ABSTRACT

The main reason for Georg Cantor's introduction of set theory was to provide a framework for tying down notions of infinity. Cantor's theory turned out with hindsight to have made heavy use of the axiom of choice, hardly surprising if dealings with infinite sets are involved. So it is customary to distinguish between those parts which use axiom of choice (AC) and those which do not. The theory with AC is richer, but even without it one can derive many of Cantor's most remarkable results. Ernst Schr0der's proof at around the same time contained an error which was ultimately corrected. Bernstein's proof, published in 1898, was the first correct proof avoiding use of AC. Some require a piece of mathematical knowledge from outside the confines of set theory. A complex number is said to be transcendental if it is not algebraic.