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palavras-chave<br />
resumo<br />
geometria plana,sistema axiomático, axioma, termo primitivo, interpretação,<br />
modelo<br />
Um sistema axiomático é uma estrutura lógica organizada constituída por<br />
termos primitivos, axiomas, termos definidos, um sistema de regras de<br />
inferência e teoremas ou proposições. Um modelo para um sistema axiomático<br />
é uma interpretação desse sistema, consistindo na atribuição de significados<br />
particulares aos termos primitivos, de modo a que os axiomas, lidos à luz<br />
desta interpretação, se tornem proposições verdadeiras.<br />
No presente trabalho apresentamos uma hierarquização de geometrias planas,<br />
desde as geometrias abstractas até às geometrias neutras ou absolutas.<br />
Faremos uma breve referência às geometrias Euclidiana e hiperbólica planas,<br />
cujos sistemas axiomáticos são categóricos. Por fim, apresentaremos o plano<br />
cartesiano real e o semiplano de Poincaré como modelos para as geometrias<br />
Euclidiana e hiperbólica, respectivamente.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
keywords<br />
abstract<br />
plane geometry, axiomatic system, axiom, undefined term, interpretation,<br />
model<br />
An axiomatic system is an organized logical structure consisting of undefined<br />
terms, axioms, defined terms, a system of inference rules and theorems. A<br />
model for an axiomatic system is an interpretation of this system, consisting of<br />
the attribution of particular meanings to the undefined terms, in order that the<br />
axioms, read in the light of this interpretation, become true propositions.<br />
In this work we will present plane geometries, from abstract to neutral or<br />
absolute geometries. We will make a brief reference to Euclidean and<br />
hyperbolic plane geometries, whose axiomatic systems are categorical. Finally,<br />
we will present the real cartesian plane and Poincaré half plane as models for<br />
the Euclidean and hyperbolic geometries, respectively.
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