Ultimate L

λthereisanelementaryembeddingj:VoMwith{mcp}(j)=δ，j(δ)>λ，andj‘λ\inM(or，equivalently，{}^λM\subseteqM).

Infact，givensuchanembeddingj，wecandefineanormalfineUonP_δ(λ)by

A\inUiffj‘λ\inj(A).

Conversely，givenanormalfineultrafilterUonP_δ(λ)，theultrapowerembeddinggeneratedbyUisanexampleofsuchanembeddingj.Moreover，ifU_jistheultrafilteronP_δ(λ)derivedfromjasexplainedabove，thenU_j=U.

AnothercharacterizationofsupercompactnesswasfoundbyMagidor，anditwillplayakeyroleintheselectinthisreformulation，ratherthanthecriticalpoint，δappearsastheimageofthecriticalpointsoftheembeddingsunderconsideration.Thisversionseemsideallydesignedtobeusedasaguideintheconstructionofextendermodelsforsupercompactness，althoughrecentresultssuggestthatthisis，infact，aredherring.

Thekeynotionwewillbestudyingisthefollowing:

Definition.N\subseteqVisaweakextendermodelfor`δissupercompact’iffforallλ>δthereisanormalfineUonP_δ(λ)suchthat:

try{mad1();} catch(ex){}

P_δ(λ)\capN\inU，and

U\capN\inN.

ThisdefinitioncouplesthesupercompactnessofδinNdirectlywithitssupercompactnessinV.Inthemanuscript，thatNisaweakextendermodelfor`δissupercompact’isdenotedbyo^N_{mlong}(δ)=\infty.Notethatthisisaweaknotionindeed，inthatwearenotrequiringthatN=L[\vecE]forsome(long)sequence\vecEofextenders.TheideaistostudybasicpropertiesofNthatfollowfromthisnotion，inthehopesofbetterunderstandinghowsuchanL[\vecE]modelcanactuallybeconstructed.

Forexample，finenessofUalreadyimpliesthatNsatisfiesaversionofcovering:IfA\subseteqλand|A|<δ，thenthereisaB\inP_{δ}(λ)\capNwithA\subseteqB.Butinfactasignificantlystrongerversionofcoveringholds.Toproveit，wefirstneedtorecallaniceresultduetoSolovay，whousedittoshowthat{\sfSCH}holdsaboveasupercompact.

Solovay’sLemma.Letλ>δberegular.ThenthereisasetXwiththepropertythatthefunctionf:a\mapsto\sup(a)isinjectiveonXand，foranynormalfinemeasureUonP_δ(λ)，X\inU.

ItfollowsfromSolovay’slemmathatanysuchUisequivalenttoameasureonordinals.

Proof.Let\vec