Mächtigkeit

Mächtigkeit is a German noun that denotes power, might, or strength, as well as the magnitude or size of something. In everyday language it refers to the capacity to influence or control events or situations, for example the political Mächtigkeit of a state or the economic Mächtigkeit of a corporation. In philosophy, political theory, and sociology the term is used to discuss power relations, legitimacy, and authority, often in relation to other forms of control.

In mathematics, particularly in set theory, Mächtigkeit means the cardinality or size of a set. The Mächtigkeit of a finite set is a natural number equal to the number of its elements. A set is finite exactly when it has a finite Mächtigkeit; otherwise it is infinite. Infinite Mächtigkeiten are studied in terms of their relative sizes: some infinite sets are countable (abzählbare Mächtigkeit), such as the natural numbers, while others are uncountable (unabzählbare Mächtigkeit), such as the real numbers. Cantor’s theory shows that infinite Mächtigkeiten can be ordered by cardinal numbers, with ℵ0 (aleph-null) being the Mächtigkeit of the natural numbers and higher cardinals for larger sets. The concept is central to discussions of computability, analysis, and logic.

Etymology traces Mächtigkeit to macht (power) and the suffix -igkeit, forming a noun that conveys the quality or extent of power or size. In German, it is a formal term whose usage depends on context, ranging from everyday descriptions of influence to precise mathematical definitions.