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Abstract

InRussell'sRamiedTheoryofTypesrtt,twohierarchicalconceptsdominate:ordersandtypes.Theuseofordershasasaconsequencethatthelogic partofrttispredicative.Theconceptoforderhowever,isalmostdead sinceRamseyeliminateditfromrtt.ThisiswhywendChurch'ssimple theoryoftypes(whichusesthetypeconceptwithouttheorderone)atthe bottomoftheBarendregtCuberatherthanrtt.Despitethedisappearence oforderswhichhaveastrongcorrelationwithpredicativity,predicativelogic stillplaysaninuentialroleinComputerScience.Animportantexampleis theproofcheckerNuprl,whichisbasedonMartin-Lof'sTypeTheorywhich usestypeuniverses.Thosetypeuniverses,andalsodegreesofexpressionsin Automath,arecloselyrelatedtoorders.Inthispaper,weshowthatorders havenotdisappearedfrommodernlogicandcomputerscience,rather,orders playacrucialroleinunderstandingthehierarchyofmodernsystems.Inorder toachieveourgoal,weconcentrateonNuprl. Thenoveltyofourpaperliesin:1)therevivalofRussell'sorders,2)the placingofthehistoricalsystemrttunderlyingthefamousPrincipiaMathematicainacontextwithamodernsystemofcomputermathematics(Nuprl) andmoderntypetheories(Martin-Lof'stypetheoryandPTSs),and3)the presentationofacomplextypesystem(Nuprl)asasimpleandcompactPTS.

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.297.8235
Provided by: CiteSeerX
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