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keywords
abstract
plane geometry, axiomatic system, axiom, undefined term, interpretation,
model
An axiomatic system is an organized logical structure consisting of undefined
terms, axioms, defined terms, a system of inference rules and theorems. A
model for an axiomatic system is an interpretation of this system, consisting of
the attribution of particular meanings to the undefined terms, in order that the
axioms, read in the light of this interpretation, become true propositions.
In this work we will present plane geometries, from abstract to neutral or
absolute geometries. We will make a brief reference to Euclidean and
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,
we will present the real cartesian plane and Poincaré half plane as models for
the Euclidean and hyperbolic geometries, respectively.