SeveralexamplesthatservetovalidatetheAHP/ANPwithmatriceshierarchiesandnetworksaregiveninthispaper.Theyarethenfollowedbyadiscussionoftherealnumbersandhowtheyaregeneratedwithouttheneedforanabsolutezero,andhowtheydefineanabsolutescaleofmeasurementthatalsodoesnotneedanabsolutezero.IntheAHP/ANPthemeasurementofanalternativedependsonwhatotheralternativesitiscomparedwith.Theresultisthatrankcanchangeifalternativesareaddedordeleted,somethingthatdoesnotoccurinone-at-a-timeratingofthealternativesbycomparingthemwithanideal.Anexampleisprovidedtoshowthatthisisnaturalandneednotinvolvenewcriteriaorchangeinjudgments.AbriefdiscussionofUtilityTheory,theothermulti-criteriatheory,whichusesintervalscalestomeasureintangiblesandsomeofitsproblemsandparadoxes,isgiven.ThereferencesattheendincludemostofthepapersthatareadversetotheAHl'withbriefcommentsaboutseveralofthemgiveninthepaper.
