Victor Pambuccian

Axiomatizing geometric constructions

In this survey paper, we present several results linking quantifier-free axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.

Axiomatizations of Hyperbolic and Absolute Geometries

A survey of finite first-order axiomatizations for hyperbolic and absolute geometries. 1. Hyperbolic Geometry Elementary Hyperbolic Geometry as conceived by Hilbert To axiomatize a geometry one needs a language in which to write the axioms, and a logic by means of which to deduce consequences from those axioms. Based on the work of Skolem, Hilbert and Ackermann, Gödel, and Tarski, a consensus had been reached by the end of the first half of the 20th century that, as Skolem had emphasized since 1923, “if we are interested in producing an axiomatic system, we can only use first-order logic” ([21, p. 472]). The language of first-order logic consists of the logical symbols , , , , , a denumerable list of symbols called individual variables, as well as denumerable lists of -ary predicate (relation) and function (operation) symbols for all natural numbers , as well as individual constants (which may be thought of as 0-ary function symbols), together with two quantifiers, and which can bind only individual variables, but not sets of individual variables nor predicate or function symbols. Its axioms and rules of deduction are those of classical logic. Axiomatizations in first-order logic preclude the categoricity of the axiomatized models. That is, one cannot provide an axiom system in first-order logic which admits as its only model a geometry over the field of real numbers, as Hilbert [31] had done (in a very strong logic) in his Grundlagen der Geometrie. By the Löwenheim-Skolem theorem, if such an axiom system admits an infinite model, then it will admit models of any given infinite cardinality.

A logical look at characterizations of geometric transformations under mild hypotheses

Abstract For several characterizations of geometric transformations — which state that a map, which satisfies certain conditions like injectivity, surjectivity, bijectivity and preserves certain geometric notions γi, must preserve another notion v as well — we provide the definitional counterpart, i.e. a definition that satisfies certain syntactic constraints of the notion v in terms of the notions γi. 2000 MSC: 51M05 51B10 03C40

Ternary operations as primitive notions for constructive plane geometry III

This paper continues the investigations begun in [6] and continued in [7] about quantifier-free axiomatizations of plane Euclidean geometry using ternary operations. We show that plane Euclidean geometry over Archimedean ordered Euclidean fields can be axiomatized using only two ternary operations if one allows axioms that are not first-order but universal Lw1,w sentences. The operations are: the transport of a segment on a halfline that starts at one of the endpoints of the given segment, and the operation which produces one of the intersection points of a perpendicular on a diameter of a circle (which intersects that diameter at a point inside the circle) with that circle. MSC: 03F65, 51M05, 51M15.

Constructive axiomatizations of plane absolute, Euclidean and hyperbolic geometry

In this paper we provide quantifier-free, constructive axiomatizations for 2-dimensional absolute, Euclidean, and hyperbolic geometry. The main novelty consists in the first-order languages in which the axiom systems are formulated.

ZUM STUFENAUFBAU DES PARALLELENAXIOMS

Euclids parallel postulate is shown to be equivalent to the conjunction of the following two weaker postulates: “Any perpendicular to one side of a right angle intersects any perpendicular to the other side” and “For any acute angle Oxy, the segmentPQ — whereP is a point onOx, Q a point onOy andPQ ⊥ Oy — grows indefinitely, i. e. can be made longer than any given segment”.

Positive existential definability of parallelism in terms of betweenness in Archimedean ordered affine geometry

We prove that one can define the relation ‖, with ab ‖ cd to be read as ‘a = b or c = d or ab and cd are parallel lines (or coincide)’ positively existentially in Lω1ω1 in terms of 6= and the ternary relation B of betweenness, with B(abc) to be read as ‘b lies between a and c’ in Archimedean ordered affine geometry. We also show that a self-map of an Archimedean ordered translation plane or of a flat affine plane which preserves both B and ¬B must be a surjective affine mapping. Mathematics Subject Classification: 51G05, 51F20, 51F05.

Zur Existenz gleichseitiger Dreiecke in H-Ebenen

It is shown, by indicating how to construct one with ruler and gauge, that there are equilateral triangles in absolutes planes which need not satisfy the circle axiom. However, it is not possible to construct an equilateral triangle with given base in absolute planes, even if they satisfy bachmannsLotschnittaxiom or the Archimedean axiom.

Forms of the Pasch axiom in ordered geometry

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom (IP) does not imply its outer form (OP). We also show that OP can be properly split into IP and the weak Pasch axiom (WP) (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A Reverse Analysis of the Sylvester-Gallai Theorem

Reverse analyses of three proofs of the Sylvester-Gallai theorem lead to three different and incompatible axiom systems. In particular, we show that proofs respecting the purity of the method, using only notions considered to be part of the statement of the theorem to be proved, are not always the simplest, as they may require axioms which proofs using extraneous predicates do not rely upon.