History

The question of the infinite has plagued mankind for centuries. In Western mathematical history, the Greeks were first to address the issue. ''Infinity'' was an uncomfortable subject for them, because a number without limits is imperfect. In viewing the world around them, however, they thought that time and space are unending. Infinite (or infinitely small) quantities were debated; for example, some believed that when dividing matter, eventually you would reach a particle that could not be divided, while others felt that you could continue cutting particles indefinitely. Paradoxes were invented, such as the famous Zeno's Paradox.

The Greeks used the concept of infinity as they developed geometry, but to avoid the controversial ''number,'' they allowed infinity to be potential rather than actual--that is, it could always be expanded, but never completed; you can always add 1 to a number to get a greater number, but you couldn't treat ''infinity'' as an entity.

With time, the infinite would remain controversial, but continue to be used as new mathematical ideas were developed. Limits and expansions of series relied on the idea of the unending. ''Infinitesimals'' (infinitely small parts) were essential to the creation of calculus. But critics were uneasy with the seemingly inconsistent use of infinity as quantity to be used in calculations, or dropped as insignificant when convenient. In 1544, the irrationals were declared not to be real numbers because they are ''hidden in a cloud of infinity.'' Renowned mathematician Gauss opposed the use of infinity because it was nothing more than a ''manner of speaking.''

Then Cantor came onto the scene, and while his work settled age old mysteries for many people, it did not remove the controversy involved with the infinite.