Grundgesetze

Grundgesetze refers to Gottlob Frege’s two-volume work Grundgesetze der Arithmetik (Basic Laws of Arithmetic), published in 1893 (Part I) and 1903 (Part II). It is a central attempt in the logicist project to ground arithmetic in pure logic, showing how number concepts and arithmetic truths could be derived from logical axioms and inferences. Frege developed a formal system and introduced the idea that numbers are grounded in the extensions of concepts: for a given concept F, the extension of F collects all objects that fall under F, and numerical facts would then be consequences of logical laws about these extensions.

The work presents a comprehensive axiomatic framework, with a large group of axioms—the Grundgesetze—covering principles of function application, identity, and predication. A key feature is Basic Law V, which links the extension of a concept to the identity of the extension of another concept: two extensions are identical if and only if they have exactly the same members. Frege aimed to derive arithmetic, including the natural numbers and basic operations, from such logical principles alone, treating mathematical truths as logical consequences.

Grundgesetze is also infamous for exposing a fundamental problem in Frege’s program. In 1902 Bertrand Russell pointed out a contradiction arising from Basic Law V when considering the extension of the concept “not belonging to itself,” leading to Russell’s paradox. The inconsistency undermined the project of deriving all mathematics from unmodified logic as Frege had proposed, and Part II did not salvage the program.

Despite the failure of Frege’s program, Grundgesetze is regarded as a landmark in the history of logic. It influenced subsequent foundational discussions, spurred development in formal logic and type theory, and shaped how later logicians approached the relationship between logic and mathematics.