**Burkman wrote:****Bobabounty wrote:****Burkman wrote:**

-13

Total = -10

+∞

x0

I win.

Soon after we

learn to count in our early childhood we begin to consider the task of

seeing how high we can count. Much like trying to hold our breath

forever, it is not long afterwards that we learn of the futility and

contemplate the idea of infinity, the

seemingly largest number that can ever be imagined. That supposed

number symbolized by a strange sideways-8 symbol infinity, like

a Moebius strip with a neverending surface. Most of us leave it at

that never pondering if there is more to it, or even realizing that we

as mere humans are allowed to contemplate infinity. It turns out that

infinity has an incredibly rich and structured existance. That infinity symbol may be fine for

casual use or even modern physics, but it is woefully imprecise and

ill-defined for mathematics. For that a different symbol will become

important, the aleph, aleph, the first letter of the

Hebrew alphabet.

The modern concepts of

infinity are primarily due to the lifelong compassion of German

mathmatician Georg Cantor (1845-1918).

His theory of the transfinite cardinal

numbers has demystified the wonderfully rich structure and

complexity which exists within the realm of infinity. He dared to

leave the mental safety of the potential

infinity discussed by his pears and instead mused upon the

actual infinity itself, as something that

really does exist.

The difficulties [with] infinity

depended upon adherence to one definite axiom, namely, that a whole

must have more terms than a part...

--Bertrand

Russel

The concept of infinity in one form or another

permeates almost every branch of modern mathematics, most areas of

philosophy, and has even had revolutionary impacts on religion (much

like the idea that the Earth is not the center of the Universe).

Infinity has plagued the greatest minds in those fields with

perplexity and and uncertainty, and in Cantor's case flirtation with

insanity. It has brought into question the whole of axiomatic

geometry due to it's use in Euclid's Parallel

Postulate. The fields of integral and differential calculus

are still primarily taught from the perspective of Leibiniz's theory

of the infinitesimal (infinitely small), despite questionable

reasoning about the properties of actual and potential infinities.

Almost all logical paradoxes revolve about the subtleties of infinity.

Surely, the notion of infinity is one of the greatest musings of man,

always lying just beyond our mastery.

Cardinality

When we stay within the finite world we

easily understand how to measure the size of a set, namely how many

members the set contains. Cardinality is

the term mathematicians use for this size, or as Cantor described it,

the set's power. The cardinality of a set m_set can be

represented by the symbol m_card. The idea of the double

overbar is that the concept of cardinality is a double abstraction or

generalization. The first abstraction, m_ord, called the

ordinal number, is an abstraction from the

nature of the elements of the set. For instance, it doesn't matter

whether we have a set of letters, a set of books, or a set of

vegetables. The second abstraction, m_card, is that

of ignoring the order of things. For instance the set {1,2,3} has the

same cardinality as the set {2,3,1}, or for that matter as the set

{red, green, blue}. Two sets are said to be similar if they have the same cardinality.

For finite sets we can represent the

cardinality using ordinary natural numbers. If some set contains five

different elements then we can say that the cardinality is 5.

But for infinite sets we need a new kind of number, the transfinite number, meaning "beyond finite".

If we take the set of all the natural

numbers, 0, 1, 2, and so on indefinitely, then it is easy to see that

this set is infinite in size. The cardinality of this set is

represented by the symbol aleph 0 (aleph-zero), where the

subscript is meant to indicate that this is the least transfinite cardinal. In other words there

can be no other infinite set which would have a smaller cardinality;

for if there were it would cease to be infinite and would instead be

finite. It may not be obvious yet whether there exist any other

transfinite cardinal numbers, but we will soon learn that there

are.

A major property of all infinite sets

with cardinality aleph 0 is that they are denumerable. When we also include all finite sets

this property is called countable.

Informally this means that there exists a well defined process by

which we can count or enumerate all of the elements of the set. It

does not matter whether this process ever terminates, but simply given

any single element of the set at random we are always guaranteed to

eventually enumerate or reach it. No matter how large an integer you

pick, if we start counting at zero and continue up one by one, then we

will eventually reach the number selected.

Rational Numbers and Other Denumerable Sets

We have already seen how the set of all

the natural numbers is itself denumerably infinite. We may then

consider other kinds of infinite sets and ask what their cardinality

may be. Consider the set of all rational numbers, those numbers which

may be represented by fractions or ratios, x/y, of two integers.

The rational numbers are in some sense quite a bit different from the

natural numbers. One important property is that of being everywhere dense. This means that no matter what

two rational numbers you pick, no matter how close to each other,

there will always be more rational numbers between them. In fact,

there is always an infinite number of rationals between any other

pair. This denseness certainly does not hold for two natural numbers.

You may then be surprised to learn that the cardinality of the

rational numbers is exactly the same as that of the natural numbers;

in effect, there are the same number of rational numbers as there are

whole numbers, no more, no less.

Diagonalization of the Rational NumbersThis result at first sight just

doesn't seem right. There are after all an infinite number of

rationals between just 0 and 1. How then can we be expected to

believe that there are as many whole numbers as there are rationals?

One way to demonstrate this is through a process called

diagonalization. This demonstration centers on a technique of

arranging the rationals in a table, part of which is shown at the

right. We arrange the rationals in a two-dimensional grid where we

vary the numerator along one axis and the demoninator along the other.

Clearly if this grid is extended out in both directions infinitely

then it will include every possible rational number. We can then

enumerate the rationals by following the pattern shown by the blue

arrows, going up and down each diagonal. As you can see no matter

what rational number you pick, we will eventually reach it. Note that

although our diagram includes some duplicate rational numbers (such as

1/1 and 2/2), the more formal proof shows that they do not change our

final conclusion: that the cardinality of the set of all rational

numbers is aleph 0. This rather unexpected

result is but just one of the ways in which the nature of infinity

will surprise us.

Continuity and the Real Numbers

Consider the set of all irrational

numbers, a subset of the real numbers excluding any whole or rational

numbers. The set of all irrational numbers is not denumerable. This

means that we finally have an infinite set whose cardinality is

not aleph 0.

This can be shown by using another

diagonalization technique. First, to simplify things lets just

concentrate on all the irrationals between 0 and 1. Now lets

assume that the irrationals Attempted diagonalization of all the real numbers are denumberable; this would then mean

that there would be some way to list all of them such that we would

not leave any number out. Part of one such example list appears on

the diagram at the left. Since this is clearly an infinte set and

since the decimal representation of every irrational must contain an

infinte number of digits, the diagram really extends indefinitely in

both directions. Now take each digit along some diagonal of this

list; in this example 0.13497.... This then gives us one of the

irrational numbers. But if we now alter every single digit, say by

adding 1 (wrapping 9's back to 0's), then we get another irrational

number 0.24508.... But this new number can not possibly appear

anywhere in our list since it must always have at least one digit with

the wrong value no matter which row we may compare it against. We

have just produced an irrational number which is not in our list, and

therefore our original assumption that the irrational numbers are

denumerable must be wrong.

At this point we have discovered some

new level of infinity, and infinity which is somehow greater than the

infinity of all the integers. For now lets call this new transfinite

cardinal number by the symbol r_card. We also have the

relation aleph 0 < r_card. It was Cantor's

believe that this was the next immediate cardinal number, that is

r_card = aleph 1, but before we can

discuss whether that is true or not we must discuss a little more

theory.

Before leaving our discussion of the

irrational numbers it is worth taking a look at what we have before us

now. The set of irrational numbers, or sets which are isomorphic to

it, is traditionally called The Continuum.

We saw how the set of rational numbers had the property of being

everywhere dense. What sets the irrational numbers apart is the

property of continuity, or being continuous.

Many of us have a strange sense for what continuous means but would

struggle to express the rules of continuity. Those who have studied

integral calculus or the theory of functions may have been introduced

to the concept in terms of some infinitesimally small error factor,

usually labeled epsilon. Although that definition serves the limited

needs of the various applied math and engineering fields well, there

is in fact a more fundamental definition of being continuous.

A natural question to ask is what is the

cardinality of continuous spaces of different dimensions. Most of us

are quite confortable considering the sequence of real numbers being

expressed as the points on a continuous line, a 1-dimensional space.

We may thus say that the number of points on a line is r_card. But consider higher

dimensions: the 2-dimensional plane or the 3-dimensional Cartesian

space.

Spaces of different dimensions

Clearly it would seem that as we keep

adding another dimension the number of points would be of a higher

power, a larger level of infinity. After all there are an uncountably

infinite number of lines within a plane, and a uncountably infinite

number of planes in a 3-dimenisional volume. But here once again

infinity surprises our intuiition. The entire collection of points

within a space of any dimension, an n-dimensional space, as

long as n is finite is the same number of points as we find

along a simple volumeless straight line, that of r_card. The proof is actually

quite simple. Take a point on a two-dimensional plane (x,y). We can

take the digits which we would use to write down x and y and simply

interleave them.

Proof of dimensional similarity

This interleaving technique results in

real number for every possible point, and no two points on the plane

map to the same number. This same argument can be extended to any

number of dimensions, as long as we have a finite number of

dimensions.

We have already seen something like this

before; the previous diagonalization technique we used to enumerate

the rational numbers. We had essentially constructed the rationals by

creating a 2-dimenisional plane of integers. A general conclusion is

that the concept of dimension has no effect on the size or cardinality

of an infinite space; dimensions are cardinally meaningless.

The Continuum Hypothesis

As Cantor proved there is a series of

alephs each larger than then previous. This leads one to question

whether this series is itself complete or if there may be other levels

of infinity which lie between the alephs. The conjecture that there

are no intermediate infinities is most famously stated by Cantor's

Continuum Hypothesis, expressed by the

equation:

continuum hypothesis[center]