hilbert’s crossbar theorem希尔伯特的纵横定理

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1、Hilbert’sCrossbarTheoremJacobPacarroMath300(BryanSmith)CrossbarTheorem:IfrayADisbetweenraysACandAB,thenrayADintersectslinesegmentBC.IntroductionDavidHilbertwasaleadingpioneerintheworldofmathematicsandphysics.Hisnotableaccomplishmentsandresearchareintheareasofgeometry,algebra,infi

2、nite-dimensionalspacesandmathematicallogic(Greenberg,71).WhilestudyingEuclid’swork,HilbertrealizedthatEuclidfailedtocreateaspecificaxiomaticsystemthatallowedforthecleardefinitionoftheexistenceoflinesandpointsandtheirrelationships.Hilbertdevelopedhisaxiomaticsystemduringtheearlypa

3、rtofthetwentiethcentury,andtheseaxiomsreflectconceptsverysimilartoEuclid’sideas.Hilbert’saxiomscanbeseparatedintofivesubgroups:connection(relationshipsbetweenpoint,lineandplane),order(explainingtheideaof“between”),congruence,continuityandparallelism.Hilbert’scrossbartheoremiscate

4、gorizedasanaxiomoforderandsimplystatesthatifrayADisbetweenraysACandAB,thenrayADintersectslinesegmentBC.Oneinterpretationoftheproofgoesasfollows:5FCGADEBfigure1ProofofCrossbarTheoremThedefinitionofRayBetweennessstatesthatrayADisbetweenraysACandABprovidedrayABandrayACarenotopposite

5、raysandpointDisinteriorto

6、ntB,andisalsoonthesamesideoflineABaspointC.LetpointsEandFbeanypointssuchthattheysatisfyC*A*EandB*A*F(seefigure1).Bybetweennessaxiomone,weknowthatpointsC,A,andEaredistinctpointslyingonthesameline.Likewise,pointsB,AandFaredistinctpointslyingonthesameline.Thethirdclauseofproposition

7、3.8statesthatifDisinterioroftheangle

8、andCareonoppositesidesoflineAD,bythedefi