Infinity

The term infinity conveys the mathematical concept of large without bound, and is given the symbol ∞. As children, we learn to count, and are pleased when first we count to 10, then 100, and then 1,000. By the time we reach 1,000, we may realize that counting to 2,000, or certainly 100,000, is not worth the effort. This is partly because we have better things to do, and partly because we realize no matter how high we count, it is always possible to count higher. At this point we are introduced to the infinite, and begin to realize what infinity is and is not.

Infinity is not the largest number. It is the term we use to convey the notion that there is no largest number. We say there is an infinite number of numbers.

There are aspects of the infinite that are not altogether intuitive, however. For example, at first glance there would seem to be half as many odd (or even) integers as there are integers all together. Yet it is certainly possible to continue counting by twos forever, just as it is possible to count by ones forever. In fact, we can count by tens, hundreds, or thousands, it does not matter. Once the counting has begun, it never ends.

What of fractions? It seems that just between zero and one there must be as many fractions as there are positive integers. This is easily seen by listing them, 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8,.... But there are multiples of these fractions as well, for instance, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, and 8/8. Of course many of these multiples are duplicates, 2/8 is the same as 1/4 and so on. It turns out, after all the duplicates are removed, that there is the same number of fractions as there are integers. Not at all an obvious result.

In addition to fractions, or rational numbers, there are irrational numbers, which cannot be expressed as the ratio of whole numbers. Instead, they are recognized by the fact that, when expressed in decimal form, the digits to the right of the decimal point never end, and never form a repeating sequence. Terminating decimals, such as 6.125, and repeating decimals, such as 1.333¯ or 6.534¯ (the bar over the last digits indicates that sequence is to be repeated indefinitely), are rational. Irrational numbers are interesting because they can never be written down. The instant one stops writing down digits to the right of the decimal point, the number becomes rational, though perhaps a good approximation to an irrational number.

It can be proved that there are infinitely more irrational numbers than there are rational numbers, in spite of the fact that every irrational number can be approximated by a rational number. Taken together, the rational and irrational numbers form the set of real numbers.

The word infinite is also used in reference to the very small, or infinitesimal. Consider dividing a line segment in half, then dividing each half, and so on, infinitely many times. This procedure would results in an infinite number of infinitely short line segments. Of course it is not physically possible to carry out such a process; but it is possible to imagine reaching a point beyond which it is not worth the effort to proceed. We understand that the line segments will never have exactly zero length, but after a while no one fully understands what it means to be any shorter. In the language of mathematics, we have approached the limit.

Beginning with the ancient Greeks, and continuing to the turn of the twentieth century, mathematicians either avoided the infinite, or made use of the intuitive concepts of infinitely large or infinitely small. Not until the German mathematician, Georg Cantor (1845-1918), rigorously defined the transfinite numbers did the notion of infinity finally seem fully understood. Cantor defined the transfinite numbers in terms of the number of elements in an infinite set. The natural numbers have u₀ elements (the first transfinite number). The real numbers have u₁ elements (the second transfinite number). Then, any two sets whose elements can be placed in 1-1 correspondence, have the same number of elements. Following this procedure, Cantor showed that the set of integers, the set of odd (or even) integers, and the set of rational numbers all have u₀ elements; and the set of irrational numbers has u₁ elements. He was never able, however, to show that no set of an intermediate size between u₀ and u₁ exists, and this remains unproved today.