Lupactnessshouldprovablybeanultipact.ThebasicproblepactinN.
Ofcourse，theshapeoftheanswerdependsonwhatwepactnessofδinNshould“directlytraceback”toitssupercopleteandforallF:P_δ(λ)oλ，ifFisregressiveU-ae(i.e.，if\{a\pactiffforallλthereisanorpactiffforallλthereisanelepactnesswasfoundbyMagidor，anditwillplayakeyroleintheselectinthisreforpactness，althoughrecentresultssuggestthatthisis，infact，aredherring.
Thekeynotionwewillbestudyingisthefollowing:
Definition.N\subseteqVisaweakextenderpact’iffforallλ&apactnessofδinNdirectlywithitssupercopact’isdenotedbyo^N_{pact.
Solovay’sLepleteultrafilter{\mathcalV}onλ.Thedefinabilityinfactensuresthat{\mathcalV}\inV^λ/{\mathcalV}，contradiction.Wewillencounterasimilardefinablesplittingargumentinthethirdlecture.
LetXconsistofthosea\inP_δ(λ)suchthat，letting\beta=\sup(a)，wehave{mcf}(\beta)>\omega，and
a=\{\alpha<\beta\midS_\alpha\cap\betaisstationaryin\beta\}.
Thenfis1-1onXsince，bydefinition，anya\inXcanbereconstructedfrom\vecSand\sup(a).AllthatneedsarguingisthatX\inUforanynormalfinemeasureUonP_δ(λ).(ThisshowsthattodefineU-measure1sets，weonlyneedapartition\vecSofS^λ_\omegaintostationarysets.)
Letj:VoMbetheultrapowerembeddinggeneratedbyU，so
U=\{A\inP_δ(λ)\midj‘λ\inj(A)\}.
Weneedtoverifythatj‘λ\inj(X).First，notethatj‘λ\inM.Lettingau=\sup(j‘λ)，wethenhavethatM\models{mcf}(au)=λ.Since
M\modelsj(λ)\geauisregular，
itfollowsthatau<j(λ).Let\left<T_\beta\mid\beta<j(λ)ight>=j(\left<S_\alpha\mid\alpha<λight>).InM，theT_\betapartitionS^{j(λ)}_\omegaintostationarysets.Let
A=\{\beta