First-Order Axiom Systems $\mathscr{E}_{d}$ and $\mathscr{E}_{da}$ Extending Tarski's $\mathscr{E}_{2}$ with Distance and Angle Function Symbols for Quantitative Euclidean Geometry

TL;DR

This paper introduces new first-order axiom systems for Euclidean geometry that incorporate distance and angle functions, enabling direct expression of the Pythagorean theorem and unifying synthetic and analytic geometry.

Contribution

The paper presents the systems $ ext{E}_d$ and $ ext{E}_{da}$ with primitive distance and angle functions, providing a simple, quantitative foundation for Euclidean geometry with proven consistency, completeness, and decidability.

Findings

$ ext{E}_d$ is consistent, complete, and decidable.

The Axiom of Similarity implies Euclid's Parallel Postulate.

Quantitative geometric results are directly expressible in the new systems.

Abstract

Tarski's first-order axiom system $\mathscr{E}_{2}$ 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 $\mathscr{E}_{2}$, since neither distance nor area is a primitive notion in the language of $\mathscr{E}_{2}$. In this paper, we introduce an alternative axiom system $\mathscr{E}_{d}$ in a two-sorted language, which takes a two-place distance function $d$ as the only geometric primitive. We also present a conservative extension $\mathscr{E}_{da}$ of it, which also incorporates a three-place angle function $a$. The system $\mathscr{E}_{d}$ 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 $\mathscr{E}_{d}$, 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 $\mathscr{E}_{d}$ and $\mathscr{E}_{da}$, motivating the name Quantitative Euclidean Geometry. The traditional analytic geometry can be united under synthetic geometry in $\mathscr{E}_{d}$. Namely, analytic geometry is not treated as a model of $\mathscr{E}_{d}$, but rather, its statements can be expressed as first-order formal sentences in the language of $\mathscr{E}_{d}$. The system $\mathscr{E}_{d}$ is shown to be consistent, complete, and decidable. Finally, we extend the theories to hyperbolic geometry and Euclidean geometry in higher dimensions.