Infinity
Most everyone is familiar with the infinity
symbol, the one that looks like the
number eight tipped over on its side.
Infinity sometimes crops up in everyday
speech as a superlative form of the
word many. But how many is infinitely many?
How big is infinity? Does
infinity really exist? You can't count to infinity.
Yet we are
comfortable with the idea that there are infinitely many numbers to
count
with; no matter how big a number you might come up with, someone else
can
come up with a bigger one; that number plus one, plus two, times two, and
many
others. There simply is no biggest number. You can prove this with a
simple
proof by contradiction. Proof: Assume there is a largest number, n.
Consider
n+1. n+1*n. Therefore the statement is false and its contradiction,
"there is
no largest integer," is true. This theorem is valid based on the
"Validity
of Proof by Contradiction." In 1895, a German mathematician by the
name of
Georg Cantor introduced a way to describe infinity using number
sets. The number
of elements in a set is called its cardinality. For example,
the cardinality of
the set {3, 8, 12, 4} is 4. This set is finite because it
is possible to count
all of the elements in it. Normally, cardinality has
been detected by counting
the number of elements in the set, but Cantor took
this a step farther. Because
it is impossible to count the number of elements
in an infinite set, Cantor said
that an infinite set has No elements; By this
definition of No, No+1=No. He said
that a set like this is countable
infinite, which means that you can put it into
a 1-1 correspondence. A 1-1
correspondence can be seen in sets that have the
same cardinality. For
example, {1, 3, 5, 7, 9}has a 1-1 correspondence with {2,
4, 6, 8, 10}.
Sets such as these are countable finite, which means that it is
possible to
count the elements in the set. Cantor took the idea of 1-1
correspondence a
step farther, though. He said that there is a 1-1
correspondence between the
set of positive integers and the set of positive even
integers. E.g. {1, 2,
3, 4, 5, 6, ...n ...} has a 1-1 correspondence with {2, 4,
6, 8, 10, 12,
...2n ...}. This concept seems a little off at first, but if you
think about
it, it makes sense. You can add 1 to any integer to obtain the next
one, and
you can also add 2 to any even integer to obtain the next even integer,
thus
they will go on infinitely with a 1-1 correspondence. Certain infinite
sets
are not 1-1, though. Canter determined that the set of real numbers
is
uncountable, and they therefore can not be put into a 1-1 correspondence
with
the set of positive integers. To prove this, you use indirect reasoning.
Proof:
Suppose there were a set of real numbers that looks like as
follows 1st
4.674433548... 2nd 5.000000000... 3rd 723.655884543... 4th
3.547815886... 5th
17.08376433... 6th 0.00000023... and so on, were each
decimal is thought of as
an infinite decimal. Show that there is a real
number r that is not on the list.
Let r be any number whose 1st decimal
place is different from the first decimal
place in the first number, whose
2nd decimal place is different from the 2nd
decimal place in the 2nd number,
and so on. One such number is r=0.5214211...
Since r is a real number
that differs from every number on the list, the list
does not contain all
real numbers. Since this argument can be used with any list
of real numbers,
no list can include all of the reals. Therefore, the set of all
real numbers
is infinite, but this is a different infinity from No. The letter c
is used
to represent the cardinality of the reals. C is larger than No. Infinity
is a
very controversial topic in mathematics. Several arguments were made by a
man
named Zeno, a Greek mathematician who lived about 2300 years ago. Much
of
Cantor’s work tries to disprove his theories. Zeno said, " There is
no
motion because that which moved must arrive at the middle of its course
before
it arrives at the end. And, of course, it must traverse the half of
the half
before it reaches the middle, and so on for infinity." Another
argument that
he stated was that, " If Achilles (a Greek Godlike person) can
run 1000 yards
a minute, he will never overtake a turtle that runs 100 yards
a minute." Once
Achilles has advanced 1000 yards, the turtle is 100 yards
ahead of him. By the
time Achilles covers these 100 yards, the turtle is
still ahead of him, and so
on into infinity, as the following table shows.
Another argument he gives is the
one of the arrow in flight. He said, "The
tip of an arrow is in one and only
one position at each and every instance of
time; in other words, at every
instance of time, it is at rest. Hence it
never moves." Zeno assumes that a
finite part of time consists of a finite
series of successive instances.
Throughout an instance, he says, the tip
of the arrow is at one point. Imagine a
period consisting of 1,000,000 small
instances, and picture the arrow in flight
during the period. At each of the
1 million instances, the arrow is where it is,
and at the next instance, it
is somewhere else. It never moves, but somehow
accomplishes the change of
position. Thus, motion is an illusory, irregular sort
of thing-a succession
of stills, like a movie-not the smooth sort of transition
our senses picture.
All of these examples are that Cantor attempted to disprove
by forming his
own infinity theories. As of now, infinity is a tentative area
in
mathematics, because certain concepts involved with it have not of yet
been
proven to everyone’s satisfaction. This is one of the few areas
that
mathematics and science may never be able to explain completely,
because
infinity can not be measured in the classic sense.