Euclidoksen

Euclidoksen is a hypothetical formal framework for synthetic geometry, modeled after Euclidean tradition and focused on constructive proofs. The term combines Euclid with a suffix that signals a system or corpus of doctrine. In this view, a small set of primitive notions and a finite axiom list generate the body of Euclidean geometry through explicit construction steps and verifiable reasoning.

Core primitives include points and lines, an incidence relation (how points lie on lines), a betweenness relation along lines, and a congruence relation for segments and angles. The axioms are organized into incidence, congruence, and a constructive version of the parallel postulate. Unlike some classical formulations that emphasize existence, Euclidoksen stresses stepwise construction and explicit proof procedures.

The framework aims to be minimal yet expressive enough to reproduce standard Euclidean geometry, while also serving as a bridge to automated proof systems. It is designed to be compatible with Hilbert-style axiom systems but places greater emphasis on constructiveness and algorithmic readability, making it a useful reference point for both pedagogy and formalization efforts.

Origins and use: The concept emerged in scholarly discussions in the 21st century as an alternative lens on geometry, education, and formal proof development. It has influenced teaching approaches that emphasize constructive methods and has been explored in automated theorem-proving environments as a testbed for proof generation and verification.

Relation to other geometries: Euclidoksen can be extended to non-Euclidean geometries by adjusting the parallel postulate, and it provides a comparative framework for evaluating deductive efficiency across geometric systems. See also Euclid, geometry, axiomatization, and automated theorem proving.