Do They Have the Same Cardinality?
Does the set of real numbers have the same cardinality as the set of natural numbers? The cardinality of each of the sets is infinity. So, therefore, they have the same cardinality, right? Well, that is actually wrong. Georg Cantor, a German mathematician, proves this. He proved that infinity is not one size but that some infinities are more infinite than others. Cantor focused on what it would mean for the real numbers and the natural numbers to have the same cardinality. It means that they need to have one-to-one correspondence. He came up with a way to prove that they do not have a one-to-one correspondence. The cardinality of the set of real numbers is different than the cardinality of the set of natural numbers.
The main idea behind Cantor’s theory is that each real number can be expressed as an infinitely long decimal expansion. For example, 943.21342455764523555665342... is a real number. After observing this real number, it is obvious that it is infinitely long. However, if the third place after the decimal point is changed from a 3 to a 6, then the real number is a completely different real number and it also is infinitely long. This can be done wit any infinitely long, real, decimal number. So there is an infinite amount of real numbers and there is also an infinite amount of infinitely long decimal expansions of each of the infinite amount of real numbers.
Cantor proved his theory through a simple idea. His strategy was to attempt to list all of the natural numbers in one column and all of the real numbers in another column. If they had the same cardinality, then it would be possible to show a one-to-one correspondence between them. But then he would show a real decimal number that has not appeared anywhere on the infinite list. Another way to look at it is if we have all the natural numbers in one barrel and all of the real numbers in another, and remove...