First-Order Axiom Systems E d and E d a Extending Tarski’s E 2 with Distance and Angle Function Symbols for Quantitative Euclidean Geometry

Hongyu Guo ()
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Hongyu Guo: Department of Computer and Information Sciences, Texas A&M University–Victoria, Victoria, TX 77901, USA

Mathematics, 2025, vol. 13, issue 21, 1-34

Abstract: Tarski’s first-order axiom system E 2 for Euclidean geometry is notable for its completeness and decidability. However, the Pythagorean theorem—either in its modern algebraic form a 2 + b 2 = c 2 or in Euclid’s Elements —cannot be directly expressed in E 2 , since neither distance nor area is a primitive notion in the language of E 2 . In this paper, we introduce an alternative axiom system E d in a two-sorted language, which takes a two-place distance function d as the only geometric primitive. We also present a conservative extension E d a of it, which also incorporates a three-place angle function a , both formulated strictly within first-order logic. The system E d has two distinctive features: it is simple (with a single geometric primitive) and it is quantitative. Numerical distance can be directly expressed in this language. The Axiom of Similarity plays a central role in E d , effectively killing two birds with one stone: it provides a rigorous foundation for the theory of proportion and similarity, and it implies Euclid’s Parallel Postulate (EPP). The Axiom of Similarity can be viewed as a quantitative formulation of EPP. The Pythagorean theorem and other quantitative results from similarity theory can be directly expressed in the languages of E d and E d a , motivating the name Quantitative Euclidean Geometry . The traditional analytic geometry can be united under synthetic geometry in E d . Namely, analytic geometry is not treated as a model of E d , but rather, its statements can be expressed as first-order formal sentences in the language of E d . The system E d is shown to be consistent, complete, and decidable. Finally, we extend the theories to hyperbolic geometry and Euclidean geometry in higher dimensions.

Keywords: quantitative Euclidean geometry; hyperbolic geometry; distance function; angle function; axiom of similarity; Tarski’s axioms; real closed fields; completeness; decidability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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