▁to ▁hold ▁in ▁particular ▁inner ▁models , ▁such ▁as ▁in ▁the ▁construct ible ▁universe . ▁However , ▁some ▁statements ▁that ▁are ▁true ▁about ▁construct ible ▁sets ▁are ▁not ▁consistent ▁with ▁hypoth es ized ▁large ▁cardinal ▁ax i oms . ▁ ▁For cing ▁proves ▁that ▁the ▁following ▁statements ▁are ▁independent ▁of ▁Z FC : ▁▁ ▁Cont inu um ▁hypothesis ▁ ▁Diam ond ▁principle ▁ ▁Sus lin ▁hypothesis ▁ ▁Martin ' s ▁ax iom ▁( which ▁is ▁not ▁a ▁Z FC ▁ax iom ) ▁ ▁A xi om ▁of ▁Con struct ibility ▁( V = L ) ▁( which ▁is ▁also ▁not ▁a ▁Z FC ▁ax iom ). ▁ ▁Rem arks : ▁▁ ▁The ▁consist ency ▁of ▁V = L ▁is ▁prov able ▁by ▁inner ▁models ▁but ▁not ▁forcing : ▁every ▁model ▁of ▁Z F ▁can ▁be ▁trim med ▁to ▁become ▁a ▁model ▁of ▁Z FC ▁+ ▁V = L . ▁ ▁The ▁Diam ond ▁Princi ple ▁implies ▁the ▁Cont inu um ▁Hyp oth esis ▁and ▁the ▁neg ation ▁of ▁the ▁Sus lin ▁Hyp oth esis . ▁ ▁Martin ' s ▁ax iom ▁plus ▁the ▁neg ation ▁of ▁the ▁Cont inu um ▁Hyp oth esis ▁implies ▁the ▁Sus lin ▁Hyp oth esis . ▁ ▁The ▁construct ible ▁universe ▁satisfies ▁the ▁General ized ▁Cont inu um ▁Hyp oth esis , ▁the ▁Diam ond ▁Princi ple , ▁Martin ' s ▁A xi om ▁and ▁the ▁K ure pa ▁Hyp oth esis . ▁ ▁The ▁failure ▁of ▁the ▁K ure pa ▁hypothesis ▁is ▁equ icons istent ▁with ▁the