Related papers: Towards an Independent Version of Tarski's System …
A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms…
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…
Qualitative spatial models based on Goodman-style mereology and pseudo-topology often pose problems for advanced geometric reasoning, as they lack true Euclidean geometry and fully developed topological spaces. We address this issue by…
A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W.…
In this paper we will develop an axiomatic foundation for the geometric study of straight edge, protractor, and compass constructions, which while being related to previous foundations, will be the first to have all axioms written and all…
We report on a project to use a theorem prover to find proofs of the theorems in Tarskian geometry. These theorems start with fundamental properties of betweenness, proceed through the derivations of several famous theorems due to Gupta and…
The aim of this paper is to develop a new axiomatization of planar geometry by reinterpreting the original axioms of Euclid. The basic concept is still that of a line segment but its equivalent notion of betweenness is viewed as a…
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…
In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R^n with n >= 1…
It is shown that Tarski's set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski's axioms slightly, and in fact by…
We introduce a system of software tools that can automatically prove or discover geometric inequalities. The system, called GeoGebra Discovery, consisting of an extended version of GeoGebra, a controller web service realgeom, and the…
We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We…
We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are…
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…
Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of…
This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically…
We consider the Cartan extension of Riemann geometry as the basis upon which to build the Sciama--Kibble completion of Einstein gravity, developing the most general theory in which torsion and metric have two independent coupling constants:…
We survey the status of decidabilty of the consequence relation in various axiomatizations of Euclidean geometry. We draw attention to a widely overlooked result by Martin Ziegler from 1980, which proves Tarski's conjecture on the…
Two axioms of order geoemtry are the poset axioms of transitivity and antisymmetry of the relation "is in front of" when looking from a point. From these axioms, by looking from an interval instead of a point, further well-known axioms of…
We improve upon Huntington's affine geometry by showing that his independence proofs can be, in some cases, simplified. We carry out a systematic investigation of the strict notion of betweenness that Huntington employs (the three arguments…