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Before the 19th century, the idea of the infinite was dismissed by mathematicians. Although they knew that there were infinite sets, they felt that nothing interesting could be said about them. Something was either infinite or not, and that was all that could be said.
Towards the end of that century, the German mathematician Georg Cantor (1845 - 1918) began to think somewhat about the idea of the infinite in mathematics. He tried, for example, to come up with a definition of the concept of the infinite. (Previously, such definitions as ‘bigger than any number’ were used, but that is in fact a circular definition, because the set of numbers is infinite, so that it amounts to a definition of infinite as ‘as big as infinite’.) He also set out to categorize infinite sets.
One of the first things that Cantor realized was that some infinite sets were bigger than others. He defined when two sets were to be of the same size: his definition was based on the intuitive behaviour of the numbers which were already familiar. Two sets are ‘equinumerous’ (equal in number) if there is a mapping between them which is a bijection, that is, a function where the two non-equal elements in the first set have non-equal images, and where every element in the second set has an element which the function maps to it. For example, the sets 1,2 and 5,7 are equinumerous, because the function mapping 1 to 5 and 2 to 7 is a bijection.
The smallest infinite set is that of all the natural numbers, and any set which is equinumerous with them is called countable or enumerable, because it is possible to write any such set as an infinite list (as the bijection between it and the natural numbers effectively gives you a first element, and a second element, and so on). Many kinds of numbers are countable, such as the integers and the rational numbers, while others are not (the way that the real numbers are shown to be uncountable is in algebraic numbers). It came as a big shock that there were such things as uncountable sets; previously it had seemed that you must be able to list the elements of any set. The reaction was so strong that many mathematicians condemned Cantor\'s results.
The same ideas give a non-circular definition of the infinite. An infinite set is defined to be one which is equinumerous with some subset of itself (other than the whole thing). For example, the set of natural numbers is equinumerous with the set of even numbers.
Today, however, the infinite is very much part of mathematics, and much of the work in set theory in this century has been to do with the various properties of infinite sets; see axiom of choice for a discussion of one of the most important. SMcL
Further reading R. Rucker, Infinity and the Mind. |
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