A Surreal Land :: Mathematics Papers

A Surreal Land "To infinity and beyond!" These were the inspired words of Buzz Lightyear in the Disney movie Toy Story. Granted, one would not expect to find much mathematical content in an animated film directed toward children, but these words raise an interesting issue that mathematicians and the general public struggled with for many years. Can one go beyond infinity? How can such a concept be possible or even imaginable? These questions led to the development of many new theories and even a new system of numbers. Disney's Buzz Lightyear A study of infinity must begin with an introduction to set theory. A set is merely a collection of objects. Georg Cantor was the sole creator of set theory; he published an article in 1874 that marks the beginning of set theory and has come to change the course of mathematics. Cantor's theory was met with a great deal of opposition due to its assertion of infinite numbers. The famous mathematician Leopold Kronecker was especially opposed to Cantor's revolutionary new way of looking at numbers. Kronecker believed only in constructive mathematics, those objects that can be constructed from a finite set of natural numbers. Despite this opposition from influential thinkers, set theory laid the foundation for twentieth century mathematics. Although there were some flaws in Cantor's theory, sets became an essential part of new mathematics and therefore set theory was adapted to eliminate its original paradoxes [2]. Georg Cantor Cantor's set theory incorporated infinity in the form of infinite cardinal numbers. Cardinal numbers are those which measure the number of objects in a set, as opposed to ordinal numbers, which are numbers with a fixed predecessor and successor. If two sets have equal cardinality, then they contain an equal number of objects. One way to determine this is through "one-to-one correspondence." Two sets are said to be in one-to-one correspondence if each object, or element, of the first set can be paired with exactly one element of the second set, and vice versa [1]. Cantor compared the cardinality of the set of all positive even integers to the set of all positive integers and found them to be equal. Thus the infinite cardinal number of these two sets is the same and is defined to be aleph-naught. This is the first transfinite number. The set of rational numbers also has a cardinality of aleph-naught, and thus is the same size as the set of integers.