First-Order Axiom Systems Ed and Eda Extending Tarski’s E2 with Distance and Angle Function Symbols for Quantitative Euclidean Geometry

2025-10-30

Mathematics

AI Summary

Tarski’s first-order Euclidean geometry system $mathscr{E}_2$ is complete and decidable, but lacks primitive predicates for distance and angle, preventing direct first-order formulation of quantitative theorems such as the Pythagorean Theorem. Method: We introduce two sorted first-order extensions—Ed (with a binary distance function $d$) and Eda (further augmented with a ternary angle function $a$)—grounded on similarity axioms that intrinsically encode proportion theory and the parallel postulate. Contribution/Results: These extensions enable direct first-order expression of metric concepts—including distance, angle, and the Pythagorean Theorem—while remaining conservative over $mathscr{E}_2$, preserving its simplicity and logical properties. We prove consistency, completeness, and decidability for both systems. The framework unifies synthetic and analytic geometry within a single formal language and generalizes naturally to hyperbolic geometry and higher-dimensional Euclidean spaces.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Tarski’s first-order axiom system E2 for Euclidean geometry is notable for its completeness and decidability. However, the Pythagorean theorem—either in its modern algebraic form a2+b2=c2 or in Euclid’s Elements—cannot be directly expressed in E2, since neither distance nor area is a primitive notion in the language of E2. In this paper, we introduce an alternative axiom system Ed in a two-sorted language, which takes a two-place distance function d as the only geometric primitive. We also present a conservative extension Eda of it, which also incorporates a three-place angle function a, both formulated strictly within first-order logic. The system Ed 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 Ed, 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 Ed and Eda, motivating the name Quantitative Euclidean Geometry. The traditional analytic geometry can be united under synthetic geometry in Ed. Namely, analytic geometry is not treated as a model of Ed, but rather, its statements can be expressed as first-order formal sentences in the language of Ed. The system Ed is shown to be consistent, complete, and decidable. Finally, we extend the theories to hyperbolic geometry and Euclidean geometry in higher dimensions.

Problem

• Extends Tarski's geometry with distance functions for quantitative Euclidean geometry
• Enables direct expression of Pythagorean theorem through similarity axioms
• Provides complete decidable system unifying synthetic and analytic geometry approaches

Innovation

• Extends Tarski's geometry with distance function
• Adds angle function as conservative extension
• Quantitative formulation enables Pythagorean theorem expression