Cantorische

Cantorische is a German adjective meaning Cantorian, relating to the work of Georg Cantor and his theory of infinity. In English mathematical writing the equivalent term is Cantorian. In German-language texts Cantorische is used to qualify objects, results, or concepts associated with Cantor’s ideas.

Common examples include Cantorische Mengen, known in English as Cantor sets, and Cantors Diagonalisierungsbeweis, the Cantor diagonal argument. The diagonal argument demonstrates that the set of real numbers cannot be put into a one-to-one correspondence with the natural numbers, establishing the uncountability of the reals and a fundamental result in set theory.

The Cantor set is a standard construction: start with the closed interval [0,1], remove the open middle third, and repeat indefinitely. The resulting set is uncountable, perfect (every point is a limit point), nowhere dense, and has Lebesgue measure zero. The Cantor function, another Cantorian object, is a continuous, non-decreasing map from [0,1] to [0,1] that is flat on the intervals removed in the Cantor construction yet increases from 0 to 1; it serves as a classical example in real analysis and measure theory.

Beyond these, Cantor’s work laid the foundations for transfinite numbers, cardinalities, and the modern theory of infinity, introducing concepts such as aleph numbers and the continuum hypothesis. The term Cantorische thus encompasses a broad family of Cantorian ideas, objects, and methods used in mathematical discourse.