In my novel *White Light, *I describe a mountain that is higher than infinity.4
This mountain, called Mount On, Consists of alternating cliffs and meadows. The curious
thing about it is that even after one has climbed ten cliffs, a thousand cliffs, infinitely
many cliffs . . . there are always more cliffs. The climbers of Mount On are able to make
some progress because they are able to execute a procedure called a "speed-up." By'
using speed-ups they are able, for instance, to zip past the first infinity of cliffs in two
hours.

How is this done? The idea is to climb the first cliff in one hour, the next cliff in half an hour, the one after that in a quarter of an hour, and, in general, the nth cliff in 1/ hours. Since 1 + 1/2 + 1/4 + 1/8 + . . . sums to 2, we see that after two hours our climbers have passed infinitely many cliffs. But there are more, many more.

In this section we will climb up through the transfinite numbers, which are usually
called *ordinal numbers, *or just *ordinals. *Typically, one describes some
ordinal a by giving an example of an ordered set *M *such that if one could count
*M *in the correct order, *then *one would count up to a. a is then viewed as
the *abstract order type of M, *called for short. The ordinal is gotten
from the ordered set *M *by ignoring the actual appearance of the individual
members of *M *and instead concentrating on the arrangement, or order, of these
members.

The transfinite ordinal numbers can be thought of as arising through counting. There
are two principles for generating ordinal numbers: I) if you have the ordinal number a,
then you can find a *next *ordinal, called *a *+ 1; II) if you have some definite
sequence of increasing ordinals a, then you can find a last ordinal which is greater than
all the a's, called lim(a).

We also need a first ordinal to start with, called 0. (Strictly speaking, the second
principle for generating ordinals gives us 0, since zero is the first ordinal after the empty
sequence.) In any case, once we have zero, the first principle can be repeatedly applied
to get the ordinal numbers 0, 1, 2, . . . . Now, to get past the infinite sequence of finite
ordinals we use principle II to get *lim(n), *usually called *, *pronounced
"omega." (Omega is also sometimes called "alef-null.")

Omega is the last letter in the Greek alphabet (they put zeta somewhere in the middle), which may be why Cantor chose to use it as the number after all the finite numbers. The word "omega" is somewhat familiar, as it appears in the Book of Revelations, where God is quoted as saying, "I am the Alpha and the Omega," meaning, "I am the beginning and the end."

Now we have 0, 1, 2,. . . . Using principle I repeatedly we get the sequence 0, 1, 2, . . . , +1, + 2, + 3 . . . . To go further, we use principle II to form lim( + *n), *which is
usually called + or * 2.

You might wonder why lim( + n), + , and * 2 should all be the
same. It turns out that there is a definite way to define addition and multiplication of
infinite ordinals so that everything works out. let me briefly explain. The ordinal number
a + *b *is obtained by counting out to a and then counting *b *steps further. The
ordinal number a . *b *is obtained by counting up to a, *b *times in a row. That
is, a ** b *is obtained by sticking together *b *copies of a, treating this as an
ordered set *M, *and then abstracting to get the ordinal number= *a *
b.*

*
A*s long as one sticks to finite ordinals, these operations are the same as the
ordinary' plus and times, and they are commutative. However, once we start working
with infinite ordinals, commutativity no longer holds. Thus,

1 + * *is just the same as , but * *+ 1 is the next number after *.
*Again,

* * *2 is two omegas placed next to each other, which gives an ordinal

* + , *but 2 * is omega twos placed next to each other, which makes

an ordered set with ordinal number .

Moving on with the ordinals, by using principle I repeatedly, we now have 0, 1,
2, . . ., + 1, + 2, . . . , *2, *2 + 1, * 2 + 2, . . . . It is evident that lim(*2 +
n) should be the ordinal * 2 + also known as * 3. Continuing in
this vein, we can arrive at * *n *for each finite *n, *and using principle II we
form lim(*n), which should be Omega copies of omega, that is, ** , *also
known as.

In order to see how to pass from to to and
finally on to ), it will be useful to look at the pictures in Figure 41.1 will now
describe four sets of points, *M1, M2, M3, *and *M. *In each case, the set of
points can be thought of as lying between zero and one on the real number line, and
each *Mi * will be such that if we view it as an ordered set and abstract to form the
ordinal number *Mi, *we arrive at . One may wonder how to fit an ordered
set with ordinal number in between zero and one on the real line, since the space
available seems to be finite. The trick is based on Zeno's paradox: if you start out going
from zero to one, but do it by first going halfway, and then going half the remaining
distance, and then half the remaining distance, and so on . . . always going just half of
what's left . . . then it will take you omega steps to get there. This is basically what the
first picture shows.

I have written next to each picture the real number coordinates of the points involved
if we think of the interval as the unit interval on the real line. These set definitions are
not particularly important for us, but what *is *important is to realize that a
transfinite arrangement of points can be fitted into a finite space. By using Zenonian
squeezing we can sort of see an ordering of type all at once! Transfinite
ordinals are not really so inconceivable after all.

It turns out that one can actually fit *any *countable ordinal into a picture of this
nature. (In the next subsection we will look at the uncountable ordinals.) But the
illustration of is already something of a mess, and had I not used the
arrow symbolism, the picture of would be so lacking in detail as to be
quite uninformative. We will look at a different technique for picturing ordinals
shortly.

But first let me describe just how far we want to go in this section. One way of
characterizing is that it is an ordinal a such that + a = a. This can be
seen by thinking of as being + + + + . . . Clearly, putting an " + "
in front of this symbol changes nothing. In point of fact, is the *first
*such ordinal.

What about the first ordinal a such that * a = a? If we take to be
*** . . . ,we can see that placing an " *" in front of
changes nothing. Or, assuming that the familiar laws of exponents hold for ordinals (and they do), we can reason that ,
since 1 + = . As before, it is possible to prove that is the
*first *ordinal a such that *a = a.

The first ordinal a such that = a is called ,
pronounced "epsilon zero." Simply by manipulating symbols, we would expect
that ** **has the following form:

But we can describe epsilon zero a little better. Suppose that is de- fined to mean "an exponentiated stack of a many b's." is pronounced,

*"b *tetrated to the a." The name tetration is used since ** tetra **is the
Latin root for

As further examples, which is just under eight trillion.
, which is the number that is written
by putting a one and then *ten billion *zeros (as opposed to, say, a million, which is
written by putting a one and then ** six **zeros.) Now, is just
). And , which is kind of hard to get at.
One way of visualizing this number is to go back to the picture of , and to
imagine replacing each of the dots on the line by the symbol

In any case), the point of all this is that is . And the countable ordinals don't stop there. If, for instance), one could make some sense of the following symbol one would have an even larger ordinal.

Absolute infinite is all about. Then you try to formalize your glimpse, and you end up with a new system for naming ordinals . . . which eventually peters out. . . . 5

But this is just the beginning. Let us look at a different type of picture of
. Suppose that we let *PN *be the set of all polynomials in *x
*with natural number coefficients. Examples of members of *PN *would be x, x + 3, +163. Now, suppose that given two
polynomials p(x) and *q(x) *from *PN, *we define the following order relation:
p(x) < bep q(x) if and only if the graph of the polynomial q
eventually manages to get above, and stay above, the graph of the polynomial *p.
*(The letters "bep" stand for "by-end-pieces.")

In Figure 42 1 have illustrated the fact that 0 < bep1 < bep2 < bepx + 1 < bep2x *x *tetrated to the three, ).

Strictly speaking *PPN *be the set of all pseudo-polynomials
formed by using natural number coefficients and tetration), as well as exponentiation),
then it is not difficult to see that *PPN *is when *PPN *is ordered by

We could really restrict our attention to functions from the natural numbers into the natural numbers. It is amusing to represent ordinal numbers as stacks of such functions, filling in the lines between graph points in the natural way. In order to make the pictures look nice, you leave out parts of the lines to avoid crossings.

The famous mathematician David Hilbert used to illustrate his popular lectures with
stories about a hotel with infinitely many rooms.8 This mythical hotel, usually called
Hilbert's Hotel, is supposed to have omega rooms: Room 0, Room 1, Room 2,... , Room
*n, *and so on. As in the last section, it is convenient to start counting with 0.

To fix the ideas, I have drawn a picture of Hilbert's Hotel in Figure 45. In order to fit it on the page, I have assumed that each floor is equipped with a science-fictional space condenser, a device that makes each succeeding story two-thirds as high as the one before. The shrinking field also affects the guests. Thus, although the ceilings on Floor 3 are only two or three feet high, the space condenser on that story shrinks the guests to one or two feet), and they are perfectly comfortable. 1 will leave it as an exercise for the reader to check that if the first floor is ten feet high, and each successive floor is two- thirds as high as the one before, then the total height of the hotel's stories is thirty feet.

One of the most paradoxical things about Hilbert's Hotel is that even after it fills up,
more and more people can be squeezed in, without making anyone share a room! Say,
for instance, that * *guests have arrived, and every room is occupied with a guest
*n *in each Room *n. *Now, say that one more guest arrives: Guest ** .
**How to fit him in?

Fine! But what if there had been an ** infinite **number of new guests?
Even such a procession of

To get a better idea of alef-one, let's go back to the idea of people climbing a mountain as high as all the ordinals. How hard will it be for them to get to alef-one?

We will assume that, in the marvelous land where Mount On rises, the climbers can
attain any desired finite speed. Then, as we mentioned at the beginning of the
"Transfinite Numbers" section, they can scale the infinity of cliffs leading up to *
*in a finite time. If they keep accelerating so that the nth cliff only takes 1/
hours, then they reach * *after two hours. By repeating this they could reach *
+ *in four hours, or, if they did everything four times as fast, they could actually get
to * *+ in one hour. Turning to Figure 46, we can see how our climbers could
even reach * *in one hour. The idea is to devote to each of the cliffs
the allotted finite interval of time. (Thus, for instance, the stretch between *2 and * *
*2 + 1 is to be covered during the time interval between 3/4 hour and **13/16
**hour.)

One last picture of alef-one. Go back to Figure 42. In this picture we saw how various
ordinals can be represented as sequences of functions ordered according to steepness
(the * f *in S that is steeper, then S must have
at least alef-one members.

it is now time to give a formal definition of alef-one. This definition hinges upon the
notion of cardinality. Given two ordinals *A *and *B, *we say that *A has the
same cardinality as B *if there exists a one-to-one map from *A *onto *B.
*(When I speak of a map from *A *onto *B, *I really mean a map from the set
of ordinals less than *A *onto the set of ordinals less than *B.)*

*
I*n our discussion of Hilbert's Hotel we learned, for instance, that * + *has
the same cardinality as . For there is a way of matching up, in a one-to-one fashion, the
members of {O, 1, 2,. . . *, *+ 1, * *+ 2,. . .} with the members of {0,1, 2, . . . }. That is, there is a one-to-one map from * + *onto *. *Now, alef-one is
defined to be the first ordinal with cardinality greater than *. *Alef-one is the first
ordinal that cannot be mapped one-to-one onto .

In general, we say that the ordinal number *A *is a *cardinal number *if
arid only if A does not have the same cardinality as any *B *less than *A *. All
of the natural numbers are cardinals. There is, for instance, no way to find a *one-to-
one *map from three onto two. (Keep in mind that we commonly identify the number
*N *with the set {0, 1,. . . , *N *- l} of N numbers less than *N.)*

*
*

The infinite cardinals are also called *alefs. *In general, alef-a, or , means
the ath infinite cardinal. Thus, alef-null, or , is just , the first (0th) infinite
cardinal. It turns out that just as we can always find more ordinals, we can always find
more cardinals. After come and so on. After

a while we can even find a number such that = . One way of getting such a is indicated below.

According to the Reflection Principle, it is impossible ever to conceive of an end to the
ordinals. We do have a symbol , the big , which we use to stand for the
Absolute Infinity that lies beyond all the ordinals. But is inconceivable. The Reflection
Principle makes this precise by saying that any description *D *of that we
might come up with will apply to some ordinal *a *short of .

is called Absolute Infinity because it is not a relative notion. The line of
ordinals leading out to contains *all *the ordinals, all the possible stages of
counting. It is because every possible ordinal occurs before that is not
really a definite ordinal number. Confusing talk! If you would like to read more about
and the transfinite numbers plan to take Excursion I.