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.