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Georg Cantor and Infinity

Georg Cantor was an important mathematician who lived during the late 19th century.  He devised a method for comparing different sizes of infinite quantities and demonstrated that there were differnet sizes of infinite sets.

The foundation of Cantor’s ideas is the concept of a one-to-one correspondence.  This is a concept which is deeply rooted in the human psyche, as is shown by the fact that the word digit – meaning number – also means finger.  This one-to-one correspondence between enumerating objects by comparing them to the fingers and/or toes of the human body is the presumed origin of our base ten number system.

What Cantor did was to compare infinite sets by establishing a one-to-one correspondence bewteen the elements of the two sets.  If it was possible to establish a one-to-one correspondence, then Cantor concluded that the sets were the same size.  In 1873, Cantor published a paper showing that the rational numbers (which includes all fractions and whole numbers) is actually the same size as the counting numbers (1, 2, 3, …and so on).  Any set that is the same size as the counting numbers is called “countable.”

Another example of this is the set of even numbers.  The set of even numbers is the same size as the set of counting numbers.  It might seem as though there would be fewer even numbers, because the set of counting numbers includes all the odds, but the set of even numbers doesn’t!

But, you can create a one-to-one correspondence between the coutning numbers and the even numbers.

1 (in the counting numbers) matches up with 2 (in the even numbers)

2 (in the counting numbers) matches up with 4 (in the even numbers)

3 matches up with 6

4 matches up with 8 and so on.  Any counting number N will match up with 2N in the set of even numbers.

For every counting number, there is a corresponding even number – therefore the sets are the same size.  Cantor called this size – aleph null, the smallest infinite size.  Cantor showed that the Real Numbers, represented by the number line and including all rational and irrational numbers, are NOT countable – in fact, there are more Real Numbers than Counting Numbers.  Cantor’s famous Diagonal Proof is actually fairly simplistic.

Most mathematicians of the late 19th century hated Cantor’s ideas, and he didn’t receive much support intellectually while he was struggling with his work.  Cantor often became depressed toward the end of his life and spent time in and out of sanitariums, eventually dying in 1918.  By this time, however, many younger mathematicians looked up to Cantor and relied heavily on his work and ideas.

Today Cantor is regarded as one of the most important mathematicians to ever work.  His ideas regarding infinity are deeply philosophical and have had a profound impact on mathematics, science and philosophy.