This page will, very briefly, discuss the concept of infinity as it is understood in mathematics.
In junior high, you may have learned that the "natural numbers" are those integers you get by starting from zero:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...
There is no biggest natural number. There are natural numbers so big that you wouldn't be able to live long enough to write them down, but they still exist in the mathematical sense, and they have all the properties of other natural numbers, including being able to be multiplied by other numbers and to be added to.
Thus, if you wanted to answer the question, "How many natural numbers are there?", you would have to say that the number of them is infinite.
But the number of even numbers is infinite. The number of integers starting from 3 is infinite.
If that isn't obvious, there is an easy way to show that it is true, by pairing off the numbers in these groups with the natural numbers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ... 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ...
So "infinity" isn't really a number like 25, since infinity/2 is still infinity, and infinity-3 is still infinity.
As a result, many great mathematicians, for example Karl Friedrich Gauss, took the position that while one can think of an object like the natural numbers always reaching towards infinite size, thinking about infinity as something that has been reached, as if it were existing in a finished form was contrary to the principles of mathematics.
Of course, one could just remember that certain rules don't apply to infinity. Projective geometry can be considered to be a mathematical discipline that takes this into account.
But it was the mathematician Georg Cantor who first tried to define infinity, and work out just what rules did apply to it. He did so by examining mathematical objects that were gradually developed over the entire course of the history of mathematics.
We've met the natural numbers. Other numbers have been added to mathematics. A debt can be represented, for accounting purposes, as a negative number, and for some purposes in electronics, an imaginary number can be used to represent a magnetic field as an electric field.
We will not be looking at these directional extensions to the natural numbers here. Instead, we are first interested in fractions. Fractions were known to the ancient Egyptians, although they handled them in a way that seems clumsy to us.
Integers let us count things; fractions let us measure out continuous substances by length or weight. A board can be five and three-sixteenths inches wide. A pitcher can contain one and a third quarts of milk.
Since a "fraction" of something also means a part of it, the numbers that include fractions in them are called the rational numbers. This includes numbers like one and one-half, or minus three and one-half, in addition to the fraction one-half.
The ancient Greeks discovered something very disturbing, though, about fractions.
To make a fraction, one divides a unit up into equal parts. These parts can be any integer number of parts, starting with 2: you can use halves, thirds, quarters, fifths, and so on. Or thousandths, or millionths. As we've seen, there is no limit to how big an integer can be.
So, it was natural to think that every distance along a line one foot long was some fraction of a foot. Certainly, it was true that you could get as close as you liked to any distance, just by making the parts smaller.
But the ancient Greeks discovered, as a result of their exploration of geometry, that some distances existed which could not be expressed exactly with fractions.
The length of the diagonal of a square is the square root of two times its side. This is a consequence of the Pythagorean theorem, which I won't prove here; but if you have any doubts, the diagram below should explain it:
What number multiplied by itself makes two?
If it was a fraction, then we would have the following:
a a 2 - * - = - , so a*a = 2*b*b. b b 1
But if a and b are both integers, then a must be even, and dividing a*a by 2 produces an even result, since that is (a/2), an integer, times a, an even number, and so b must be even, which means a/b is not - and can never possibly be - reduced to lowest terms. And that can always be done with any real fraction.
The square root of two is an algebraic number, and much later it was found that there were other possible numbers, called transcendental numbers that could correspond to exact distances on a line. Pi, the ratio of the circumference of a circle to its diameter, is an example of a transcendental number.
If we take the positive and negative forms of all the numbers that can represent distances or weights or volumes, we have the real numbers.
The real numbers "fill" a line; the rational numbers do not, yet however small a piece of the line you take, as long as it has some length, it will have rational numbers in it.
Does this mean there are more real numbers than rational numbers, or not? Clearly, the number of rational numbers between any two different numbers is already infinite, since there is no limit to how many small parts the space may be divided into.
Cantor answered these questions.
The next integer after 25 is 26. But there is no such thing as "the next rational number after 1/2". However close you look, you always find more rational numbers.
But that doesn't mean there are more rational numbers than there are natural numbers. By putting the rational numbers on a list in a different order, one can make them line up with the natural numbers, as follows:
N R N R ----------- --------- 0 0 11 4 1 1 12 -4 2 -1 13 1 1/2 3 2 14 -1 1/2 4 -2 15 2/3 5 1/2 16 -2/3 6 -1/2 17 1/5 7 3 18 -1/5 8 -3 19 5 9 1/3 20 -5 10 -1/3 ...
Each number is followed by its negative version, except zero, which doesn't have a different negative version. The fractions, if considered as improper fractions, are ordered by the sum of the numerator and denominator, starting with the fraction with the largest numerator: thus, we go through 4/1, 3/2, 2/3, 1/5 in order. (Earlier, 2/2 was omitted, since that equals 1, which is already on the list.)
Could we make a list of all the real numbers that way? Cantor is famous for finding a proof that we couldn't.
No matter what our list looks like, it would fall prey to this problem, illustrated by an attempt to list all the real numbers on the half open interval [0,1), which means all real numbers greater than or equal to zero and also less than one:
.27098611325... .40213927064... .08113961678... .53719442618... .82122547689... .17218943291...
No number that differs in every decimal place from
.201129...
the number you get by running down the diagonal of this list, could possibly be on the list.
There is a slight oversimplification here: it is possible to write .499999.... as a real number, but that is actually the same real number as .5. This problem can be dealt with, though, so it doesn't spoil the proof.
Since there are ten possible values for any decimal digit, one could just choose a digit from 1 to 8 when making a number different in every position from the diagonal number, thus ensuring one has made a number which can't have a duplicate written a different way.
Or one could be even more tidy, by coding the numbers with duplicate representations to all the strings of digits involved in the problem, for example, like this:
number digit string number digit string ------ ------------- ------ ------------- 0 .0 .15 .7 .1 .9999999... .16 .6999999... .2 .1 .17 .8 .3 .0999999... .18 .7999999... .4 .2 .19 .9 .5 .1999999... .21 .8999999... .6 .3 .22 .11 .7 .2999999... .23 .1099999... .8 .4 .24 .12 .9 .3999999... .25 .1199999... .11 .5 .26 .13 .12 .4999999... .27 .1299999... .13 .6 .28 .14 .14 .5999999... .29 .1399999...
and a similar technique allows us to carry out the Cantor diagonal proof even using base 2 radix fractions.
This meant that the real numbers have a higher cardinality than the natural numbers, while the rational numbers have the same cardinality as the natural numbers.
If you consider the set of subsets of the natural numbers, including all infinite subsets, this set has the same cardinality as the real numbers. This was proven by Cantor by noting that if you wrote a real number in binary form, and used 0 for the first digit after the decimal point if 0 was not in the set, but 1 if it was, and used 0 for the second digit after the decimal point if 1 was not in the set, but 1 if it was, and so on, then you would have exactly one string of digits corresponding to each possible subset of the natural numbers. (Again, the problem that .1000... and .0111... are numerically equal can be solved by coding strings to numbers.)
Simple geometrical transformations can show that all the real numbers have the same cardinality as the real numbers in a short line segment.
Also, it can be shown that the number of points in a cube is the same as the number of points in a line segment, again by looking at the strings of digits that represent them:
the three real numbers that correspond to a point in a cube extending over the range [0,1) along each axis:
X: .23149602... Y: .78029152... Z: .53144026...
can be interleaved to form just one number, corresponding to a point on the line segment [0,1):
.275 383 101 424 994 610 052 226...
When we deal in infinities, instead of just having one kind of "size", that concept seems to divide into three tiers:
Cantor proved that if you started with the number of natural numbers, the most basic kind of infinity, which he called aleph-null, by taking the set of all possible subsets of the natural numbers, you created a set of higher cardinality. And he went on to prove that if you repeated this operation, taking the set of all possible subsets of the set of all possible subsets of the natural numbers, you went to a set of still higher cardinality, and so on, forever. Or, to be precise, aleph-null times.
Some people suggested that he ought to include in his system the infinity that would result from doing it a number of times with a higher cardinality, but since Cantor did not feel he knew how to give meaning to such cardinalities, or even to show that they really existed, he resisted this. The book Infinity and the Mind, by Rudy Rucker is a popular book which, among other things, discusses some of the larger infinities of this type, which other mathematicians have later discussed, and which have names such as inaccessible cardinals, Mahlo cardinals, and hyper-Mahlo cardinals.
In addition to calling the simplest infinity aleph-null, he used the term aleph-one to refer to the next higher kind of infinity, aleph-two for the one after that, and so on.
Although he could not find an infinity between the number of natural numbers and the number of real numbers (called the cardinality of the continuum, and sometimes noted as a Gothic-style c), he could not prove there wasn't one.
By 1963, it was proved that whether or not there was such an intermediate cardinality (whether or not c failed to equal aleph-one) most of set theory would still work. One axiom of set theory, called the Axiom of Choice, would have to be true if the Generalized Cantor Hypothesis was true: the Generalized Cantor Hypothesis is that there are no distinct forms of infinity between any of the cardinalities formed by repeatedly taking the set of all possible subsets of a set of the previous cardinality.
While an infinity between that of the natural numbers and the real numbers seems like something bizarre, the strict Cantorian view of sets would also lead to some unusual consequences. One, called the Banach-Tarski paradox, states that it should be possible to dissect a volume into a finite number of parts (each one, however, of infinite complexity) and put it back together to fill a larger volume. And it is believed that we will never be able to decide which view of sets is true by exhibiting either unusual phenomenon.