mængdelæren

Mængdelæren, or set theory in English, is a fundamental branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything, such as numbers, points, or even other sets. The basic idea of a set is intuitive, but its formalization led to significant developments in mathematics.

The founding figure of mængdelæren is the German mathematician Georg Cantor, who developed it in the late 19th century. Cantor introduced concepts like the cardinality of a set, which measures the number of elements it contains, and showed that infinite sets can have different sizes. This was a revolutionary idea that challenged existing mathematical paradigms.

Key concepts in mængdelæren include elements, subsets, unions, intersections, and complements. An element is an individual object within a set. A subset is a set whose elements are all contained within another set. The union of two sets combines all elements from both, while the intersection contains only elements common to both. The complement of a set consists of all elements not in that set, within a given universal set.

Mængdelæren provides a rigorous foundation for many areas of mathematics, including calculus, algebra, and topology. It offers a precise language and framework for defining mathematical objects and reasoning about their properties. Despite its foundational importance, the development of naive set theory also revealed paradoxes, such as Russell's paradox, which highlighted the need for axiomatic set theory, like Zermelo-Fraenkel set theory, to avoid contradictions.