Grundgesamtheiten

Grundgesamtheit is a German term used in mathematics, logic and statistics to denote the universe of discourse in a given context. It refers to the collection of all objects under consideration for a problem, inquiry, or theory. In naive set theory, the Grundgesamtheit would be the universal set containing all objects, typically denoted U.

In formal set theory, the idea of a universal set as a set leads to paradoxes, most famously Russell’s paradox. Consequently, standard axiom systems such as ZF do not include a set of all sets. Some frameworks, notably Von Neumann–Bernays–Gödel (NBG) and Morse–Kelley (MK) class theories, treat the collection of all sets as a proper class rather than a set. In these contexts the Grundgesamtheit can be discussed as a universal object, but it is not itself a set.

In statistics, Grundgesamtheit has a different but related meaning: the population, i.e., the complete set of elements from which a sample is drawn. The Grundgesamtheit may be finite or infinite, and a sampling frame is a practical subset accessible to data collection. The term emphasizes the scope of inference for a given study.

The plural form, Grundgesamtheiten, appears when referring to multiple universes or populations across different contexts or studies. See also universal set, proper class, population and sample, Russell’s paradox, and foundational set theories such as NBG and MK.