# Essay, Research Paper: Infinity

## Mathematics

Free Mathematics research papers were donated by our members/visitors and are presented free of charge for informational use only. The essay or term paper you are seeing on this page
was not produced by our company and should not be considered a sample of our research/writing service. We are neither affiliated with the author of this essay nor responsible for its content. If you need high quality, fresh and competent research / writing done on the subject of Mathematics, use the professional writing service offered by our company.

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.

15

0

**Good or bad? How would you rate this essay?**

Help other users to find the good and worthy free term papers and trash the bad ones.

Help other users to find the good and worthy free term papers and trash the bad ones.

# Get a Custom Paper on Mathematics:

**Free papers** will not meet the guidelines of your specific project. If you need a custom **essay on
Mathematics: **, we can write you a high quality authentic essay. While **free essays** can be traced by Turnitin (plagiarism detection program),
our **custom written papers** will pass any plagiarism test, guaranteed. Our writing service will save you time and grade.

## Related essays:

2

1

**Mathematics**/ Looks Can Be Deceiving

Paradoxes are sometimes composed of contradictory ideas presented together,
ultimately leading to an unworkable situation. Paradoxes, however, are not
simply ambiguous questions. Paradoxes are the es...

4

1

**Mathematics**/ Modular Arithmetic

Modular arithmetic can be used to compute exactly, at low cost, a set of simple
computations. These include most geometric predicates, that need to be checked
exactly, and especially, the sign of det...

7

4

**Mathematics**/ Pascal`s Triangle

Blasé Pacal was born in France in 1623. He was a child prodigy and was
fascinated by mathematics. When Pascal was 19 he invented the first calculating
machine that actually worked. Many other people ...

8

7

**Mathematics**/ Pi Number

A little known verse in the bible reads “And he made a molten sea, ten cubits
from the one brim to the other; it was round all about, and his height was five
cubits; and a line of thirty cubits did c...

2

2

**Mathematics**/ Postulates And Theorems

P1-Ruler Postulate. P2-seg. add. postulate. P3-Protractor postulate. P4-angle
add. postulate. P5- A line contains at least two points; a plane contains at
least 3 points not all in one line; space co...