sort of insane logician's
extrapolation of ordinary mathematical induction.
But now I see that induction up to epsilon_0 can be thought of as
induction on trees, and it seems like an obviously correct principle.
Of course Peano's axioms also seem obviously correct, so I don't know
that Gentzen's proof makes me *more sure* Peano arithmetic is
consistent. But, it's interesting.
Induction up to epsilon_0 also lets you prove other stuff you
can't prove with just Peano arithmetic. For example, it lets you
prove that every Goodstein sequence eventually reaches zero!
Huh?
To write down a Goodstein sequence, you start with any natural
number and write it in "recursive base 2", like this:
2^{2^2 + 1} + 2^1
Then you replace all the 2's by 3's:
3^{3^3 + 1} + 3^1
Then you subtract 1 and write the answer in "recursive base 3":
3^{3^3 + 1} + 1 + 1
Then you replace all the 3's by 4's, subtract 1 and write the
answer in recursive base 4. Then you replace all the 4's by
5's, subtract 1 and write the answer in recursive base 5. And so on.
You can try some examples using the applet on this site:
1) National Curve Bank, Goodstein's theorem,
http://curvebank.calstatela.edu/goodstein/goodstein.htm
You'll see that for any starting number bigger than 3, the
Goodstein sequence seems to keep growing forever. So, it's
shocking to learn that they all eventually go to zero!
For example, if you start with the number 4, you get this sequence:
4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...
Believe it or not, this goes to zero after about 7 x 10^{121210694}
steps! For a proof, see Kevin Buzzard's calculations in the
Addendum at the bottom of this article.
In fact, it takes nothing but algebra, careful reasoning and
persistence to work out when any given Goodstein sequence reaches
zero. But, proving they *all* eventually reach zero uses induction
up to epsilon_0.
How can that be?
The point is that these numbers in "recursive base n" look a lot
like ordinals in Cantor normal form. If we translate them into
ordinals by replacing n by omega, the ordinals keep getting smaller
at each step, even when the numbers get bigger!
For example, when we do the translation
2^{2^2 + 1} + 2 |-> omega^{omega^omega + 1} + omega^1
3^{3^3 + 1} + 1 + 1 |-> omega^{omega^omega + 1} + 1 + 1
we see the ordinal got smaller even though the number got bigger.
Since epsilon_0 is well-ordered, the ordinals must bottom out at zero
after a finite number of steps - that's what "induction up to epsilon_0"
tells us. So, the numbers must too!
In short, induction up to epsilon_0 lets us prove all Goodstein
sequences eventually reach zero. But Kirby and Paris showed
something much deeper: they showed that you *need* induction
up to epsilon_0 to get the job done.
And, they noted a big consequence of this fact. Suppose Peano
arithmetic is consistent. Then you can't do induction up to epsilon_0
in this system - since if you could, Gentzen's work would let this
system prove itself consistent, and Goedel's theorem would kick in
and say it's *not*. But since you *need* induction up to epsilon_0
to prove all Goodstein sequences go to zero, you can't prove this
in Peano arithmetic.
So, the fact that Goodstein sequences go to zero is obvious if
you think about it the right way, but it's not provable in Peano
arithmetic.
I don't know any results in mathematical physics that use induction
up to epsilon_0, but these could be one - after all, trees show up
in the theory of Feynman diagrams. That would be pretty interesting.
There's a lot more to say about this, but I hear what you're asking:
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,..., epsilon_0 + omega^3,... , epsilon_0 + omega^4,...
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 want 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}},...
epsilon_1 is the limit of this:
epsilon_0 + 1, omega^{epsilon_0 + 1}, omega^{omega^{epsilon_0 + 1}},...
In other words, it's the *union* of all these well-ordered sets.
In what sense is epsilon_1 the "first really interesting ordinal" after
epsilon_0? I'm not sure! Maybe it's the first one that can't be
built out of 1, omega and epsilon_0 using finitely many additions,
multiplications and exponentiations. Does anyone out there know?
Anyway, the next really interesting ordinal I know after epsilon_1 is
epsilon_2. It's 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}},...
Maybe now you get the pattern. In general, epsilon_alpha is the
alpha-th solution of
x = omega^x
and 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,..., epsilon_3,..., epsilon_4, ...
and eventually
epsilon_omega,..., epsilon_{omega+1},..., epsilon_{omega+2},...
and eventually
epsilon_{omega^2},..., epsilon_{omega^3},..., epsilon_{omega^4},...
and eventually
epsilon_{omega^omega},..., epsilon_{omega^{omega^omega}},...
As you can see, this gets boring after a while - it's suspiciously
similar to the beginning of our trip through the ordinals, with
them now showing up as subscripts under this "epsilon" notation.
But this is misleading: we're moving much faster now. 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}}
Anyway... so finally we *got* to this South Dakota Tractor Museum,
driving pretty darn fast at this point, about 85 miles an hour...
and guess what?
Oh - wait a minute - it's sort of interesting here:
epsilon_{epsilon_0},..., epsilon_{epsilon_1},..., epsilon_{epsilon_2}, ...
and now we reach
epsilon_{epsilon_omega}
and then
epsilon_{epsilon_{omega^omega}},...,
epsilon_{epsilon_{omega^{omega^omega}}},...
and then as we keep speeding up, we see:
epsilon_{epsilon_{epsilon_0},...
epsilon_{epsilon_{epsilon_{epsilon_0}}},...
epsilon_{epsilon_{epsilon_{epsilon_{epsilon_0}}}},...
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 little anecdote, in fact.
But that's 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 a
really big one... and when you find one, you see it's part of a new
pattern, and that gets boring too...
Anyway, 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}}}},...
our notation breaks down, since this is the first solution of
x = epsilon_x
We could make up a new name for this ordinal, like eta_0.
Then we could play the whole game again, defining eta_{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!
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 now these etas. It's gonna keep happening! For
each type of ordinal, our notation runs 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, let's use
phi_gamma for the gamma-th type of ordinal, and phi_gamma(alpha) for
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)
We can even define this stuff when gamma itself is infinite.
For a more precise definition see the Wikipedia article cited below...
but I hope you get the rough idea.
This defines a lot of really big ordinals, called the "Veblen hierarchy".
There's a souped-up version of Cantor normal form that can handle
every ordinal that's a finite sum of guys in the Veblen hierarchy:
you can write them *uniquely* as finite sums of the form
phi_{gamma_1}(alpha_1) + ... + phi_{gamma_k}(alpha_k)
where each term is less than or equal to the previous one, and each
alpha_i is not a fixed point of phi_{gamma_i}.
But as you might have suspected, not *all* ordinals can be written
in this way. For one thing, every ordinal we've reached so far is
*countable*: as a set you can put it in one-to-one correspondence
with the integers. There are much bigger *uncountable* ordinals -
at least if you believe you can well-order uncountable sets.
But even in the realm of the countable, we're nowhere near done!
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 - the faster we drive, the faster it chases after us.
But, it's interesting to hear what happens next. At this point we
reach something a bit like the Badlands on the western edge of South
Dakota - something a bit spooky!
It's called the Feferman-Schuette ordinal, Gamma_0. This is just
the limit, or union if you prefer, of all the ordinals mentioned
so far: all the ones you can get from the Veblen hierarchy. You
can also define Gamma_0 by a fixed point property: it's the smallest
ordinal x with
phi_x(0) = x
Now, we've already seen that induction up to different ordinals
gives us different amounts of mathematical power: induction up
to omega is just ordinary mathematical induction as formalized by
Peano arithmetic, but induction up to epsilon_0 buys us more -
it lets us prove the consistency of Peano arithmetic!
Logicians including Feferman and Schuette have carried out a detailed
analysis of this subject. They know a lot about how much induction
up to different ordinals buys you. And apparently, induction up to
Gamma_0 lets us prove the consistency of a system called "predicative
analysis". I don't understand this, nor do I understand the claim
I've seen that Gamma_0 is the first ordinal that cannot be defined
predicatively - i.e., can't be defined without reference to itself.
Sure, saying Gamma_0 is the first solution of
phi_x(0) = x
is non-predicative. But what about saying that Gamma_0 is the union
of all ordinals in the Veblen hierarchy? What's non-predicative
about that?
If anyone could explain this in simple terms, I'd be much obliged.
As you can see, I'm getting out my depth here. That's pretty typical
in This Week's Finds, but this time - just to shock the world -
I'll take it as a cue to shut up. So, I won't try to explain the
outrageously large Bachmann-Howard ordinal, or the even more
outrageously large Church-Kleene ordinal - the first one that can't
be written down using *any* computable system of notation. You'll
just have to read the references.
For a fun introduction to the infinite, try
2) Rudy Rucker, Infinity and the Mind: The Science and Philosophy
of the Infinite, Princeton University Press, Princeton, 2004.
Rucker also wrote a novel, White Light, about a mathematician who
actually goes to the transfinite realms and climbs some transfinite
mountains.
For online self-study, I urge you to start by reading the Wikipedia
article on ordinal numbers, then the article on ordinal arithmetic,
and then the one on large countable ordinals - they're really well-written:
3) Wikipedia, Ordinal numbers,
http://en.wikipedia.org/wiki/Ordinal_number
Ordinal arithmetic,
http://en.wikipedia.org/wiki/Ordinal_arithmetic
Large countable ordinals,
http://en.wikipedia.org/wiki/Large_countable_ordinals
The last one has a tempting bibliography, but warns us that most
books on this subject are hard to read and out of print. Apparently
nobody can agree on notation for ordinals beyond the Veblen hierarchy,
either.
Gentzen proved the consistency of Peano arithmetic in 1936:
4) Gerhard Gentzen, Die Widerspruchfreiheit der reinen Zahlentheorie,
Mathematische Annalen 112 (1936), 493-565. Translated as "The
consistency of arithmetic" in M. E. Szabo ed., The Collected Works
of Gerhard Gentzen, North-Holland, Amsterdam, 1969.
Goodstein's theorem came shortly afterwards:
5) R. Goodstein, On the restricted ordinal theorem, Journal of
Symbolic Logic, 9 (1944), 33-41.
but Kirby and Paris proved it independent of Peano arithmetic
only in 1982:
6) L. Kirby and J. Paris, Accessible independence results for Peano
arithmetic, Bull. London. Math. Soc. 14 (1982), 285-93.
That marvelous guy Alan Turing wrote his PhD thesis at Princeton
under the logician Alonzo Church. It was about ordinals and their
relation to logic:
7) Alan M. Turing, Systems of logic defined by ordinals, Proc.
London Math. Soc., Series 2, 45 (1939), 161-228.
This is regarded as his most difficult paper. The idea is to
take a system of logic like Peano arithmetic and throw in an
extra axiom saying that system is consistent, and then another
axiom saying *that* system is consistent, and so on ad infinitum -
getting a new system for each ordinal. These systems are recursively
axiomatizable up to (but not including) the Church-Kleene ordinal.
These ideas were later developed much further....
But, reading original articles is not so easy, especially if you're
in Shanghai without access to a library. So, what about online stuff -
especially stuff for the amateur, like me?
Well, this article is great fun if you're looking for a readable
overview of the grand early days of proof theory, when Hilbert was
battling Brouwer, and then Goedel came and blew everyone away:
8) Jeremy Avigad and Erich H. Reck, "Clarifying the nature of the
infinite": the development of metamathematics and proof theory,
Carnegie-Mellon Technical Report CMU-PHIL-120, 2001. Also
available as http://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf
But, it doesn't say much about the newer stuff, like the idea that
induction up to a given ordinal can prove the consistency of a logical
system - the bigger the ordinal, the stronger the system. For work
up to 1960, this is a good overview:
9) Solomon Feferman, Highlights in proof theory, in Proof Theory,
eds. V. F. Hendricks et al, Kluwer, Dordrecht (2000), pp. 11-31.
Also available at http://math.stanford.edu/~feferman/papers.html
For newer stuff, try this:
10) Solomon Feferman, Proof theory since 1960, prepared for the
Encyclopedia of Philosophy Supplement, Macmillan Publishing Co.,
New York. Also available at
http://math.stanford.edu/~feferman/papers.html
Also try the stuff on proof theory, trees and categories mentioned
in "week227", and this book mentioned in "week70" - now you can get
it free online:
11) Jean-Yves Girard, Y. Lafont and P. Taylor, Proofs and Types,
Cambridge Tracts in Theoretical Computer Science 7, Cambridge U.
Press, 1989. Also available at
http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html
Finally, sometime I want to get ahold of this book by someone who
always enlivened logic discussions on the internet until his death in
April this year:
12) Torkel Franzen, Inexhaustibility: A Non-Exhaustive Treatment,
Lecture Notes in Logic 16, A. K. Peters, Ltd., 2004.
The blurb sounds nice: "The inexhaustibility of mathematical
knowledge is treated based on the concept of transfinite
progressions of theories as conceived by Turing and Feferman."
Okay, now for a bit about the icosahedron - my favorite Platonic solid.
I've been thinking about the "geometric McKay correspondence" lately,
and among other things this sets up a nice relationship between the
symmetry group of the icosahedron and an amazing entity called E8.
E8 is the largest of the exceptional Lie groups - it's 248-dimensional.
It's related to the octonions (the number "8" is no coincidence) and
it shows up in string theory. It's very beautiful how this complicated
sounding stuff can be seen in distilled form in the icosahedron.
I have a lot to say about this, but you're probably worn out by our
road trip through the land of big ordinals. So for now, try "week164"
and "week230" if you're curious. Let's talk about something less
stressful - the early history of the icosahedron.
I spoke about the early history of the dodecahedron in "week63".
It's conjectured that the Greeks got interested in this shape
from looking at crystals of iron pyrite. These aren't regular
dodecahedra, since normal crystals can't have 5-fold symmetry -
though "quasicrystals" can. Instead, they're "pyritohedra".
The Greeks' love of mathematical perfection led them to the
regular dodecahedron....
... and it also led them to invent the icosahedron:
13) Benno Artmann, About the cover: the mathematical conquest of
the third dimension, Bulletin of the AMS, 43 (2006), 231-235.
Also available at
http://www.ams.org/bull/2006-43-02/S0273-0979-06-01111-6/
According to Artmann, an ancient note written in the margins of a copy
of Euclid's Elements says the regular icosahedron and octahedron
were discovered by Theaetetus!
If you're a cultured sort, you may know Theaetetus through Plato's
dialog of the same name, where he's described as a mathematical
genius. He's also mentioned in Plato's "The Sophist". He probably
discovered the icosahedron between 380 and 370 BC, and died at an
early age in 369. Euclid wrote his construction of the icosahedron
that we find in Euclid's Elements:
14) Euclid, Elements, Book XIII, Proposition 16, online version
due to David Joyce at
http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII16.html
Artmann says this was the first time a geometrical entity appeared
in pure thought before it was seen! An interesting thought.
Book XIII also contains a complete classification of the Platonic
solids - perhaps the first really interesting classification
theorem in mathematics, and certainly the first "ADE classification":
15) Euclid, Elements, Book XIII, Proposition 18, online version
due to David Joyce at
http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII18.html
If you don't know about ADE classifications, see "week62".
I got curious about this "ancient note written in the margins of a
copy of Euclid" that Artmann mentions. It seemed too good to be true.
Just for fun, I tried to track down the facts about this, using only
my web browser here in Shanghai.
First of all, if you're imagining an old book in a library somewhere
with marginal notes scribbled by a pal of Theaetetus, dream on.
It ain't that simple! Our knowledge of Euclid's original Elements
relies on copies of copies of copies... and centuries of detective
work, with each detective having to root through obscure journals
and dim-lit library basements to learn what the previous detectives
did.
The oldest traces of Euclid's Elements are pathetic fragments of
papyrus. People found some in a library roasted by the eruption
of Mount Vesuvius in 79 AD, some more in a garbage dump in the
Egyptian town of Oxyrhynchus (see "week221"), and a couple more in
the Fayum region near the Nile. All these were written centuries
after Euclid died. For a look at one, try this:
16) Bill Casselman, One of the oldest extant diagrams from Euclid,
http://www.math.ubc.ca/~cass/Euclid/papyrus/
The oldest nearly complete copy of the Elements lurks in a museum
called the Bodleian at Oxford. It dates back to 888 AD, about a
millennium after Euclid.
More copies date back to the 10th century; you can find their stories
here:
17) Thomas L. Heath, editor, Euclid's Elements, chap. V: the text,
Cambridge U. Press, Cambridge, 1925. Also available at
http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+5
18) Menso Folkerts, Euclid's Elements in Medieval Europe,
http://www.math.ubc.ca/~cass/Euclid/folkerts/folkerts.html
All these copies are somewhat different. So, getting at Euclid's
original Elements is as hard as sequencing the genome of Neanderthal
man, seeing a quark, or peering back to the Big Bang!
A lot of these copies contain "scholia": comments inserted by
various usually unnamed copyists. These were collected and
classified by a scholar named Heiberg in the late 1800s:
19) Thomas L. Heath, editor, Euclid's Elements, chap. VI: the scholia,
Cambride U. Press, Cambridge, 1925. Also available at
http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+6
One or more copies contains a scholium about Platonic solids in
book XIII. Which copies? Ah, for that I'll have to read Heiberg's
book when I get back to UC Riverside - our library has it, I'm
proud to say.
And, it turns out that another scholar named Hultsch argued
that this scholium was written by Geminus of Rhodes.
Geminus of Rhodes was an astronomer and mathematician who may have
lived between 130 and 60 BC. He seems like a cool dude. In his
Introduction to Astronomy, he broke open the "celestial sphere",
writing:
... we must not suppose that all the stars lie on one surface,
but rather that some of them are higher and some are lower.
And in his Theory of Mathematics, he proved a classification theorem
stating that the helix, the circle and the straight line are the only
curves for which any portion is the same shape as any other portion
with the same length.
Anyway, the first scholium in book XIII of Euclid's Elements, which
Hultsch attributes to Geminus, mentions
... the five so-called Platonic figures which, however, do not
belong to Plato, three of the five being due to the Pythagoreans,
namely the cube, the pyramid, and the dodecahedron, while the
octahedron and the icosahedron are due to Theaetetus.
So, that's what I know about the origin of the icosahedron!
Someday I'll read more, so let me make a note to myself:
20) Benno Artmann, Antike Darstellungen des Ikosaeders, Mitt.
DMV 13 (2005), 45-50. (Here the drawing of the icosahedron in
Euclid's elements is analysed in detail.)
21) A. E. Taylor, Plato: the Man and His Work, Dover Books, New
York, 2001, page 322. (This discusses traditions concerning
Theaetetus and Platonic solids.)
22) Euclid, Elementa: Libri XI-XIII cum appendicibus, postscript
by Johan Ludvig Heiberg, edited by Euangelos S. Stamatis,
Teubner BSB, Leipzig, 1969. (Apparently this contains information
on the scholium in book XIII of the Elements.)
Now for something a bit newer: categorification and quantum mechanics.
I've said so much about this already that I'm pretty much talked out:
23) John Baez and James Dolan, From finite sets to Feynman diagrams,
in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Bjoern
Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.
24) John Baez and Derek Wise, Quantization and Categorification,
Quantum Gravity Seminar lecture notes, available at:
http://math.ucr.edu/home/baez/qg-fall2003/
http://math.ucr.edu/home/baez/qg-winter2004/
http://math.ucr.edu/home/baez/qg-spring2004/
As I explained in "week185", many basic facts about harmonic
oscillators, Fock space and Feynman diagrams have combinatorial
interpretations. For example, the commutation relation between
the annihilation operator a and the creation operator a*:
aa* - a*a = 1
comes from the fact that if you have some balls in a box, there's one
more way to put a ball in and then take one out than to take one out
and then put one in! This way of thinking amounts to using finite
sets as a substitute for the usual eigenstates of the number operator,
so we're really "categorifying" the harmonic oscillator: giving it a
category of states instead of a set of states.
Working out the detailed consequences takes us through Joyal's
theory of "structure types" or "species" - see "week202" - and
on to more general "stuff types". Some nice category and
2-category theory is needed to make the ideas precise. For a
careful treatment, see this thesis by a student of Ross Street:
25) Simon Byrne, On Groupoids and Stuff, honors thesis,
Macquarie University, 2005, available at
http://www.maths.mq.edu.au/~street/ByrneHons.pdf and
http://math.ucr.edu/home/baez/qg-spring2004/ByrneHons.pdf
However, none of this work dealt with the all-important *phases*
in quantum mechanics! For that, we'd need a generalization of
finite sets whose cardinality can be be complex. And that's what
my student Jeffrey Morton introduces here:
26) Jeffrey Morton, Categorified algebra and quantum mechanics,
Theory and Application of Categories 16 (2006), 785-854. Available at
http://www.emis.de/journals/TAC/volumes/16/29/16-29abs.html; also
available as math.QA/0601458.
He starts from the beginning, explains how and why one would
try to categorify the harmonic oscillator, introduces the
"U(1)-sets" and "U(1)-stuff types" needed to do this, and shows
how the usual theorem expressing time evolution of a perturbed
oscillator as a sum over Feynman diagrams can be categorified.
His paper is now *the* place to read about this subject. Take
a look!
-----------------------------------------------------------------------
Quote of the Week:
Said Conrad Cornelius O'Donald O'Dell,
My very young friend who is learning to spell,
"The A is for Ape. And the B is for Bear.
The C is for Camel. The H is for Hare.
The M is for Mouse. And the R is for Rat.
I know all the twenty-six letters like that...
... through to Z is for Zebra. I know them all well.
So now I know everything anyone knows
From beginning to end. From the start to the close.
Because Z is as far as the alphabet goes."
Then he almost fell flat on his face on the floor
When I picked up the chalk and drew one letter more!
A letter he never had dreamed of before!
And I said, "You can stop, if you want, with the Z
Because most people stop with the Z
But not me!
In the places I go there are things that I see
That I never could spell if I stopped with the Z.
I'm telling you this 'cause you're one of my friends.
My alphabet starts where your alphabet ends!" - Doctor Seuss
-----------------------------------------------------------------------
Addenda: I thank Tim Chow, Phillip Helbig, Rudy Rucker, Jeffrey Winkler
and especially Kevin Buzzard for interesting comments.
Jeffrey Winkler wrote:
Are the ridiculously infinite forms of infinity you discuss in your
article ever used for anything?
If they only refer to themselves, if the only thing these infinite sets
refer to is other infinite sets then it's pointless self-reflexive
recursive circular reasoning. A set that has the same number of elements
as the number of apples on your table, or the number of points in a line
or a plane, or in infinitely dimensional Hilbert space, could be said to
refer to something, but give an example of something, other than one of
the sets you're talking about, which would have epsilon_0 elements. Where
in mathematics or physics would you ever use such a concept? If you never
would, then what's the point? In other words, is there anything where you
could have epsilon_0 "many" of something, other than the sets themselves,
and if there isn't, then are these actually numbers? If nothing is ever
"that many", then is it a number? Of course, we've expanded the concept
of "number" to include complex numbers, quaternions, octonions, vectors,
tensors, matrices, etc. where they don't literally correspond to "how
many" something is, yet in all those examples, they have uses in various
fields of mathematics and physics, other than just when talking about
themselves, so there is a reason for inventing them.
Jeffery
I replied:
Jeffery Winkler wrote:
> Are the ridiculously infinite forms of infinity you discuss in your
> article ever used for anything?
Without epsilon_0 you can't prove that Goodstein sequences converge
to zero - an obviously true fact. As I mentioned, the main use of
these ordinals is to measure the strength of axiom systems. But, I
didn't write about these ordinals because they're useful. I wrote
about them because they're fun.
They're not "ridiculously infinite", though. The ordinals I mentioned
are all countable ordered sets, and you can describe them all
*explicitly* as subsets of the rational numbers.
More precisely: any one of the ordinals I mentioned, up to and including
the Feferman-Schuette ordinal (and quite a ways beyond), is isomorphic
as an ordered set to a subset of the rational numbers. Moreover, you
can write a computer program that will decide whether or not any given
fraction is in this subset. As a consequence, you can also write a
computer program that lists the fractions in this set.
It's pretty obvious how to do this for omega^2:
http://math.ucr.edu/home/baez/omega_squared.png
But you can do it for any one of the ordinals I mentioned! David
Madore has drawn a picture of epsilon_0, for example.
So, if you really object to these ordinals as "ridiculously infinite",
you must have some doubts about the legitimacy of computable subsets
of the rational numbers as valid objects of study. That seems like
an extreme position.
The Church-Kleene ordinal is much larger than any of the ordinals
I discussed in detail. It's still countable. Any ordinal below it
can be described in a computable way - but it itself can't. So, if
you believe that only computable mathematical entities are worth
studying, you might want to stop shy of this one. I stopped *far*
short of this one.
> If they only refer to themselves, if the only thing these infinite
> sets refer to is other infinite sets then it's pointless self-reflexive
> recursive circular reasoning.
I hope you see that this is not true for the countable ordinals I
was discussing. I deliberately refrained from mentioning the large
cardinals that logicians often discuss, precisely because I share
your distaste for such stuff.
I realized after I posted "week236" that some people might think I was
talking about mystical entities, when I was actually talking about very
concrete things.
Best,
jb
And now for Kevin Buzzard's wonderful calculation! In the original
version of this Week's Finds, I wrote:
> if you start with the number 4, you get this Goodstein sequence:
>
> 4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...
>
> and apparently it takes about 3 x 10^{60605351} steps to reach zero!
I got this figure from the "National Curve Bank" website mentioned
above, but I got the details wrong: they said the sequence "can
increase for approximately 2.6 * 10^{60605351} steps", whatever
that means.
Kevin Buzzard then sent me an email in which he worked out the
number himself. After correcting a few small mistakes, we seem
to have settled on a different answer.
Kevin writes:
You write this as if it were some kind of mystery. I remember working
out this number explicitly when I was a graduate student! There is
some nice form for it, as I recall. Let's see if I can reconstruct
what I did.
If I've understood the sequence correctly, it should be (where "n)"
at the beginning of a line denotes we're working in base n on this
line, so strictly speaking it's probably the n-1st term in the sequence)
2) 2^2 = 4
3) 3^3-1 = 2.3^2+2.3+2 = 26 [note: base 3, ends in 2, and 3+2=5]
4) 2.4^2+2.4+1 = 41 [note: base 4, ends in 1, and 4+1=5]
5) 2.5^2+2.5 = 60 [we're at a limit ordinal here, note 3+2=4+1=5]
6) 2.6^2+2.6-1 = 2.6^2+6+5 = 83 [note: base 6, ends in 5]
7) 2.7^2+7+4 [note: base 7, ends in 4]
8) 2.8^2+8+3 [note: base 8, ends in 3, so we next get a limit ordinal at...]
.
.
11) 2.11^2+11
12) 2.12^2+12-1 = 2.12^2+11
13) 2.13^2+10
.
.
.
23) 2.23^2 (as 23 = 12+11 = 13+10= ...)
24) 24^2+23.24+23
.
.
.
47) 47^2+23.47
48) 48^2+22.48+47
.
.
.
95) 95^2+22.95
96) 96^2+21.96+95
.
.
.
and now we spot a pattern: we're just doubling---getting a limit ordinal
at bases 24-1, 48-1, 96-1 and so on. Let's look again at those limit
ordinals:
47) 47^2+23.47
95) 95^2+22.95
.
.
.
24*2^t-1) (24*2^t-1)^2+(24-t)*(24*2^t-1)
.
.
.
so the last one with a square in it will be the case t=24, corresponding
to
r) r^2
where
r = 24 * 2^24 - 1 = 402653183.
All those 24s, but I'm sure you'll not get carried away. Let's define
n = r+1 = 24 * 2^24
and continue on. At the next step, the ordinal decreases sharply:
n) n^2-1 = (n-1)n + (n-1)
n+1) (n-1)(n+1) + (n-2) [note: now back to the usual tricks]
.
.
.
2n-1) (n-1)(2n-1) [the next limit, at base 2n-1]
2n) (n-2)(2n) + (2n-1)
.
.
.
4n-1) (n-2)(4n-1)
4n) (n-2)(4n)+(4n-1)
.
.
.
and the limit ordinals we're running into now (and we're going to
run into about n of them, which is a lot), are
2n-1) (n-1)(2n-1)
4n-1) (n-2)(4n-1)
8n-1) (n-3)(8n-1)
.
.
.
n2^s-1) (n-s)(n2^s - 1)
.
.
.
and finally when s = n-1
m) m
where m = n 2^{n-1} - 1. The sequence now looks like
m+1) (m+1)-1 = m
m+2) m-1
m+3) m-2
.
.
.
2m+1) 0
So the sequence becomes zero at base n2^n - 1, where n = 24 * 2^24.
If 2^2 is the first term in the sequence, I guess this is
the (n2^n - 2)th term. I make this about 6.9*10^{121210694} -
curses, you got something else! Actually, I have about the square
of what you wrote and hence I have most likely made a slip. On the other
hand you can see that it's not a mystery at all, it's just an
elementary exercise. It really helps you learn about why
the countable ordinals are well-ordered too: as you continue working
out the numbers, you always have this impending sense of doom
telling you that your gut feeling that the sequence tends to
infinity might just be wrong...
Kevin
So, in simple terms, the 4th Goodstein sequence starts out by
shooting up faster and faster, reaching almost n^2 by the nth term,
where
n = 24 * 2^24 = 402653184
By the (2n)th term it reaches almost 2n^2.
Then it grows by steps of n-2, and
by the (4n)th term it reaches almost 4n^2.
Then it grows by steps of n-3, and
by the (8n)th term it reaches almost 8n^2.
And so on... it's slowing down now.
After about the (2^{n-2}n)th term it levels off completely,
and by the (2^{n-1}n)th term it equals almost 2^{n-1}n.
Note these numbers are the same.
From this point on the sequence decreases by one each time, and it
hits zero at about the (2^n n)th term.
So, it spends the last half of its life decreasing by one each
time. Right before that, it spends a quarter of its life remaining
constant. Right before that, it spends an eighth of its life growing
by one each time. Right before that it, it spends a sixteenth of its
life growning by two each time. And so on, but not indefinitely:
this description covers the phase after the nth term.
In particular, it reaches its maximum and then levels off after about
quarter of its life is done. It spends about the next quarter of its
life being constant, and the next half going down one step at a time.
Let's be a bit more precise. For this let's call the ith term of
the 4th Goodstein sequence a_i, where we count things so that
a_1 = 4, a_2 = 26, ...
Then a_i first reaches its maximum value at exactly
i = (1/4) 24 2^{24} 2^{24 2^{24}} - 2 ~ 1.72 x 10^{121210694}
Its value at this point is exactly
a_i = (1/2) 24 2^{24} 2^{24 2^{24}} - 1 ~ 3.45 x 10^{121210694}
It then stays constant for about twice as many more terms, and
then goes to zero one step at a time, hitting zero precisely when
i = 24 2^{24} 2^{24 2^{24}} - 2 ~ 6.89 x 10^{121210694}
Believe me, it's a lot more fun to figure this stuff out than to
read it.
I was worried that Kevin and I had made a mistake until we found
this thesis which gets the same answer (though the author starts
the sequence at a_0 = 4, so it looks one off from ours):
27) Justin T. Miller, On the Independence of Goodstein's Theorem,
Masters thesis, University of Arizona, 2001. Also available as
http://www.u.arizona.edu/~miller/thesis/thesis.html
Tim Chow had some comments about why the Feferman-Schuette ordinal
is considered the first "impredicative" ordinal. He wrote:
John Baez wrote:
>Logicians including Feferman and Schuette have carried out a detailed
>analysis of this subject. They know a lot about how much induction
>up to different ordinals buys you. And apparently, induction up to
>Gamma_0 lets us prove the consistency of a system called "predicative
>analysis". I don't understand this, nor do I understand the claim
>I've seen that Gamma_0 is the first ordinal that cannot be defined
>predicatively - i.e., can't be defined without reference to itself.
>Sure, saying Gamma_0 is the first solution of
>
>phi_x(0) = x
>
>is non-predicative. But what about saying that Gamma_0 is the union
>of all ordinals in the Veblen hierarchy? What's non-predicative
>about that?
The situation is somewhat akin to the situation with the Church-Turing
thesis, in that one is tentatively equating an informal notion
(predicativity or computability) with a precise mathematical notion.
Therefore there is no definitive answer to your question, and Feferman
himself has articulated potential objections to the "standard view"
that Gamma_0 marks the boundary of predicativity.
Having said that, I'll also say that one of the reasons for the standard
view is that Gamma_0 marks the boundary of "autonomous progressions" of
arithmetical theories. The book by Torkel Franzen that you cited is
probably the most accessible introduction to this subject. Roughly
speaking, the idea is that if anyone fully accepts first-order Peano
arithmetic PA, then implicitly he accepts its consistency Con(PA), as
well as Con(PA+Con(PA)), etc. If one tries to articulate exactly what
is "implicitly" involved in accepting PA in this sense, then one can
make a plausibility argument that Gamma_0 is a natural stopping point.
I think you have a better shot at grasping the underlying intuition via
this approach than by staring at Gamma_0 itself and trying to figure out
what is non-predicative about its definition.
I replied asking if "etc." means there's one theory like this
per ordinal. I also asked for more clues about this "plausibility
argument", and noted:
There's also someone named Nik Weaver who has debated Feferman
on this subject:
http://www.cs.nyu.edu/pipermail/fom/2006-April/010472.html
http://www.math.wustl.edu/~nweaver/conceptualism.html
He seems to claim that Gamma_0 and even larger ordinals have predicative
definitions. However, I'm too ignorant to follow this debate.
Usually in physics I have a sense for when people are being reasonable
even if I don't follow the details. In this debate I can't even
do that.
Tim Chow replied:
Let's look more closely at what the notion of "one theory like this
per ordinal" means. There's no difficulty figuring out what "Con(PA)"
means or how to express that statement in the first-order language
of arithmetic. Ditto with "Con(PA+Con(PA))". However, once you start
ascending the ordinal hierarchy, a difficulty appears. The language
of arithmetic doesn't let you talk about "ordinals" directly---that's a
set-theoretical concept. In order to express a statement like "Con(T)"
for some theory T, you need at minimum to be able to give some sort of
"recursive description" or "recursive axiomatization" of T (where here
I use the word "recursive" in the technical sense of recursive function
theory) in the first-order language of arithmetic. This observation
already yields the intuition that we're not going to be able to ascend
beyond the Church-Kleene ordinal, because we won't even be able to
figure out how to *say* "T is consistent" for a theory T that requires
that many iterations to reach from PA.
There are other problems, though, that potentially get in the way before
we reach the Church-Kleene ordinal. Once we realize that what we need is
a system of "ordinal notations" to "fake" the relevant set theory, we may
(if we are predicativists) worry about issues such as:
1. As we ascend the ordinal hierarchy, isn't it illegitimate to make a
jump to an ordinal alpha unless we've already proved, at the level of some
ordinal beta that we've already reached, that an ordinal of type alpha
exists?
2. And isn't it illegitimate to create sets by quantification over things
other than the natural numbers themselves and sets that we've already
created?
Condition 1 goes by the name of "autonomy" and condition 2 goes by the
name of "ramification." If one formalizes these notions in a certain
plausible manner, then one arrives at Gamma_0 as the least upper bound
of theories that you can get to, starting with (for example) PA.
One can of course wonder whether 1 and 2 above really capture the concept
of "predicativity." Some secondary evidence has accumulated of the
following form: Some argument that intuitively seems to be predicative
but that is not immediately seen to be provable in the Feferman-Schuette
framework is shown, after some work, to indeed be provable below Gamma_0.
It's still possible, of course, for someone---you mentioned Nik Weaver---
to come along and argue that our intuitive notion of predicativism, fuzzy
though it is, can't possibly be identified with the level Gamma_0. The
reason you can't seem to decide immediately whether Weaver's position is
nonsensical or not is probably because the critical questions are not
mathematical but philosophical, and of course it's usually harder to
arrive at definitive answers in philosophy than in mathematics.
Finally, let me record a number of papers that treat notations for
ordinals above the Feferman-Schuette ordinal. In the discussion of
this Week's Finds on sci.math.research, Dave Renfro pointed out a
really nice readable paper on large countable ordinals which goes
far beyond what I discussed:
28) Hilbert Levitz, Transfinite ordinals and their notations: for
the uninitiated, available at http://www.cs.fsu.edu/~levitz/research.html
This paper introduced the "Schuette Klammersymbole", which generalize
the Veblen hierarchy:
29) Kurt Schuette, Kennzeichnung von Orgnungszahlen durch rekursiv
erklaerte Funktionen, Math. Ann 127 (1954), 15-32.
These papers discuss a general concept of "ordinal notation system",
which includes the Schuette Klammersymbole and also something
called the "n-ary Veblen hierarchy":
30) Anton Setzer, An introduction to well-ordering proofs in Martin-
Loef's type theory, in Twenty-Five Years of Constructive Type Theory,
eds. G. Sambin and J. Smith, Clarendon Press, Oxford, 1998, pp. 245-263.
Also available at http://www.cs.swan.ac.uk/~csetzer/index.html
Anton Setzer, Ordinal systems, in Sets and Proofs, Cambridge U. Press,
Cambridge, 2011, pp. 301-331. Also available at
http://www.cs.swan.ac.uk/~csetzer/index.html
This paper has a nice expository section on generalizations of the
Veblen hierarchy:
31) Jean H. Gallier, What's so special about Kruskal's theorem and
the ordinal Gamma_0? A survey of some results in proof theory,
sec. 7, A glimpse at Veblen hierarchies, Ann. Pure Appl. Logic 53
(1991), 199-260. Also available at
http://www.cis.upenn.edu/~jean/gallier-old-pubs.html
This paper is very useful, since it compares different notations:
32) Larry W. Miller, Normal functions and constructive ordinal notations,
J. Symb. Log. 41 (1976), 439-459.
You can get it through JSTOR if you have access to that.
This webpage gives a nice definition of "ordinal notation system"
as a coalgebra of a certain functor - nice if you understand categories,
that is:
33) Peter Hancock, Ordinal notation systems,
http://homepages.inf.ed.ac.uk/v1phanc1/ordinal-notations.html
Kevin Watkins pointed out this website, which contains several papers
on ordinal notations:
34) Harold Simmons, Abstracts of papers and notes,
http://www.cs.man.ac.uk/~hsimmons/DOCUMENTS/papersandnotes.html
Finally, the Wikipedia article on "large countable ordinals" has
some references to books which are, alas, out of print.
-----------------------------------------------------------------------
Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html