The axioms of constructive geometry

Quantifier-Free Axioms For Constructive Affine Plane Geometry

The purpose of this paper is to state a set of axioms for plane geometry which do not use any quantifiers, but only constructive operations. The relevant definitions and general theorems are stated; for reasons of space the proofs are only sketched. Quantifier-free arithmetic and quantifier-free algebra have been the subjects of several investigations, beginning at least with the early importan...

Constructive Geometry

Euclidean geometry, as presented by Euclid, consists of straightedge-and-compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathematics” leads to the development of a first-order theory ECG of the “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid ...

Tarski Geometry Axioms

This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous a...

Constructive Reals in Coq: Axioms and Categoricity

We describe a construction of the real numbers carried out in the Coq proof assistant. The basis is a set of axioms for the constructive real numbers as used in the FTA (Fundamental Theorem of Algebra) project, carried out at Nijmegen University. The aim of this work is to show that these axioms can be satisfied, by constructing a model for them. Apart from that, we show the robustness of the s...

Tarski Geometry Axioms - Part II