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

whose 1st decimal place is different from the first decimal place in the first number,

Canter determined that the set of real numbers is uncountable, and they therefore can not

with {2, 4, 6, 8, 10}. Sets such as these are countable finite, which means that it is

decimal. Show that there is a real number r that is not on the list. Let r be any number

everyone’s satisfaction. This is one of the few areas that mathematics and science may

. Normally, cardinality has been detected by counting the number of

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

of transition our senses picture. All of these examples are that Cantor attempted to

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