- 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]
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