Infinity

A detailed Summary of 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 be

elements in the set, but Cantor took this a step farther. Because it is impossible to count

arrow is in one and only one position at each and every instance of time; in other words, at

any list of real numbers, no list can include all of the reals. Therefore, the set of all real

Another argument that he stated was that, " If Achilles (a Greek Godlike person)

will go on infinitely with a 1-1 correspondence. Certain infinite sets are not 1-1, though.

the following table shows. Position Achilles Turtle 1 0 1000 2 1000 1100 3 1100 1110 4

step farther, though. He said that there is a 1-1 correspondence between the set of positive

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Approximate Pages = 4 (250 words per page double spaced)

Category: Miscellaneous

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