How big is infinity? Does infinity really exist?

Authors Avatar

Infinity

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,

Join now!

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 ...

This is a preview of the whole essay