A Constructive Version of Tarski's Geometry

TL;DR

This paper develops three constructive versions of Tarski's Euclidean geometry theory, modifying axioms to ensure unique, continuous point existence, linking formal geometry with ruler-and-compass constructions.

Contribution

It introduces three modified, constructive versions of Tarski's geometry theory that connect formal axioms with practical ruler-and-compass constructions.

Findings

Constructive axioms ensure unique, continuous point existence.

Points proven to exist can be constructed with ruler and compass.

The modified theories retain the same theorems as Tarski's original theory.

Abstract

Euclid's reasoning is essentially constructive. Tarski's elegant and concise first-order theory of Euclidean geometry, on the other hand, is essentially non-constructive, even if we restrict attention (as we do here) to the theory with line-circle and circle-circle continuity in place of first-order Dedekind completeness. Here we exhibit three constructive versions of Tarski's theory. One, like Tarski's theory, has existential axioms and no function symbols. We then consider a version in which function symbols are used instead of existential quantifiers. The third version has a function symbol for the intersection point of two non-parallel, non-coincident lines, instead of only for intersection points produced by Pasch's axiom and the parallel axiom; this choice of function symbols connects directly to ruler-and-compass constructions. All three versions have this in common: the axioms have been modified so that the points they assert to exist are unique and depend continuously on parameters. This modification of Tarski's axioms, with classical logic, has the same theorems as Tarski's theory, but we obtain results connecting it with ruler-and-compass constructions as well. In particular, points constructively proved to exist can be constructed with ruler and compass, uniformly in parameters; the same is true with non-constructive proofs if several constructions are allowed for different cases.