Controversy over Cantor's theory

Lore & Background

Cantor's first proof that infinite sets can have different cardinalities was published in 1874, demonstrating that the set of natural numbers and the set of real numbers have different cardinalities. This proof used the theorem that a bounded increasing sequence of real numbers has a limit, which could be proved using Cantor's or Richard Dedekind's construction of the irrational numbers. Because Leopold Kronecker did not accept these constructions, Cantor was motivated to develop a new proof.

In 1891, Cantor published a much simpler proof using his diagonal argument, which does not depend on considering the irrational numbers. This argument proves that there exists an infinite set with a larger number of elements than the set of natural numbers. Cantor generalized this argument to an arbitrary set A and the set of all functions from A to {0,1}, leading to Cantor's theorem: the power set P(A) has greater cardinality than A.

Cantor's set theory was controversial at the start, but later became largely accepted. Most modern mathematics textbooks implicitly use Cantor's views on mathematical infinity. For example, a line is generally presented as the infinite set of its points, and it is commonly taught that there are more real numbers than rational numbers.

Reader's Guide

Cantor's theory of infinite sets fundamentally changed mathematics by establishing that infinities come in different sizes. His diagonal argument and Cantor's theorem are cornerstones of set theory, showing that the power set of any set has a strictly greater cardinality than the set itself. This work, though initially controversial—particularly due to opposition from Leopold Kronecker—eventually became standard. Cantor's views on mathematical infinity are now implicitly used in most modern mathematics textbooks, such as treating a line as the infinite set of its points. The argument for Cantor's theorem uses only five axioms of set theory and has been refined over time. Cantor's later definition of 'greater than' correctly defined the concept without relying on his well-ordering principle, which he had earlier regarded as a 'law of thought'. His legacy endures in the widespread acceptance of transfinite numbers and the hierarchy of infinite cardinalities.

Did You Know?

• Cantor's first proof that infinite sets can have different cardinalities was published in 1874.
• Cantor's 1891 proof uses his diagonal argument and does not depend on considering the irrational numbers.
• Cantor's theorem implies that the power set P(A) has greater cardinality than A.
• The argument for Cantor's theorem uses only five axioms of set theory.