As we all know, "infinity" is not a number: most people will say it is concept, not a number. But what sort of concept? What are its parameters? Most people cannot give a definition, but will say something imprecise, like "its the thing bigger than all the numbers" or "it what you get when you put all the numbers together," or, worse, "it's what you get when you divide by zero."
The person who showed us how to put away all this imprecise, silly nonsense was Georg Cantor. Born to a German merchant family in St. Petersburg in 1845, sixteen years before the emancipation of the serfs in Russia, and about the time Tolstoy was flunking out of college. But he wouldn't spend his life in Russia; at 11 years of age, his family returned to pre-unification Germany.
His family, like many from the non-noble classes with higher aspirations, took advantage of the advancement opportunities that the expanding economies and liberalizing mentalities of 19th century Europe offered. Because his family was modestly wealthy, they had the opportunity to enroll Georg in schools teaching broad curricula in the humanities and science, as opposed to the trade/training schools most would attend in his day. These were the Realschule of aristocratic, pre-Bismarckian Germany.
He showed considerable skill in mathematics, and found himself, with his family's backing, able to attend the top schools (rubbing elbows with the likes of Kronecker and the great Karl Weierstrass). He finally landed a spot at Halle, a university that dates to 1502.
Cantor single-handedly developed the notion of cardinality, today regarded as a cornerstone mathematics---this notion is as valuable to modern mathematics as division is.
Definition: Two sets have the same cardinality if there is a one-to-one correspondence between them.
Now there are one-to-one correspondences between sets of finite size, provided the number of elements in the sets are the same: any two sets of fifty elements, say, have the same cardinality. For instance, there is a (very natural) one-to-one correspondence between the States in the US and the junior senators in the US senate.
In math, the notion of cardinality correlates to some, but not all, of our intuitive ideas about "amount." Take for instance the natural numbers and the primes. It would seen that there are far fewer primes than there are numbers, right? Well there is a one-to-one correspondence:
N. Number | Prime
---------------------------------
1 | 2
2 | 3
3 | 5
4 | 7
5 | 11
6 | 13
7 | 17
etc. | etc.
Each natural number appears once and only one on the left side of the list, and each prime appears once and only once on the right side. The list itself demonstrates a one-to-one correspondence. Thus the cardinality of the primes is the same as the cardinality of the natural numbers.
Other one-to-one correspondences are possible, such as between the natural numbers and the integers:
N. Number | Integer
---------------------------------
1 | 0
2 | 1
3 | -1
4 | 2
5 | -2
6 | 3
7 | -3
8 | 4
9 | -4
etc. | etc.
In this correspondence, we can actually write down a formula: the nth natural number corresponds to the integer n/2 if n is even, and to -(n-1)/2 if n is odd.
There are also many correspondences between natural numbers and fractions. The correspondence below is NOT one-to-one:
N. Number | Fraction
---------------------------------
1 | 1
2 | 1/2
3 | 2/2
4 | 1/3
5 | 2/3
6 | 3/3
7 | 1/4
8 | 2/4
9 | 3/4
10 | 4/4
11 | 1/5
etc. | etc.
Why? There are repeats. We can make a one-to-one correspondence between the natural numbers and all fractions (strictly) between 0 and 1 by removing all the repeats. We obtain
N. Number | Fraction
---------------------------------
1 | 1/2
2 | 1/3
3 | 2/3
4 | 1/4
5 | 3/4
6 | 1/5
7 | 2/5
8 | 3/5
9 | 4/5
10 | 1/6
11 | 5/6
12 | 1/7
13 | 2/7
etc. | etc.
In class we demonstrated a one-to-one correspondence between the natural numbers and ALL positive fractions. It would seem, so far, that there is only one infinity.
But Cantor showed us otherwise. His proof was by contradiction: that is, assume there IS a one-ton-one correspondence between, say, all natural numbers and all real numbers between 0 and 1. This would mean there is some list of the sort
N. Number | Fraction
---------------------------------
1 | .123456789123...
2 | .101001000100001.....
3 | .1112131415161......
4 | .141518191819.....
5 | .1111111111111.....
6 | .8989898989898989....
7 | .0000001000000001000000.....
etc. | etc.
Given such a list, one first forms the diagonal decimal number: the number whose first digit is the first digit of the first number, whose second digit is the second digit of the second number, and so forth. For the list above, the diagonal is .10155191....
Then one create the modified diagonal, in which each digit of the diagonal is increase by 1, except the digit "9" which becomes the digit "1." In our case, the modified diagonal is .21266202....
It is a simple matter to show that the modified diagonal is not on the list: it differs from the first number on the list in the first digit, from the second number in the second digit, and so on.
Therefore, given any list of decimals, you can find a number that is not on the list. This means there is NO complete list of decimals, and therefore NO one-to-one correspondence between the decimals and the natural numbers.
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