Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2511.08494

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

Hongyu Guo
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.LOcs.LOmath.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hongyu Guo. 2025-11-11. 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. https://doi.org/10.3390/math13213462

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bluebirds and mockingbirds cannot produce a fixed-point combinator

Let $B$ be the bluebird combinator with reduction rule $Bxyz \to_{w} x\left(yz\right)$, let $M$ be the mockingbird combinator with reduction rule $Mx \to_{w} xx$, and let $I$ be the identity bird combinator with reduction rule $Ix \to_{w} x$. A fixed-point combinator, called a sage bird by Smullyan, is a closed term $Y$ such that, for a fresh variable $x$, $Yx$ is equivalent to $x\left(Yx\right)$ under these reduction rules. For a fixed variable $x$, we construct an invariant $\mathrm{Tr}_{x}\left(u\right)$ of a $BMI$-term $u$ with respect to $\to_{w}$. This invariant traces the occurrences of $x$ in the leftmost-innermost reduction sequence of $u$. We then prove that $\mathrm{Tr}_{x}\left(Yx\right) \neq \mathrm{Tr}_{x}\left(x^{r}\left( Yx \right)\right)$ for every $x$-free $BMI$-term $Y$ and every $r\geq 1$. Consequently, there exists no fixed-point combinator in $BMI$-combinatory logic. This provides a negative answer to the problem posed by Smullyan in 1985.

math.LO↗
arXiv2026

Pointwise provable equality and the failure of composition

Montagna (1989) and Di Paola--Montagna (1991) claim that the algebraic systems $S'$ and $S'_T$, respectively, are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two program indices that are pointwise provably equal in $T$ but yield inequivalent composites when each is run after the same program. Montagna's $S'$ is the case $T=\mathrm{PA}$. The failure already occurs for partial maps from $ω$ to itself. Weak totality and the proposed range assignment also depend on the choice of representatives. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $Π^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $Σ^0_1$-sound and the universal relation otherwise.

math.LO↗
arXiv2026

Compactness via Consistency Properties

We will use consistency properties to characterize strongly compact cardinals, first showing an adequate Model Existence Theorem for larger fragments.

math.LO↗