\voffset=-.15in
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$\ $\vskip .3in
\centerline{``Identity Crises and Strong Compactness''}
\vskip .25in
\centerline{by}
\vskip .25in
\centerline{Arthur W. Apter${}^{*, {***}}$}
\centerline{Department of Mathematics}
\centerline{Baruch College of CUNY}
\centerline{New York, New York 10010}
\vskip .125in
\centerline{and}
\vskip .125in
\centerline{James Cummings${}^{{**}, {***}}$}
\centerline{Department of Mathematics}
\centerline{Carnegie-Mellon University}
\centerline{Pittsburgh, Pennsylvania 15213}
\vskip .25in
\centerline{February 24, 1998}
\centerline{(revised September 7, 1998,
February 14, 1999, and August 13, 1999)}
\vskip .25in
\no Abstract: Combining techniques of the first author and
Shelah with ideas of Magidor, we show how to get
a model in which, for fixed but arbitrary finite $n$, the
first $n$ strongly compact cardinals $\kappa_1, \ldots,
\kappa_n$
are so that $\kappa_i$ for $i = 1, \ldots, n$ is both
the $i^{{\hbox{\rm th}}}$ measurable cardinal and
$\kappa^+_i$ supercompact. This generalizes an unpublished
theorem of Magidor and answers a question of Apter and
Shelah.
\hfil
\vskip .25in
\no ${}^*$Supported by the Volkswagen-Stiftung (RiP-program
at Oberwolfach). In addition, this research was partially
supported by PSC-CUNY Grant 667379. \hb
\no ${}^{**}$Supported by the Volkswagen-Stiftung (RiP-program
at Oberwolfach). In addition, this research was partially
supported by NSF Grant DMS-9703945. \hb
\no ${}^{***}$Both authors wish to express their gratitude
to Menachem Magidor for his explanations to them given at
the January 7-13, 1996 meeting in Set Theory held at the
Mathematics Research Institute, Oberwolfach on his
method of forcing to make the first $n$ measurable and
strongly compact cardinals coincide, for any finite $n$.
\hb
\no 1991 Mathematics Subject Classification:
Primary 03E35, 03E55. \hb
\no Key Words and Phrases: Strongly compact cardinal,
supercompact cardinal, measurable cardinal, identity
crisis, reverse Easton iteration.
\vfill\eject
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\def\underleftarrow{\mathpalette\underleftarrow@}
\def\underleftarrow@#1#2{\vtop{\ialign{$##$\cr
\hfil#1#2\hfil\cr
\noalign{\kern -1\p@\nointerlineskip}
#1{\leftarrow}\mkern-6mu\cleaders\hbox{$#1\mkern-2mu{-}\mkern-2mu$}\hfill
\mkern-6mu{-}\cr}}}
\def\underleftrightarrow{\mathpalette\underleftrightarrow@}
\def\underleftrightarrow@#1#2{\vtop{\ialign{$##$\cr
\hfil#1#2\hfil\cr
\noalign{\kern -1\p@\nointerlineskip}
#1{\leftarrow}\mkern-6mu\cleaders\hbox{$#1\mkern-2mu{-}\mkern-2mu$}\hfill
\mkern-6mu{\to}\cr}}}
\def\sqrt#1{\radical"270370 {#1}}
\def\dots{\relax\ifmmode\let\next=\ldots\else\let\next=\tdots@\fi\next}
\def\tdots@{\unskip\ \tdots@@}
\def\tdots@@{\futurelet\next\tdots@@@}
\def\tdots@@@{$\mathinner{\ldotp\ldotp\ldotp}\,
\ifx\next,$\else
\ifx\next.\,$\else
\ifx\next;\,$\else
\ifx\next:\,$\else
\ifx\next?\,$\else
\ifx\next!\,$\else
$ \fi\fi\fi\fi\fi\fi}
\def\text{\relax\ifmmode\let\next=\text@\else\let\next=\text@@\fi\next}
\def\text@@#1{\hbox{#1}}
\def\text@#1{\mathchoice
{\hbox{\everymath{\displaystyle}\def\textfonti{\the\textfont1 }%
\def\textfontii{\the\textfont2 }\textdef@@ T#1}}
{\hbox{\everymath{\textstyle}\def\textfonti{\the\textfont1 }%
\def\textfontii{\the\textfont2 }\textdef@@ T#1}}
{\hbox{\everymath{\scriptstyle}\def\textfonti{\the\scriptfont1 }%
\def\textfontii{\the\scriptfont2 }\textdef@@ S\rm#1}}
{\hbox{\everymath{\scriptscriptstyle}\def\textfonti{\the\scriptscriptfont1 }%
\def\textfontii{\the\scriptscriptfont2 }\textdef@@ s\rm#1}}}
\def\textdef@@#1{\textdef@#1\rm \textdef@#1\bf
\textdef@#1\sl \textdef@#1\it}
\def\rmfam{0}
\def\textdef@#1#2{\def\next{\csname\expandafter\eat@\string#2fam\endcsname}%
\if S#1\edef#2{\the\scriptfont\next\relax}%
\else\if s#1\edef#2{\the\scriptscriptfont\next\relax}%
\else\edef#2{\the\textfont\next\relax}\fi\fi}
\scriptfont\itfam=\tenit \scriptscriptfont\itfam=\tenit
\scriptfont\slfam=\tensl \scriptscriptfont\slfam=\tensl
\mathcode`\0="0030
\mathcode`\1="0031
\mathcode`\2="0032
\mathcode`\3="0033
\mathcode`\4="0034
\mathcode`\5="0035
\mathcode`\6="0036
\mathcode`\7="0037
\mathcode`\8="0038
\mathcode`\9="0039
\def\Cal{\relax\ifmmode\let\next=\Cal@\else
\def\next{\errmessage{Use \string\Cal\space only in math mode}}\fi\next}
\def\Cal@#1{{\fam2 #1}}
\def\bold{\relax\ifmmode\let\next=\bold@\else
\def\next{\errmessage{Use \string\bold\space only in math
mode}}\fi\next}\def\bold@#1{{\fam\bffam #1}}
\mathchardef\Gamma="0000
\mathchardef\Delta="0001
\mathchardef\Theta="0002
\mathchardef\Lambda="0003
\mathchardef\Xi="0004
\mathchardef\Pi="0005
\mathchardef\Sigma="0006
\mathchardef\Upsilon="0007
\mathchardef\Phi="0008
\mathchardef\Psi="0009
\mathchardef\Omega="000A
\mathchardef\varGamma="0100
\mathchardef\varDelta="0101
\mathchardef\varTheta="0102
\mathchardef\varLambda="0103
\mathchardef\varXi="0104
\mathchardef\varPi="0105
\mathchardef\varSigma="0106
\mathchardef\varUpsilon="0107
\mathchardef\varPhi="0108
\mathchardef\varPsi="0109
\mathchardef\varOmega="010A
\font\dummyft@=dummy
\fontdimen1 \dummyft@=\z@
\fontdimen2 \dummyft@=\z@
\fontdimen3 \dummyft@=\z@
\fontdimen4 \dummyft@=\z@
\fontdimen5 \dummyft@=\z@
\fontdimen6 \dummyft@=\z@
\fontdimen7 \dummyft@=\z@
\fontdimen8 \dummyft@=\z@
\fontdimen9 \dummyft@=\z@
\fontdimen10 \dummyft@=\z@
\fontdimen11 \dummyft@=\z@
\fontdimen12 \dummyft@=\z@
\fontdimen13 \dummyft@=\z@
\fontdimen14 \dummyft@=\z@
\fontdimen15 \dummyft@=\z@
\fontdimen16 \dummyft@=\z@
\fontdimen17 \dummyft@=\z@
\fontdimen18 \dummyft@=\z@
\fontdimen19 \dummyft@=\z@
\fontdimen20 \dummyft@=\z@
\fontdimen21 \dummyft@=\z@
\fontdimen22 \dummyft@=\z@
\def\fontlist@{\\{\tenrm}\\{\sevenrm}\\{\fiverm}\\{\teni}\\{\seveni}%
\\{\fivei}\\{\tensy}\\{\sevensy}\\{\fivesy}\\{\tenex}\\{\tenbf}\\{\sevenbf}%
\\{\fivebf}\\{\tensl}\\{\tenit}\\{\tensmc}}
\def\dodummy@{{\def\\##1{\global\let##1=\dummyft@}\fontlist@}}
\newif\ifsyntax@
\newcount\countxviii@
\def\newtoks@{\alloc@5\toks\toksdef\@cclvi}
\def\nopages@{\output={\setbox\z@=\box\@cclv \deadcycles=\z@}\newtoks@\output}
\def\syntax{\syntax@true\dodummy@\countxviii@=\count18
\loop \ifnum\countxviii@ > \z@ \textfont\countxviii@=\dummyft@
\scriptfont\countxviii@=\dummyft@ \scriptscriptfont\countxviii@=\dummyft@
\advance\countxviii@ by-\@ne\repeat
\dummyft@\tracinglostchars=\z@
\nopages@\frenchspacing\hbadness=\@M}
\def\magstep#1{\ifcase#1 1000\or
1200\or 1440\or 1728\or 2074\or 2488\or
\errmessage{\string\magstep\space only works up to 5}\fi\relax}
{\lccode`\2=`\p \lccode`\3=`\t
\lowercase{\gdef\tru@#123{#1truept}}}
\def\scaledocument#1{\mag=#1\relax}
\def\scaletype#1{\mag=#1\relax
\hsize=\expandafter\tru@\the\hsize
\vsize=\expandafter\tru@\the\vsize
\dimen\footins=\expandafter\tru@\the\dimen\footins}
\def\maxfootnotes#1{\dimen\footins=#1\relax}
\def\scalefont#1#2\andcallit#3{\edef\font@{\the\font}#1\font#3=
\fontname\font\space scaled #2\relax\font@}
\def\Mag@#1#2{\ifdim#1<1pt\multiply#1 #2\relax\divide#1 1000 \else
\ifdim#1<10pt\divide#1 10 \multiply#1 #2\relax\divide#1 100\else
\divide#1 100 \multiply#1 #2\relax\divide#1 10 \fi\fi}
\def\scalelinespacing#1{\Mag@\baselineskip{#1}\Mag@\lineskip{#1}%
\Mag@\lineskiplimit{#1}}
\def\wlog#1{\immediate\write-1{#1}}
\catcode`\@=\active
\catcode`@=11
\def\binrel@#1{\setbox\z@\hbox{\thinmuskip0mu
\medmuskip\m@ne mu\thickmuskip\@ne mu$#1\m@th$}%
\setbox\@ne\hbox{\thinmuskip0mu\medmuskip\m@ne mu\thickmuskip
\@ne mu${}#1{}\m@th$}%
\setbox\tw@\hbox{\hskip\wd\@ne\hskip-\wd\z@}}
\def\overset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
\mathbin{\mathop{\kern\z@#2}\limits^{#1}}\else\ifdim\wd\tw@>\z@
\mathrel{\mathop{\kern\z@#2}\limits^{#1}}\else
{\mathop{\kern\z@#2}\limits^{#1}}{}\fi\fi}
\def\underset#1\to#2{\binrel@{#2}\ifdim\wd\tw@<\z@
\mathbin{\mathop{\kern\z@#2}\limits_{#1}}\else\ifdim\wd\tw@>\z@
\mathrel{\mathop{\kern\z@#2}\limits_{#1}}\else
{\mathop{\kern\z@#2}\limits_{#1}}{}\fi\fi}
\def\circle#1{\leavevmode\setbox0=\hbox{h}\dimen@=\ht0
\advance\dimen@ by-1ex\rlap{\raise1.5\dimen@\hbox{\char'27}}#1}
\def\sqr#1#2{{\vcenter{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
\hrule height.#2pt}}}
\def\square{\mathchoice\sqr34\sqr34\sqr{2.1}3\sqr{1.5}3}
\def\force{\hbox{$\|\hskip-2pt\hbox{--}$\hskip2pt}}
\catcode`@=\active
%.....
%mathchar.tex
%.....
\catcode`\@=11
\def\bold{\relaxnext@\ifmmode\let\next\bold@\else
\def\next{\Err@{Use \string\bold\space only in math mode}}\fi\next}
\def\bold@#1{{\bold@@{#1}}}
\def\bold@@#1{\fam\bffam#1}
\def\hexnumber@#1{\ifnum#1<10 \number#1\else
\ifnum#1=10 A\else\ifnum#1=11 B\else\ifnum#1=12 C\else
\ifnum#1=13 D\else\ifnum#1=14 E\else\ifnum#1=15 F\fi\fi\fi\fi\fi\fi\fi}
\def\bffam@{\hexnumber@\bffam}
%\mathchardef\boldGamma="0\bffam@00
%\mathchardef\boldDelta="0\bffam@01
%\mathchardef\boldTheta="0\bffam@02
%\mathchardef\boldLambda="0\bffam@03
%\mathchardef\boldXi="0\bffam@04
%\mathchardef\boldPi="0\bffam@05
%\mathchardef\boldSigma="0\bffam@06
%\mathchardef\boldUpsilon="0\bffam@07
%\mathchardef\boldPhi="0\bffam@08
%\mathchardef\boldPsi="0\bffam@09
%\mathchardef\boldOmega="0\bffam@0A
% change to msam and msbm files 1.1.92
\font\tenmsx=msam10
\font\sevenmsx=msam7
\font\fivemsx=msam5
\font\tenmsy=msbm10
\font\sevenmsy=msbm7
\font\fivemsy=msbm7
\newfam\msxfam
\newfam\msyfam
\textfont\msxfam=\tenmsx
\scriptfont\msxfam=\sevenmsx
\scriptscriptfont\msxfam=\fivemsx
\textfont\msyfam=\tenmsy
\scriptfont\msyfam=\sevenmsy
\scriptscriptfont\msyfam=\fivemsy
\def\msx@{\hexnumber@\msxfam}
\def\msy@{\hexnumber@\msyfam}
\mathchardef\boxdot="2\msx@00
\mathchardef\boxplus="2\msx@01
\mathchardef\boxtimes="2\msx@02
\mathchardef\square="0\msx@03
\mathchardef\blacksquare="0\msx@04
\mathchardef\centerdot="2\msx@05
\mathchardef\lozenge="0\msx@06
\mathchardef\blacklozenge="0\msx@07
\mathchardef\circlearrowright="3\msx@08
\mathchardef\circlearrowleft="3\msx@09
\mathchardef\rightleftharpoons="3\msx@0A
\mathchardef\leftrightharpoons="3\msx@0B
\mathchardef\boxminus="2\msx@0C
\mathchardef\Vdash="3\msx@0D
\mathchardef\Vvdash="3\msx@0E
\mathchardef\vDash="3\msx@0F
\mathchardef\twoheadrightarrow="3\msx@10
\mathchardef\twoheadleftarrow="3\msx@11
\mathchardef\leftleftarrows="3\msx@12
\mathchardef\rightrightarrows="3\msx@13
\mathchardef\upuparrows="3\msx@14
\mathchardef\downdownarrows="3\msx@15
\mathchardef\upharpoonright="3\msx@16
\let\restriction\upharpoonright
\mathchardef\downharpoonright="3\msx@17
\mathchardef\upharpoonleft="3\msx@18
\mathchardef\downharpoonleft="3\msx@19
\mathchardef\rightarrowtail="3\msx@1A
\mathchardef\leftarrowtail="3\msx@1B
\mathchardef\leftrightarrows="3\msx@1C
\mathchardef\rightleftarrows="3\msx@1D
\mathchardef\Lsh="3\msx@1E
\mathchardef\Rsh="3\msx@1F
\mathchardef\rightsquigarrow="3\msx@20
\mathchardef\leftrightsquigarrow="3\msx@21
\mathchardef\looparrowleft="3\msx@22
\mathchardef\looparrowright="3\msx@23
\mathchardef\circeq="3\msx@24
\mathchardef\succsim="3\msx@25
\mathchardef\gtrsim="3\msx@26
\mathchardef\gtrapprox="3\msx@27
\mathchardef\multimap="3\msx@28
\mathchardef\therefore="3\msx@29
\mathchardef\because="3\msx@2A
\mathchardef\doteqdot="3\msx@2B
\let\Doteq\doteqdot
\mathchardef\triangleq="3\msx@2C
\mathchardef\precsim="3\msx@2D
\mathchardef\lesssim="3\msx@2E
\mathchardef\lessapprox="3\msx@2F
\mathchardef\eqslantless="3\msx@30
\mathchardef\eqslantgtr="3\msx@31
\mathchardef\curlyeqprec="3\msx@32
\mathchardef\curlyeqsucc="3\msx@33
\mathchardef\preccurlyeq="3\msx@34
\mathchardef\leqq="3\msx@35
\mathchardef\leqslant="3\msx@36
\mathchardef\lessgtr="3\msx@37
\mathchardef\backprime="0\msx@38
\mathchardef\risingdotseq="3\msx@3A
\mathchardef\fallingdotseq="3\msx@3B
\mathchardef\succcurlyeq="3\msx@3C
\mathchardef\geqq="3\msx@3D
\mathchardef\geqslant="3\msx@3E
\mathchardef\gtrless="3\msx@3F
\mathchardef\sqsubset="3\msx@40
\mathchardef\sqsupset="3\msx@41
\mathchardef\vartriangleright="3\msx@42
\mathchardef\vartriangleleft ="3\msx@43
\mathchardef\trianglerighteq="3\msx@44
\mathchardef\trianglelefteq="3\msx@45
\mathchardef\bigstar="0\msx@46
\mathchardef\between="3\msx@47
\mathchardef\blacktriangledown="0\msx@48
\mathchardef\blacktriangleright="3\msx@49
\mathchardef\blacktriangleleft="3\msx@4A
\mathchardef\vartriangle="3\msx@4D
\mathchardef\blacktriangle="0\msx@4E
\mathchardef\triangledown="0\msx@4F
\mathchardef\eqcirc="3\msx@50
\mathchardef\lesseqgtr="3\msx@51
\mathchardef\gtreqless="3\msx@52
\mathchardef\lesseqqgtr="3\msx@53
\mathchardef\gtreqqless="3\msx@54
\mathchardef\Rrightarrow="3\msx@56
\mathchardef\Lleftarrow="3\msx@57
\mathchardef\veebar="2\msx@59
\mathchardef\barwedge="2\msx@5A
\mathchardef\doublebarwedge="2\msx@5B
\mathchardef\angle="0\msx@5C
\mathchardef\measuredangle="0\msx@5D
\mathchardef\sphericalangle="0\msx@5E
\mathchardef\varpropto="3\msx@5F
\mathchardef\smallsmile="3\msx@60
\mathchardef\smallfrown="3\msx@61
\mathchardef\Subset="3\msx@62
\mathchardef\Supset="3\msx@63
\mathchardef\Cup="2\msx@64
\let\doublecup\Cup
\mathchardef\Cap="2\msx@65
\let\doublecap\Cap
\mathchardef\curlywedge="2\msx@66
\mathchardef\curlyvee="2\msx@67
\mathchardef\leftthreetimes="2\msx@68
\mathchardef\rightthreetimes="2\msx@69
\mathchardef\subseteqq="3\msx@6A
\mathchardef\supseteqq="3\msx@6B
\mathchardef\bumpeq="3\msx@6C
\mathchardef\Bumpeq="3\msx@6D
\mathchardef\lll="3\msx@6E
\let\llless\lll
\mathchardef\ggg="3\msx@6F
\let\gggtr\ggg
\mathchardef\circledS="0\msx@73
\mathchardef\pitchfork="3\msx@74
\mathchardef\dotplus="2\msx@75
\mathchardef\backsim="3\msx@76
\mathchardef\backsimeq="3\msx@77
\mathchardef\complement="0\msx@7B
\mathchardef\intercal="2\msx@7C
\mathchardef\circledcirc="2\msx@7D
\mathchardef\circledast="2\msx@7E
\mathchardef\circleddash="2\msx@7F
\def\ulcorner{\delimiter"4\msx@70\msx@70 }
\def\urcorner{\delimiter"5\msx@71\msx@71 }
\def\llcorner{\delimiter"4\msx@78\msx@78 }
\def\lrcorner{\delimiter"5\msx@79\msx@79 }
\def\yen{{\mathhexbox@\msx@55 }}
\def\checkmark{{\mathhexbox@\msx@58 }}
\def\circledR{{\mathhexbox@\msx@72 }}
\def\maltese{{\mathhexbox@\msx@7A }}
\mathchardef\lvertneqq="3\msy@00
\mathchardef\gvertneqq="3\msy@01
\mathchardef\nleq="3\msy@02
\mathchardef\ngeq="3\msy@03
\mathchardef\nless="3\msy@04
\mathchardef\ngtr="3\msy@05
\mathchardef\nprec="3\msy@06
\mathchardef\nsucc="3\msy@07
\mathchardef\lneqq="3\msy@08
\mathchardef\gneqq="3\msy@09
\mathchardef\nleqslant="3\msy@0A
\mathchardef\ngeqslant="3\msy@0B
\mathchardef\lneq="3\msy@0C
\mathchardef\gneq="3\msy@0D
\mathchardef\npreceq="3\msy@0E
\mathchardef\nsucceq="3\msy@0F
\mathchardef\precnsim="3\msy@10
\mathchardef\succnsim="3\msy@11
\mathchardef\lnsim="3\msy@12
\mathchardef\gnsim="3\msy@13
\mathchardef\nleqq="3\msy@14
\mathchardef\ngeqq="3\msy@15
\mathchardef\precneqq="3\msy@16
\mathchardef\succneqq="3\msy@17
\mathchardef\precnapprox="3\msy@18
\mathchardef\succnapprox="3\msy@19
\mathchardef\lnapprox="3\msy@1A
\mathchardef\gnapprox="3\msy@1B
\mathchardef\nsim="3\msy@1C
\mathchardef\napprox="3\msy@1D
\mathchardef\varsubsetneq="3\msy@20
\mathchardef\varsupsetneq="3\msy@21
\mathchardef\nsubseteqq="3\msy@22
\mathchardef\nsupseteqq="3\msy@23
\mathchardef\subsetneqq="3\msy@24
\mathchardef\supsetneqq="3\msy@25
\mathchardef\varsubsetneqq="3\msy@26
\mathchardef\varsupsetneqq="3\msy@27
\mathchardef\subsetneq="3\msy@28
\mathchardef\supsetneq="3\msy@29
\mathchardef\nsubseteq="3\msy@2A
\mathchardef\nsupseteq="3\msy@2B
\mathchardef\nparallel="3\msy@2C
\mathchardef\nmid="3\msy@2D
\mathchardef\nshortmid="3\msy@2E
\mathchardef\nshortparallel="3\msy@2F
\mathchardef\nvdash="3\msy@30
\mathchardef\nVdash="3\msy@31
\mathchardef\nvDash="3\msy@32
\mathchardef\nVDash="3\msy@33
\mathchardef\ntrianglerighteq="3\msy@34
\mathchardef\ntrianglelefteq="3\msy@35
\mathchardef\ntriangleleft="3\msy@36
\mathchardef\ntriangleright="3\msy@37
\mathchardef\nleftarrow="3\msy@38
\mathchardef\nrightarrow="3\msy@39
\mathchardef\nLeftarrow="3\msy@3A
\mathchardef\nRightarrow="3\msy@3B
\mathchardef\nLeftrightarrow="3\msy@3C
\mathchardef\nleftrightarrow="3\msy@3D
\mathchardef\divideontimes="2\msy@3E
\mathchardef\varnothing="0\msy@3F
\mathchardef\nexists="0\msy@40
\mathchardef\mho="0\msy@66
\mathchardef\thorn="0\msy@67
\mathchardef\beth="0\msy@69
\mathchardef\gimel="0\msy@6A
\mathchardef\daleth="0\msy@6B
\mathchardef\lessdot="3\msy@6C
\mathchardef\gtrdot="3\msy@6D
\mathchardef\ltimes="2\msy@6E
\mathchardef\rtimes="2\msy@6F
\mathchardef\shortmid="3\msy@70
\mathchardef\shortparallel="3\msy@71
\mathchardef\smallsetminus="2\msy@72
\mathchardef\thicksim="3\msy@73
\mathchardef\thickapprox="3\msy@74
\mathchardef\approxeq="3\msy@75
\mathchardef\succapprox="3\msy@76
\mathchardef\precapprox="3\msy@77
\mathchardef\curvearrowleft="3\msy@78
\mathchardef\curvearrowright="3\msy@79
\mathchardef\digamma="0\msy@7A
\mathchardef\varkappa="0\msy@7B
\mathchardef\hslash="0\msy@7D
\mathchardef\hbar="0\msy@7E
\mathchardef\backepsilon="3\msy@7F
\def\Bbb{\relaxnext@\ifmmode\let\next\Bbb@\else
\def\next{\Err@{Use \string\Bbb\space only in math mode}}\fi\next}
\def\Bbb@#1{{\Bbb@@{#1}}}
\def\Bbb@@#1{\noaccents@\fam\msyfam#1}
\catcode`\@=12
%\font\teneuf=eufm10
%\font\fiveeuf=eufm5
%\font\seveneuf=eufm7
%\font\sc=cmcsc10
% msxm and msym fonts changed for msam and msbm fonts
% 01.01.92
\font\tenmsy=msbm10
\font\sevenmsy=msbm7
\font\fivemsy=msbm5
\font\tenmsx=msam10
\font\sevenmsx=msam7
\font\fivemsx=msam5
%\font\sevenmcyr=mcyr7
%\font\tenmcyr=mcyr10
%%\font\tenmcyb=mcyb10
%%\font\eightmcyb=mcyb8
\newfam\msyfam
\def\msy{\fam\msyfam\tenmsy}
\textfont\msyfam=\tenmsy
\scriptfont\msyfam=\sevenmsy
\scriptscriptfont\msyfam=\fivemsy
\newfam\msxfam
\def\msx{\fam\msxfam\tenmsx}
\textfont\msxfam=\tenmsx
\scriptfont\msxfam=\sevenmsx
\scriptscriptfont\msxfam=\fivemsx
%\newfam\mcyrfam
%\def\mcyr{\fam\mcyrfam\tenmcyr}
%\textfont\mcyrfam=\tenmcyr
%\scriptfont\mcyrfam=\sevenmcyr
%\newfam\euffam
%\def\euf{\fam\euffam\teneuf}
%\textfont\euffam=\teneuf
%\scriptfont\euffam=\seveneuf
%\scriptscriptfont\euffam=\fiveeuf
\def\bba{{\msy A}}
\def\bbb{{\msy B}}
\def\bbc{{\msy C}}
\def\bbd{{\msy D}}
\def\bbe{{\msy E}}
\def\bbf{{\msy F}}
\def\bbg{{\msy G}}
\def\bbh{{\msy H}}
\def\bbi{{\msy I}}
\def\bbj{{\msy J}}
\def\bbk{{\msy K}}
\def\bbl{{\msy L}}
\def\bbm{{\msy M}}
\def\bbn{{\msy N}}
\def\bbo{{\msy O}}
\def\bbp{{\msy P}}
\def\bbq{{\msy Q}}
\def\bbr{{\msy R}}
\def\bbs{{\msy S}}
\def\bbt{{\msy T}}
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\def\bbz{{\msy Z}}
%
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H^5_\del]}
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\define\cP{\Cal P}
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\def\st{\text{such that }}
\def\fs{\text{finitely satisfiable\ }}
%\def\bgam{{\bold\gamma}}
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\def\auto{\text{automorphism }}
\def\tp{\text{tp}}
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\def\varp{\varphi}
\def\Gam{\Gamma}
\def\gam{\gamma}
\def\al{\aleph}
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\def\Lam{\Lambda}
\def\sig{\sigma}
\def\Sig{\Sigma}
\def\Del{\Delta}
\def\del{\delta}
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\def\gI{{\frak\$I}}
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%\define\cf{\; \text{cf} \; }
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%\define\wl{\text{wlog}}
%\define\cP{\Cal P}
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\def\raro{\rightarrow}
%\define\bW{\bold{W}}
%\define\bZ{\bold{Z}}
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%\define\bt{\bold{t}}
%\define\bT{\bold{T}}
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%\define\cS{\Cal{S}}
%\define\cB{\Cal{B}}
%\define\cI{\Cal{I}}
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%\define\bA{\bold{A}}
%\define\cD{\Cal{D}}
\def\uhr{\upharpoonright}
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\def\rseq{\rangle}
\def\ksi{\xi}
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\def\ov{\overline}
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\def\rng{\rangle}
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\def\cov{{\rm cov}}
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\def\Om{\Omega}
\def\Def{\text{Definition}}
\def\st{\text{such that }}
\def\fs{\text{finitely satisfiable\ }}
%\def\bgam{{\bold\gamma}}
%\def\bGam{{\bold\Gamma}}
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\def\cont{\text{continuous }}
\def\seq{\text{sequence }}
\def\Pf{\text{\underbar{Proof }}}
\def\auto{\text{automorphism }}
\def\tp{\text{tp}}
\def\acl{\text{acl}}
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\def\varp{\varphi}
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\def\gam{\gamma}
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%\def\lorop{\mathop\lor}
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\def\G{\Gamma}
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\def\B{{\cal B}}
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\def\K{{\cal K}}
\def\H{{\cal H}}
\def\u{{\cal U}}
\def\ord{{\hbox{\rm Ord}}}
\def\zfc{{\hbox{\rm ZFC}}}
\def\tc{{\hbox{\rm TC}}}
\def\gch{{\hbox{\rm GCH}}}
\def\dom{{\hbox{\rm dom}}}
\def\max{{\hbox{\rm max}}}
\def\min{{\hbox{\rm min}}}
\def\ath{\alpha^{\hbox{\rm th}}}
\def\bth{\beta^{\hbox{\rm th}}}
\def\pkl{P_\kappa(\lambda)}
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\def\lr{L[\Re]}
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\def\f{\varphi}
\def\som{{[S]}^{< \omega}}
\def\pcond{\la s, S \ra}
\def\bind{\indent\indent\indent\indent}
\def\vpk{V^{P_\k}}
\def\pic{\underset i \in C \to{\prod} G_i}
\def\pic0{\underset i \in C_0 \to{\prod} G_i}
\def\bigp{P^0_{\d, \l} \ast (P^1_{\d, \l}[\dot S] \times
P^2_{\d, \l}[\dot S])}
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\voffset=-.15in\hoffset=0.00in\vsize=7.65in
\no \S 0 Introduction and Preliminaries
As is well-known, the notion of strong compactness is a
singularity in the hierarchy of large cardinals. The
fundamental work of Magidor [Ma] shows that the least
strongly compact cardinal $\k$ can either be the least
supercompact cardinal or the least measurable cardinal,
in which case $\k$ isn't even $2^\k$ supercompact. A
generalization of this result by Kimchi and Magidor
[KiM] shows that the (possibly proper) classes of
supercompact and strongly compact cardinals can coincide
except at measurable limit points or that the first $n$
(for $n \in \omega$) strongly compact cardinals can be the
first $n$ measurable cardinals.
The purpose of this paper is to show that the techniques of
[AS97a] and [AS97b] can be combined with unpublished ideas of
Magidor to produce a model in which the first $n$
(for $n \in \omega$) strongly compact cardinals are not only
the first $n$ measurable cardinals, but each is a little
supercompact. Specifically, we prove the following.
\proclaim{Theorem 1}
Con(ZFC + There are $n \in \omega$ supercompact cardinals)
$\lra$
Con(ZFC + The first $n$ strongly compact cardinals
$\k_1, \ldots, \k_n$ are the first $n$ measurable cardinals +
$2^{\k_i} = \k^{++}_i$ for $i = 1, \ldots, n$ + Each
$\k_i$ is $\k^+_i$ supercompact for $i = 1, \ldots n$).
\endproclaim
A bit of history is perhaps in order now.
As was just noted, in the early 1970s,
Magidor in [Ma] showed the consistency,
relative to the existence of a strongly compact cardinal,
of the least strongly compact cardinal being the
least measurable cardinal.
In the spring of 1983, Woodin,
in response to a question put to him by the first author,
showed the consistency (see [CW]),
relative to the existence of a cardinal $\k$ which is
$\k^{+3}$ supercompact,
of the least measurable cardinal $\k$ being so that
$2^\k = \k^{++}$ and $\k$ is $\k^+$ supercompact.
In the mid 1980s, Kimchi and Magidor did the
work of [KiM].
In late 1992, Shelah and the first author began
the research leading to the results of [AS97a]
and [AS97b].
The main theorem of [AS97a] showed, roughly
speaking, the relative consistency of the
classes of strongly compact and supercompact
cardinals coinciding level by level, except
where explicitly prohibited by ZFC.
This strengthened the work of [KiM].
The main theorem of [AS97b] showed that
Menas' result of [Me] that the least
measurable limit $\k$ of either strongly
compact or supercompact cardinals isn't
$2^\k$ supercompact is best possible by
constructing, starting from a supercompact
limit of supercompact cardinals, a model
in which the least measurable limit $\k$
of both strongly compact and supercompact
cardinals is so that $2^\k = \k^{++}$ and
$\k$ is $\k^+$ supercompact.
The forcing conditions of [AS97b] were
generalizations of the forcing conditions
of [AS97a], and both provided, among other things,
an alternate way of proving Woodin's
aforementioned 1983 theorem.
This still left open the question of combining
Woodin's results with the results of Magidor
and Kimchi and Magidor, i.e., obtaining a model
in which the least measurable cardinal $\k$ is both
the least strongly compact cardinal and is $\k^+$
supercompact, or in general, obtaining a model
in which the first $n$ measurable cardinals
$\k_1, \ldots, \k_n$
(for $n \in \omega$)
are the first $n$ strongly compact cardinals,
with each measurable cardinal $\k_i$ being
$\k^+_i$ supercompact.
This question went unresolved for a number of years,
despite several attempts at solving it made by
Shelah and the first author.
Then, during the January 7-13, 1996 meeting in
Set Theory held at the Mathematics Research
Institute, Oberwolfach, the first author proved
(see [A97a])
Theorem 1 for $n = 1$.
His proof, however, was non-iterative,
and the question remained of proving Theorem 1
for arbitrary finite $n$.
This question was finally resolved by the
two authors of this paper during their
stay as Research in Pairs fellows at the
Mathematics Research Institute, Oberwolfach,
June 8-21, 1997.
We take the opportunity here to make two remarks about
Theorem 1. The first is that there is
nothing special about each $\k_i$ being $\k^+_i$ supercompact
in Theorem 1. In fact, each $\k_i$ can be
$\k^{++}_i$, $\k^{+++}_i$, $\k^{+4}_i$, etc$.$ supercompact,
so long as
$2^{\k_i} > \k^{++}_i$,
$2^{\k_i} > \k^{+++}_i$,
$2^{\k_i} > \k^{+4}_i$, etc.
After completing the proof of Theorem 1, interested readers are
invited to look at the statement of Theorem 3 of [AS97b]
in order to determine for themselves what variants are
possible. The second is that
no proof is currently known, in both
Theorem 1 and the corresponding result of [KiM], when
$n$ is infinite. This will be discussed further at the
end of the paper.
The structure of this paper is as follows.
Section 0 contains our introductory comments and
preliminary remarks concerning notation, terminology, etc.
Section 1 contains a discussion of a certain modification
of the main forcing notion of [AS97a] (given at the end of
[AS97b]) that will be critical in the proof of Theorem 1.
Section 1 also contains a discussion of the main forcing
notion of [AS97b], which can be used as an alternative in
the proof of Theorem 1.
Section 2 shows how the forcing notions discussed in
Section 1 can be used to force a supercompact cardinal to
have a certain special kind of supercompactness embedding
that will be key in the proof of Theorem 1.
Section 3 then gives Magidor's unpublished proof,
which is quite different from the one found in [Ma],
of the consistency, relative to a supercompact
cardinal, of the least measurable cardinal
being the least strongly compact cardinal.
The ideas used in this proof will be critical in
the proof of Theorem 1.
Section 4 then gives a proof of Theorem 1 for $n=1$.
Section 5 contains a proof of Theorem 1 for arbitrary
finite $n$.
Section 6 has our concluding remarks.
We give now some preliminary information concerning
notation and terminology.
Essentially, our notation and terminology are standard, and when
this is not the case, this will be clearly noted.
For $\a < \b$ ordinals, $[\a, \b], [\a, \b), (\a, \b]$, and
$(\a, \b)$ are as in standard interval notation. If $x$ is a set,
then $\tc(x)$ is the transitive closure of $x$.
When forcing, $q \ge p$ will mean that $q$ is stronger than $p$.
For $P$ a partial ordering, $\varphi$ a formula in the forcing
language with respect to $P$, and $p \in P$, $p \|
\varphi$ will mean $p$ decides $\varphi$.
For $G$ $V$-generic over $P$, we will use both $V[G]$ and $V^{P}$
to indicate the universe obtained by forcing with $P$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$ when we actually mean $\dot x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
If $\k$ is a cardinal and $P$ is
a partial ordering, $P$ is $\k$-closed if given a sequence
$\la p_\a: \a < \k \ra$ of elements of $P$ so that
$\b < \g < \k$ implies $p_\b \le p_\g$ (an increasing chain of
length
$\k$), then there is some $p \in P$ (an upper bound to this chain)
so that $p_\a \le p$ for all $\a < \k$.
$P$ is $<\k$-closed if $P$ is $\d$-closed for all cardinals $\d <
\k$.
$P$ is $\k$-directed closed if for every cardinal $\d < \k$ and
every directed
set $\la p_\a : \a < \d \ra $ of elements of $P$
(where $\la p_\a : \a < \d \ra$ is directed if
for every two distinct elements $p_\rho, p_\nu \in
\la p_\a : \a < \d \ra$, $p_\rho$ and
$p_\nu$ have a common upper bound $p_\sigma$) there is an
upper bound $p \in P$. $P$ is $\k$-strategically closed if in the
two person game in which the players construct an increasing
sequence
$\la p_\a: \a \le\k\ra$, where player I plays odd stages and player
II plays even and limit stages, then player II has a strategy which
ensures the game can always be continued.
Note that if $P$ is $\k$-strategically closed and
$f : \k \to V$ is a function in $V^P$, then $f \in V$.
$ P$ is $< \k$-strategically closed if $P$ is $\d$-strategically
closed for all cardinals $\d < \k$.
$P$ is $\prec \k$-strategically closed if in the two
person game in which the players construct an increasing
sequence $\la p_\a : \a < \k \ra$, where
player I plays odd stages and player II plays even and limit
stages, then player II has a strategy which ensures the game
can always be continued.
Note that trivially, if $P$ is $<\k$-closed, then $P$ is
$< \k$-strategically
closed and $\prec \k $-strategically closed. The converse of
both of these facts is false.
The usual partial ordering for adding $\l$ Cohen
subsets to a regular cardinal $\k$ will be written as
${\hbox{\rm Add}}(\k, \l)$.
Standard arguments show ${\hbox{\rm Add}}(\k, \l)$ is
$\k$-directed closed. See [J],
Lemmas 19.7 and 19.8, pages
181-182, for further details.
We mention that
we are assuming complete familiarity with the notions of
measurability, strong compactness, and supercompactness.
Interested readers may consult [SRK], [Ka], or [KaM] for
further details.
We note only that all elementary embeddings witnessing the $\lambda$
supercompactness of $\k$ are presumed to come from some
fine, $\k$-complete, normal
ultrafilter $\cU$ over $P_\k (\l) = \{ x \subseteq \l: | x| < \k \}
$, and all elementary embeddings witnessing the $\l$
strong compactness of $\k$ are presumed to come from
some fine, $\k$-complete ultrafilter $\cU$ over
$P_\k(\l)$. An equivalent definition for $\k$
being $\l$ strongly compact is that there is an
elementary embedding $j : V \to M$ having critical
point $\k$ so that for any $x \subseteq M$ with
$|x| \le \l$, there is some $y \in M$ such that
$x \subseteq y$ and $M \models ``|y| < j(\k)$''.
Finally, we mention that since ideas and notions from [AS97a] and [AS97b]
are used throughout the course of this paper, it would be most
helpful to readers if copies of these papers were kept close
at hand.
In fact, at many instances during our exposition,
we will refer to proofs not given here but found
in either [AS97a] or [AS97b].
These papers, however, need not be read in their entirety
to follow this paper.
In order to facilitate the understanding of readers,
though, we will keep as much as possible to the
notations and terminology of [AS97a] and [AS97b].
\no \S 1 The Forcing Notions of [AS97a] and [AS97b]
Fix $\g < \d < \l$, $\l > \d^+$ regular cardinals in our
ground model $V$, with $\d$ inaccessible and $\l$ either
inaccessible or the successor of a cardinal of cofinality
$> \d$. We recall now the partial orderings
$P^0_{\d, \l}$
and
$P^2_{\d, \l}[S]$
of [AS97a] and [AS97b] and the version of
$P^1_{\d, \l}[S]$
given in Section 4 of [AS97b] (which is a modification
of the partial ordering
$P^1_{\d, \l}[S]$
of [AS97a]).
%and the version of
%$P^1_{\d, \l}[S]$
%of [AS97b] used in the proof of Theorem 3 of that paper.
We assume GCH holds for all
cardinals $\k \ge \d$.
As in Section 4 of [AS97b],
the first notion of forcing $P^0_{\d, \l}$ is just
the standard notion of forcing
for adding a non-reflecting stationary set of ordinals of cofinality
$\g $ to $\l$.
Specifically, $P^0_{\d,\l} = \{ p$ : For some
$\a < \l$, $p : \a \to \{0,1\}$ is a characteristic
function of $S_p$, a subset of $\a$ not stationary at its
supremum nor having any initial segment which is stationary
at its supremum, so that $\b \in S_p$ implies
$\b > \d$ and cof$(\b) = \g \}$,
ordered by $q \ge p$ iff $q \supseteq p$ and $S_p = S_q \cap
\sup (S_p)$, i.e., $S_q$ is an end extension of $S_p$.
It is well-known
that for $G$ $V$-generic over $P^0_{\d, \l}$ (see
[B] or [KiM]), in $V[G]$,
since GCH holds in $V$ for all cardinals
$\k \ge \d$, a non-reflecting stationary
set $S=S[G]=\cup\{S_p:p\in G \} \subseteq \l$
of ordinals of cofinality $\g $ has been introduced, the
bounded subsets of $\l$ are the same as those in $V$,
and cardinals, cofinalities, and GCH at cardinals
$\k \ge \d$ have been preserved.
It is also virtually immediate that $P^0_{\d, \l}$
is $\g$-directed closed, and it can be shown
(see [B], Lemma 4.15, page 436 or [KiM]) that
$P^0_{\d, \l}$
is $\prec \l$-strategically closed.
Work now in $V_1 = V^{P^0_{\d, \l}}$, letting $\dot S$
be a term always forced to denote the above set $S$.
$P^2_{\d, \l}[S]$ is the standard notion of forcing
for introducing a club set $C$ which is disjoint to $S$
(and therefore makes $S$ non-stationary).
Specifically, $P^2_{\d, \l} [S] = \{ p$ : For
some successor ordinal $\a < \l$,
$p : \a \to \{0,1\}$ is a characteristic function of
$C_p$, a club subset of $\a$, so that
$C_p \cap S = \emptyset \}$,
ordered by $ q \ge p $ iff $C_q$ is an end extension of $C_p$.
It is again well-known (see [MS]) that for $H$
$V_1$-generic over $P^2_{\d, \l}[S]$, a club set
$C = C[H] = \cup \{C_p : p \in H \}
\subseteq \l$ which is disjoint to $S$ has been introduced,
the bounded subsets of $\l$
are the same as those in $V_1$,
and cardinals, cofinalities, and GCH
for cardinals $\k \ge \d$ have been preserved.
The following lemma is proven in both [AS97a] and [AS97b].
\proclaim{Lemma 1 (Lemma 1 of [AS97a] and [AS97b])}
$\force_{P^0_{\d, \l}} ``
\clubsuit({\dot S})$'', i.e., $V_1 \models ``$There is a
sequence $\langle x_\alpha : \alpha \in S \rangle$ so that
for each $\alpha \in S$, $ x_\alpha \subseteq \alpha$ is
cofinal in
$\alpha$, and for any $A \in
{[\l]}^{\l}$, $\{\alpha \in S : x_\alpha
\subseteq A \}$ is stationary''.
\endproclaim
We fix now in $V_1$ a $\clubsuit(S)$ sequence
$X = \la x_\a : \a \in S \ra$.
We are ready to define in $V_1$
in the same manner as was done in
Section 4 of [AS97b]
the partial ordering $P^1_{\d, \l}
[S] $.
First, since each element of
$S$ has cofinality $\g$,
each $x \in X$ can be assumed to be
so that order type$(x) = \g$. Then,
$P^1_{\dell}[ S]$ is defined as the set of all
4-tuples $\la w, \a, \bar r, Z \ra$ satisfying the
following properties.
\item{1.} $w \in {[\l]}^{< \d}$.
\item{2.} $\a < \d$.
\item{3.} $ \bar r = \la r_i : i \in w \ra$ is a
sequence of functions from $\a$ to $\{0,1\}$, i.e.,
a sequence of subsets of $\a$.
\item{4.} $Z \subseteq \{x_\b : \b \in S\}$
is a set so that if $z \in Z$, then for some
$y \in {[w]}^\g$, $y \subseteq z$ and $z - y$
is bounded in the $\b$ so that $z = x_\b$.
In other words, for every $x_\b \in Z$,
$w \cap x_\b$ is cobounded in $x_\b$.
\noindent As in [AS97a], the definition of $Z$ implies
$|Z| < \d$.
The ordering on $P^1_{\dell}[S]$ is given by
$\la w^1, \a^1, \bar r^1, Z^1 \ra \le
\la w^2, \a^2, \bar r^2, Z^2 \ra$ iff the following hold.
\item{1.} $w^1 \subseteq w^2$.
\item{2.} $\a^1 \le \a^2$.
\item{3.} If $i \in w^1$, then $r^1_i
\subseteq r^2_i$.
\item{4.} $Z^1 \subseteq Z^2$.
\item{5.} If $z \in Z^1 \cap {[w^1]}^\g$ and
$\a^1 \le \a < \a^2$, then $|\{i \in z :
r^2_i(\a) = 0\}| = |\{i \in z : r^2_i(\a) = 1\}| = \g$.
The intuition behind the definition of $P^1_{\dell}[S]$
just given is essentially the same as in [AS97a]. Specifically,
we wish to be able simultaneously to make $2^\d = \l$,
destroy the measurability of $\d$, and be able to
resurrect the $< \l$ supercompactness of $\d$ if
necessary. $P^1_{\dell}[S]$ has been designed so as to
allow us to do all of these things.
The proof that $V^{P^1_{\dell}[S]}_1 \models
``\d$ is non-measurable'' is as in Lemma 3 of [AS97a].
In particular, the argument of Lemma 3 of [AS97a] will show
that $\d$ can't carry a $\g$-additive uniform ultrafilter.
We can then carry through the proof of Lemma 4 of [AS97a] to show
$P^0_{\dell} \ast (P^1_{\dell}[\dot S] \times
P^2_{\dell}[\dot S])$ is equivalent to
${\hbox{\rm Add}}(\l, 1) \ast \dot {\hbox{\rm Add}}(\d, \l)$.
The proofs of Lemma 5 of [AS97a]
and Lemma 6 of [AS97b] will then show
$P^0_{\dell} \ast P^1_{\dell}[\dot S]$ preserves cardinals
and cofinalities, is $\l^+$-c.c.,
is $< \d$-strategically closed, and is so that
$V^{P^0_{\dell} \ast P^1_{\dell}[\dot S]} \models
``2^\k = \l$ for every cardinal $\k \in [\d, \l)$''.
Although the above definition of
$P^1_{\d, \l}[S]$
(henceforth to be referred to as the ``simpler form'')
is perfectly adequate for our purposes, as mentioned at
the end of [AS97b], it will not suffice to prove Theorem
3 of [AS97b]. In order to do this, a more complicated form of
$P^1_{\d, \l}[S]$
is required.
The more complicated version of
$P^1_{\d, \l}[S]$
can then be used in the proof of Theorem 1 of this paper.
Interested readers may consult Section 1 of [AS97b]
for further details on the definition of this partial
ordering, but we will not repeat it here.
We will, however, structure our proofs so that
they are equally applicable, regardless of the version of
$P^1_{\d, \l}$ used.
%We conclude this section by noting that there are
%additional properties of the more complicated version
%of $P^1_{\d, \l}[S]$ that will be relevant to our work.
%These will be discussed in more detail in later sections.
\no \S 2 A Supercompact Cardinal with a Special Kind of
Embedding
In this section, we force and construct a supercompact
cardinal possessing a special sort of supercompactness
embedding. Such cardinals will be critical in the proof
of Theorem 1. Specifically, we prove the following.
\proclaim{Lemma 2}
Suppose $V \models ``$ZFC + GCH + $\k$ is supercompact''.
There is then a partial ordering $P^{\k, 0} \in V$ so that
$V^{P^{\k, 0}} \models ``\k$ is supercompact +
$2^\k = \k^{++}$''. In addition, there is an elementary embedding
$j^* : V^{P^{\k, 0}} \to M^{j^*(P^{\k ,0})}$
definable in $V^{P^{\k, 0}}$ witnessing the $\k^+$
supercompactness of $\k$ so that
$M^{j^*(P^{\k, 0})} \models ``\k$ isn't measurable''.
\endproclaim
\demo{Proof of Lemma 2}
Fix $f : \k \to V_\k$ a Laver function [L], i.e.,
$f$ is so that for every $x$ and every
$\l \ge |\tc(x)|$, there is a $\l$ supercompact ultrafilter
${\cal U}_{\l, x}$ with associated embedding
$j_{{\cal U}_{\l, x}} : V \to M$ so that
$j_{{\cal U}_{\l, x}}(f)(\k) = x$.
Also, let $\la \d_\a : \a \le \k \ra$ enumerate
the inaccessibles $\le \k$, and let $\g < \d_0$
be a fixed but arbitrary regular cardinal.
As Laver does in [L],
we define now simultaneously an Easton support iteration
$P^{\k, 0} = \la \la P^*_\a, \dot Q^*_\a \ra :
\a \le \k \ra$ and a sequence of ordinals
$\la \rho_\a : \a < \k \ra$, where
$\rho_\l = \underset \a < \l \to{\cup} \rho_\a$
if $\l$ is a limit ordinal. We use here in our definition
the simpler form of
$P^1_{\d, \l}[S]$
of Section 1 defined using $\g$ and the associated
$P^0_{\d, \l}$
and
$P^2_{\d, \l}[S]$
and indicate at the end of the section
the modifications needed when the more complicated form of
$P^1_{\d, \l}[S]$
is used. Specifically, the definition has $P_0$ being trivial
with $\rho_0 = 0$, and $P^*_{\a + 1} = P^*_\a \ast \dot Q^*_\a$,
where $\force_{P_\a} ``\dot Q^*_\a$ is trivial'' and
$\rho_{\a + 1} = \rho_\a$ unless one of the following holds:
\item{1.} If for all $\b < \a$,
$\rho_\b < \a$ and $\d_\a < \k$ is so that
$V \models ``\d_\a$ isn't $\d^+_\a$ supercompact'', then
$P^*_{\a + 1} = P^*_\a \ast \dot Q^*_\a$, where
$\dot Q^*_\a$ is a term for
$P^0_{\d_\a, \d^{++}_\a} \ast
P^1_{\d_\a, \d^{++}_\a}[\dot S_{\d^{++}_\a}]$,
and $\dot S_{\d^{++}_\a}$ is a term for the
non-reflecting stationary subset of $\d^{++}_\a$
introduced by $P^0_{\d_a, \d^{++}_\a}$.
If $f(\a)$ is an ordinal and $f(\a) > \rho_\a$,
then $\rho_{\a + 1} = f(\a)$. If this condition on
$f(\a)$ doesn't hold, then $\rho_{\a + 1} = \rho_\a$.
\item{2.} If for $\b < \a$,
$\rho_\b < \a$ and $\d_\a \le \k$ is so that
$V \models ``\d_\a$ is $\d^+_\a$ supercompact'', then
$P^*_{\a + 1} = P^*_\a \ast \dot Q^*_\a$, where
$\dot Q^*_\a$ is a term for
$P^0_{\d_\a, \d^{++}_\a} \ast
(P^1_{\d_\a, \d^{++}_\a}[\dot S_{\d^{++}_\a}] \times
P^2_{\d_\a, \d^{++}_\a}[\dot S_{\d^{++}_\a}])$.
If $f(\a)$ is an ordinal and $f(\a) > \rho_\a$,
then $\rho_{\a + 1} = f(\a)$. If this condition on
$f(\a)$ doesn't hold, then $\rho_{\a + 1} = \rho_\a$.
Suppose now $j : V \to M$ is an embedding witnessing
the $\k^+$ supercompactness of $\k$ so that
$M \models ``\k$ isn't $\k^+$ supercompact''.
Lemma 9 of [AS97b] shows that if $P^{\k, 0}
= P^*_\k \ast \dot Q^*_\k =
P^*_\k \ast
(\dot P^0_{\k, \k^{++}} \ast
(P^1_{\k, \k^{++}}[\dot S_{\k^{++}}] \times
P^2_{\k, \k^{++}}[\dot S_{\k^{++}}]))$
were an iteration as defined in the proofs of Theorem 1
or Theorem 3 of [AS97b], then $j : V \to M$ extends to
$j^* : V^{P^{\k, 0}} \to M^{j^*(P^{\k, 0})}$
witnessing the $\k^+$ supercompactness of $\k$
in a manner definable in $V^{P^{\k, 0}}$.
The type of iteration used in the proofs of Theorem 1
or Theorem 3 of [AS97b], however, is essentially the
one just described here. The only real difference is
that here, we use a Laver function to ``space out''
the iteration at successor stages below $\k$.
At stage $\k + 1$ in $V$, however, the partial
ordering used in the iteration is
$P^0_{\k , \k^{++} } \ast
(P^1_{\k, \k^{++} }[\dot S_{\k^{++} }] \times
P^2_{\k , \k^{++} }[\dot S_{\k^{++}} ])$,
and at stage $\k + 1$ in $M$, the partial ordering
used in the iteration is
$P^0_{\k , \k^{++} } \ast
P^1_{\k, \k^{++} }[\dot S_{\k^{++} }]$.
These occurrences at stage $\k + 1$ in $V$ and $M$
in conjunction with the definition of $P^{\k, 0}$
will then allow the arguments of Lemma 9 of
[AS97b] to go through to yield that $j$ extends to
$j^* : V^{P^{\k, 0}} \to M^{j^*(P^{\k ,0})}$.
Note that since
$P^0_{\k , \k^{++} } \ast
P^1_{\k, \k^{++} }[\dot S_{\k^{++} }]$ is used at stage
$\k + 1$ in $M$, Lemma 3 of [AS97a] and Lemma 8 of
[AS97b] show that
$M^{j^*(P^{\k, 0})} \models ``\k$ isn't measurable''.
We show now that $V^{P^{\k, 0}} \models ``\k$ is supercompact''.
To do this, we give an argument similar to the one
given in the proof of Lemma 2 of [A$\infty$].
Specifically, let $\g > \k^{++}$ be an arbitrary
cardinal, and let
$\l > 2^{{[\g]}^{< \k}}$
be a cardinal so that for some embedding $k : V \to M$
witnessing the $\l$ supercompactness of $\k$,
$k(f)(\k) = \l$. By the definition of $P^{\k, 0}$ and the
properties of $k$,
$k(P^{\k, 0}) = (P^*_\k \ast
(P^0_{\k , \k^{++} } \ast
(P^1_{\k, \k^{++} }[\dot S_{\k^{++} }] \times
P^2_{\k , \k^{++} }[\dot S_{\k^{++}} ])))
\ast \dot Q^* =
(P^*_\k \ast \dot Q^*_\k) \ast \dot Q^* =
P^{\k, 0} \ast \dot Q^* =
P^{\k, 0} \ast \dot R^* \ast \dot Q^*_{k(\k)}$,
where $\dot R^*$ is a term for the
$M^{P^{\k, 0}}$ partial ordering
$P^*_{k(\k)} / P^{\k, 0}$.
By the definition of $P^{\k, 0}$, in $M$,
$\force_{P^{\k, 0}} ``$The field of $\dot Q$ is
composed of cardinals $> \l$''. Further, by the
definition of $P^{\k, 0}$ and the fact
$M^\l \subseteq M$, it is true that in $V$ and $M$,
$\force_{P^{\k, 0}} ``$Both $\dot R^*$ and
$\dot R^* \ast \dot Q^*_{k(\k)}$ are $\l$-strategically
closed and
$\l > 2^{{[\g]}^{< \k}}$''.
And, by our earlier remarks, in both $V$ and $M$,
$\force_{P^*_\k} ``\dot Q^*_\k$ is forcing equivalent to
$\dot {\hbox{\rm Add}}(\k^{++}, 1) \ast
\dot {\hbox{\rm Add}}(\k, \k^{++})$,
a $\k$-directed closed partial ordering having size
$\k^{++}$''. Therefore,
$V^{P^*_\k \ast \dot Q^*_\k} =
V^{P^{\k, 0}} \models ``2^\k = \k^{++}$'', and the
standard arguments (see, e.g., Lemma 2 of [A$\infty$])
in turn show that
$M^{P^{\k, 0} \ast \dot R^*}$
remains $\l$-closed with respect to
$V^{P^{\k, 0} \ast \dot R^*}$
and that if $G_0 \ast G_1$ is $V$-generic over
$P^*_\k \ast \dot Q^*_\k =
P^{\k, 0}$ and $G_2$ is $V[G_0][G_1]$-generic over
$R^*$, in $V[G_0][G_1][G_2]$, we can find a master condition
$q$ extending each $p \in k '' G_1$.
If $G_3$ is $V[G_0][G_1][G_2]$-generic over $Q_{k(\k)}$ so that
$q \in G_3$, in $V[G_0][G_1][G_2][G_3]$,
there is an elementary embedding
$k^* : V[G_0][G_1] \to M[G_0][G_1][G_2][G_3]$ extending $k$.
Since in $V$,
$\force_{P^{\k, 0}} ``\dot R^* \ast \dot Q^*_{k(\k)}$
is $\l$-strategically closed'',
$V[G_0][G_1] \models ``\k$ is $\g$ supercompact''.
This proves Lemma 2.
\pbf
\hfill $\square$ Lemma 2
When the more complicated version of $P^1_{\d, \l}[S]$
and the associated versions of $P^0_{\d, \l}$ and
$P^2_{\d, \l}[S]$ are employed as the building blocks of
$P^*$, instead of working with $\d^{++}_\a$, we use
$\d^{+++}_\a$, i.e., at each non-trivial stage
in our iteration, we force with either
$P^0_{\d_\a, \d^{+++}_\a} \ast
P^1_{\d_\a, \d^{+++}_\a}[\dot S_{\d^{+++}_\a}]$ or
$P^0_{\d_\a, \d^{+++}_\a} \ast
(P^1_{\d_\a, \d^{+++}_\a}[\dot S_{\d^{+++}_\a}] \times
P^2_{\d_\a, \d^{+++}_\a}[\dot S_{\d^{+++}_\a}])$.
This is since by Lemma 6 of [AS97b], both of the just
mentioned partial orderings will collapse $\d^+_\a$.
Except for this difference, however, the proof of Lemma 2
is the same as before, making the appropriate references
to Lemmas 6 and 8 of [AS97b] as necessary.
In conclusion to this section, we note that if we assume
that $\k$ has no inaccessible cardinals above it, no use of
the Laver function $f$ is needed in the definition of
$P^{\k, 0}$. At each $\d_\a < \k$ which isn't
$\d^+_\a$ supercompact, we can force as in Case 1 of the
definition of $P^{\k, 0}$, and at each $\d_\a \le \k$
which is $\d^+_\a$ supercompact, we can force as in
Case 2 of the definition of $P^{\k, 0}$. We leave it to
any interested readers to verify that the proof of Lemma 2
becomes simpler under these circumstances. It is only
when there are large enough cardinals above $\k$
that the use of the Laver function $f$ is required in
the definition of $P^{\k, 0}$.
\no \S 3 Magidor's Unpublished Proof
We present in this section Magidor's
unpublished proof,
which doesn't use iterated
Prikry forcing, of the consistency,
relative to a supercompact cardinal,
of the least measurable cardinal being
the least strongly compact cardinal.
We state this formally now as Theorem
2.
\proclaim{Theorem 2}
Con(ZFC + $\k$ is supercompact)
$\lra$
Con(ZFC + $\k$ is both the least
strongly compact cardinal and
the least measurable cardinal).
\endproclaim
\demo{Proof of Theorem 2}
Let $V \models ``$ZFC + $\k$ is
supercompact''. Without loss of
generality, we assume in addition that
$V \models {\hbox{\rm GCH}}$.
Further, we also assume that there
are no measurable cardinals in
$V$ above $\k$.
Fix now an arbitrary regular cardinal $\g < \k$.
Let $\la \d_\a : \a < \k \ra$ this time enumerate
the measurables $< \k$. The partial ordering
$P^{\k, 1}$ we use in the proof of Theorem 2
is the Easton support iteration
$\la \la P^\k_\a, \dot Q^\k_\a \ra : \a < \k \ra$,
where $P^\k_0$ is trivial and
$\force_{P^\k_\a} ``\dot Q^\k_\a$ adds a non-reflecting
stationary set of ordinals of cofinality $\g$ to $\d_\a$''.
\proclaim{Lemma 3}
$V^{P^{\k, 1}} \models ``$No cardinal $\d < \k$ is measurable''.
\endproclaim
\demo{Proof of Lemma 3}
Let $\d < \k$ be so that
$V \models ``\d$ is measurable''. It must therefore be the
case that
$\d = \d_\a$ for some $\a < \k$. This allows us to write
$P^{\k, 1} = P^\k_\a \ast \dot Q^\k_\a \ast \dot R =
P^\k_{\a + 1} \ast \dot R$.
By the definition of $P^{\k, 1}$ and the fact that any
stationary subset of a measurable (or weakly compact)
cardinal must reflect,
$V^{P^\k_{\a + 1}} \models ``\d$ isn't measurable since
there is $S \subseteq \d$ which is a non-reflecting
stationary set of ordinals of cofinality $\g$''.
Since by the definition of $P^{\k, 1}$,
$\force_{P^\k_{\a + 1}} ``\dot R$ is $ \d'$-strategically
closed for $\d'$ the least inaccessible above $\d$'',
$V^{P^\k_{\a + 1} \ast \dot R} = V^{P^{\k, 1}} \models
``S \subseteq \d$ is a non-reflecting stationary set of
ordinals of cofinality $\g$, so $\d$ isn't measurable''. Thus,
$V^{P^{\k, 1}} \models ``$No $V$-measurable cardinal $\d < \k$ is
measurable''. The proof of Lemma 3 will therefore be
complete once we have shown there is no cardinal
$\d < \k$ so that
$\force_{P^{\k, 1}} ``\d$ is measurable''.
To do this,
we give an argument similar to the one found in the
last part of Lemma 8 of [A97b], which in turn
is essentially the same as the arguments given in
Theorem 2.1.5 of [H] and Theorem 2.5 of [KiM].
Assume that $V^{P^{\k, 1}} \models ``\d$ is measurable''.
Since we have just shown that no $V$-cardinal is
measurable in $V^{P^{\k, 1}}$, we can write
$P^{\k, 1} = P^\k_\zeta \ast \dot R$, where
$\d \not\in {\hbox{\rm field}}(P^\k_\zeta)$
and
$\force_{P^\k_\zeta} ``\dot R$ is $\d'$-strategically
closed for $\d'$ the least inaccessible above $\d$''. Thus,
$\force_{P^\k_\zeta} ``\d$ is measurable'' iff
$\force_{P^{\k, 1}} ``\d$ is measurable'',
so we show without loss of generality that
$\force_{P^\k_\zeta} ``\d$ isn't measurable''.
Note now that since $V^{P^\k_\zeta} \models ``\d$
is Mahlo'',
$V \models ``\d$ is Mahlo''.
Next, let $p \in P^\k_\zeta$ be so that
$p \force ``\dot \mu$ is a measure over $\d$''.
We show there is some $q \ge p$, $q \in P^\k_\zeta$
so that for every $X \in
{(\wp(\d))}^V
$,
$q \Vert ``X \in \dot \mu$''.
To do this, we build in $V$ a binary tree
${\cal T}$
of height $\d$, assuming no such $q$ exists.
The root of our tree is $\la p, \d \ra$.
At successor stages $\b + 1$, assuming
$\la r, X \ra$ is on the $\bth$ level of ${\cal T}$,
$r \ge p$, and $X \subseteq \d$, $X \in V$
is so that $r \force ``X \in \dot \mu$'',
we let $X = X_0 \cup X_1$ be such that
$X_0, X_1 \in V$, $X_0 \cap X_1 = \emptyset$,
and for $r_0 \ge r$, $r_1 \ge r$ incompatible,
$r_0 \force ``X_0 \in \dot \mu$'' and
$r_1 \force ``X_1 \in \dot \mu$''.
We can do this by our hypothesis of the non-existence of a
$q \in P^\k_\zeta$ as mentioned earlier.
We place both $\la r_0, X_0 \ra$ and
$\la r_1, X_1 \ra$ in ${\cal T}$ at height $\b + 1$
as the successors of $\la r, X \ra$.
At limit stages $\l < \d$, for each
branch ${\cal B}$ in ${\cal T}$ of height $\le \l$,
we take the intersection of all second coordinates
of elements along ${\cal B}$. The result is a
partition of $\d$ into $\le 2^\l$ many sets, so since
$\d$ is Mahlo in $V$, $2^\l < \d$, i.e., the
partition is into $< \d$ many sets. Since
$V^{P^\k_\zeta} \models ``\d$ is measurable'',
there is at least one element $Y$ of this partition
resulting from a branch of height $\l$ and a condition
$s \ge p$ so that $s \force ``Y \in \dot \mu$''.
For all such $Y$, we place a pair of the form
$\la s, Y \ra$ into ${\cal T}$ at level $\l$ as the
successor of each element of the branch generating $Y$.
Work now in $V^{P^\k_\zeta}$. Since $\d$ is measurable in
$V^{P^\k_\zeta}$,
$V^{P^\k_\zeta} \models ``\d$ is weakly compact''.
By construction, ${\cal T}$ is a tree having $\d$
levels so that each level has size $< \d$.
Thus, by the weak compactness of $\d$ in
$V^{P^\k_\zeta}$, we can let ${\cal B} =
\la \la r_\b, X_\b \ra : \b < \d \ra$
be a branch of height $\d$ through ${\cal T}$.
If we define for $\b < \d$ $Y_\b = X_\b -
X_{\b + 1}$, then since $\la X_\b : \b < \d \ra$
is so that $0 \le \b < \rho < \d$ implies
$X_\b \supseteq X_\rho$, for $0 \le \b < \rho < \d$,
$Y_\b \cap Y_\rho = \emptyset$. Since by the construction of
${\cal T}$, at level $\b + 1$, the two second
coordinate portions of the successor of
$\la r_\b, X_\b \ra$ are $X_{\b + 1}$ and $Y_\b$,
for the $s_\b$ so that $\la s_\b, Y_\b \ra$ is at
level $\b + 1$ of ${\cal T}$,
$\la s_\b : \b < \d \ra$ must form in
$V^{P^\k_\zeta}$ an antichain of size $\d$ in
$P^\k_\zeta$.
In $V^{P^\k_\zeta}$, $P^\k_\zeta$ is embeddable as a
subordering of the
Easton support product
$\underset \a < \zeta \to{\prod} Q^\k_\a$
as calculated in $V^{P^\k_\zeta}$.
As $V^{P^\k_\zeta} \models ``\d$ is Mahlo'',
this immediately implies that
$V^{P^\k_\zeta} \models ``P^\k_\zeta$ is $\d$-c.c.'',
contradicting that $\la s_\b : \b < \d \ra$ is in
$V^{P^\k_\zeta}$ an antichain of size $\d$.
Thus, there is some $q \ge p$ so that for every
$X \in {(\wp(\d))}^V$, $q \Vert ``X \in \dot \mu$'',
i.e., $\d$ is measurable in $V$. This contradiction
proves Lemma 3.
\pbf
\hfill $\square$ Lemma 3
\proclaim{Lemma 4}
$V^{P^{\k, 1}} \models ``\k$ is strongly compact''.
\endproclaim
\demo{Proof of Lemma 4}
Let
$\l > 2^\k = \k^+$
be an arbitrary cardinal, and let
$k_1 : V \to M$ be an embedding witnessing the $\l$
supercompactness of $\k$. $\l$ has been chosen large
enough so that
any ultrafilter over $\k$
present in $V$ is an element of $M$, so
we may assume that
$k_2 : M \to N$ is an embedding witnessing the
measurability of $\k$ definable in $M$
so that
$N \models ``\k$ isn't measurable''.
It is easily verifiable
using the embedding definition of $\l$ strong
compactness given in Section 0 that
$j = k_2 \circ k_1$ is so that
$j : V \to N$ is a $\l$ strongly compact embedding
that also witnesses the non-measurability of $\k$.
We show that $j$ extends to
$j^* : V^{P^{\k, 1}} \to N^{j^*(P^{\k, 1})}$,
thus proving Lemma 4.
To do this, write $j(P^{\k, 1})$ as
$P^{\k, 1} \ast \dot Q^\k \ast \dot R^\k$, where
$\dot Q^\k$ is a term for the portion of
$j(P^{\k, 1})$ between $\k$ and $k_2(\k)$ and
$\dot R^\k$ is a term for the rest of
$j(P^{\k, 1})$, i.e., the part above $k_2(\k)$. Note that since
$N \models ``\k$ isn't measurable'',
$\k \not\in {\hbox{\rm field}}(\dot Q^\k)$. Also, since
$M \models ``\k$ is measurable'', by elementarity,
$N \models ``k_2(\k)$ is measurable''.
Thus, the field of $\dot Q^\k$ is composed of all
$N$-measurable cardinals in the interval
$(\k, k_2(\k)]$
(so $k_2(\k) \in {\hbox{\rm field}}(\dot Q^\k)$),
and the field of $\dot R^\k$ is
composed of all $N$-measurable cardinals in the interval
$(k_2(\k), k_2(k_1(\k)))$.
Let $G_0$ be $V$-generic over $P^{\k, 1}$. We construct in
$V[G_0]$ an $N[G_0]$-generic object $G_1$ over $Q^\k$
and an $N[G_0][G_1]$-generic object $G_2$ over $R^\k$.
Since $P^{\k, 1}$ is an Easton support iteration of length
$\k$ with no forcing done at stage $\k$, the construction of
$G_1$ and $G_2$ automatically guarantees that
$j''G_0 \subseteq G_0 \ast G_1 \ast G_2$, meaning that
$j : V \to N$ extends to
$j^* : V[G_0] \to N[G_0][G_1][G_2]$.
To build $G_1$, note that since $k_2$ can be assumed
to be generated by an ultrafiler ${\cal U}$ over
$\k$,
and since in both $V$ and $M$,
$2^\k = \k^+$,
$|k_2(\k^{+})| = |k_2(2^\k)| =
|\{f : f : \k \to \k^{+}$ is a function$\}| =
|{[\k^{+}]}^{\k}| = \k^{+}$. Thus, as
$N[G_0] \models ``|Q^\k| = k_2(2^\k)$'',
we can let $\la D_\a : \a < \k^{+} \ra$ enumerate in
$V[G_0]$ the dense open subsets of $Q^\k$ present in
$N[G_0]$.
Since the $\k$ closure of $N$ with respect to either
$M$ or $V$ implies the least element of the field of
$Q^\k$ is $> \k^{+}$, the definition of $Q^\k$ as the
Easton support iteration which adds a non-reflecting
stationary set of ordinals of cofinality $\g$ to each
$N[G_0]$-measurable cardinal in the interval
$(\k, k_2(\k)]$ implies that
$N[G_0] \models ``Q^\k$ is $\prec \k^{+}$-strategically
closed''. By the fact the standard arguments
show that forcing with the $\k$-c.c$.$ partial ordering
$P^{\k, 1}$ preserves that $N[G_0]$ remains $\k$ closed
with respect to either $M[G_0]$ or $V[G_0]$,
$Q^\k$ is $\prec \k^{+}$-strategically closed in both
$M[G_0]$ and $V[G_0]$.
We can now construct $G_1$ in either $M[G_0]$ or
$V[G_0]$ as follows. Player I picks $p_\a \in D_\a$
extending $\sup(\la q_\b : \b < \a \ra)$
(initially, $q_{-1}$ is the empty condition)
and player II responds by picking $q_\a \ge p_\a$
(so $q_\a \in D_\a$). By the $\prec \k^{+}$-strategic
closure of $Q^\k$ in both $M[G_0]$ and $V[G_0]$,
player II has a winning strategy for this game, so
$\la q_\a : \a < \k^{+} \ra$ can be taken as an
increasing sequence of conditions with
$q_\a \in D_\a$ for $\a < \k^{+}$. Clearly,
$G_1 = \{p \in Q^\k : \exists \a < \k^{+}
[q_\a \ge p]\}$ is our $N[G_0]$-generic object over $Q^\k$.
It remains to construct in $V[G_0]$ the desired
$N[G_0][G_1]$-generic object $G_2$ over $R^\k$.
To do this, we first note that as $\l > 2^\k$,
$M \models ``\k$ is measurable''. This means we can write
$k_1(P^{\k, 1})$ as $P^{\k, 1} \ast \dot S^\k \ast \dot T^\k$,
where
$\force_{P^{\k, 1}} ``\dot S^\k$ adds a non-reflecting
stationary set of ordinals of cofinality $\g$ to $\k$'', and
$\dot T^\k$ is a term for the rest of $k_1(P^{\k, 1})$.
Since we have assumed
$V \models ``$No cardinal $\d > \k$ is measurable'', the
$\l$ closure of $M$ with respect to $V$ implies
$M \models ``$No cardinal $\d \in (\k, \l]$ is measurable''.
Thus, the field of $\dot T^\k$ is composed of all
$M$-measurable cardinals in the interval $(\l, k_1(\k))$,
which implies that in $M$,
$\force_{P^{\k, 1} \ast \dot S^\k} ``\dot T^\k$ is
$\prec \l^+$-strategically closed''. Further, since we can
assume $\l$ is regular, $|{[\l]}^{< \k}| = \l$, and
$2^\l = \l^+$,
and since $k_1$
can be assumed to be generated by an ultrafilter
${\cal U}$ over $P_\k(\l)$,
$|k_1(\l^+)| = |k_1(2^\l)| = |2^{k_1(\l)}| =
|\{f : f : P_\k(\l) \to \l^+$ is a function$\} =
|{[\l^+]}^\l| = \l^+$.
Work until otherwise specified in $M$.
Consider the ``term forcing'' partial ordering $T^*$
(see [C], Section 1.5, p$.$ 8)
associated with $\dot T^\k$, i.e., $\tau \in T^*$ iff
$\tau$ is a term in the forcing language with respect to
$P^{\k, 1} \ast \dot S^\k$ and
$\force_{P^{\k, 1} \ast \dot S^\k} ``\tau \in \dot T^\k$'',
ordered by $\tau \ge \sigma$ iff
$\force_{P^{\k, 1} \ast \dot S^\k} ``\tau \ge \sigma$''.
Clearly, $T^* \in M$. Also, since
$\force_{P^{\k, 1} \ast \dot S^\k} ``\dot T^\k$ is
$\prec \l^+$-strategically closed'', it can easily be verified
that $T^*$ itself is $\prec \l^+$-strategically closed in
$M$ and, since $M^\l \subseteq M$, in $V$ as well. Therefore, as
$\force_{P^{\k, 1} \ast \dot S^\k} ``|\dot T^\k| =
k_1(\l)$ and
$2^{k_1(\l)} = {(k_1(\l))}^+ = k_1(\l^+)$'',
we can assume without loss of generality that in $M$,
$|T^*| = k_1(\l)$.
This means we can let $\la D_\a : \a < \l^+ \ra$
enumerate in $V$ the dense open subsets of $T^*$
present in $M$ and argue as before to construct in
$V$ an $M$-generic object $H_2$ over $T^*$.
Note now that since
$N$ can be assumed to be given by an ultrapower
of $M$ via a normal measure ${\cal U} \in M$ over
$\k$,
Fact 2 of Section 1.2 of [C]
tells us that $k_2 '' H_2$ generates an
$N$-generic object $G^*_2$ over
$k_2(T^*)$.
By elementariness, $k_2(T^*)$ is the term
forcing whose elements are names for elements of
$k_2(\dot T^\k) = \dot R^\k$ in
$N^{k_2(P^{\k, 1} \ast \dot S^\k)}$.
Therefore, since $G^*_2$ is $N$-generic over
$k_2(T^*)$, and since
$G_0 \ast G_1$ is
$k_2(P^{\k, 1} \ast \dot S^\k)$-generic over $N$,
Fact 1 of Section 1.5 of [C]
tells us that for
$G_2 = \{i_{G_0 \ast G_1}(\tau) : \tau \in G^*_2 \}$,
$G_2$ is $N[G_0][G_1]$-generic over $R^\k$.
As $G_0$ is a set of conditions in an Easton support iteration
of length $\k$ in which a direct limit was
taken at $\k$, each condition in $G_0$
has a support which is bounded in $\k$. It follows that
in $V[G_0]$,
$j : V \to N$ extends to
$j^* : V[G_0] \to N[G_0][G_1][G_2]$.
This proves Lemma 4.
\pbf
\hfill $\square$ Lemma 4
Lemmas 3 and 4 complete the proof of Theorem 2.
\pbf
\hfill $\square$ Theorem 2
\no \S 4 The Case $n=1$
We present in this section a proof of Theorem 1 when
$n=1$. We assume that $V \models ``$ZFC + $\k$ is
supercompact''. By Lemma 2, we also assume that
$V \models ``2^\k = \k^{++}$'' and that there is a
$\k^+$ supercompactness embedding
$k^*_0 : V \to M^*$
generated by a $\k^+$ supercompact
ultrafilter over $P_\k(\k^+)$ so that
$M^* \models ``\k$ isn't measurable''.
Further, as in Section 3, we assume there are
no measurable cardinals in $V$ above $\k$.
As in Section 3,
fix an arbitrary regular cardinal $\g < \k$.
Let $\la \d_\a : \a < \k \ra$ once again enumerate
the measurables $< \k$. The partial ordering
$P^{\k, 1}$ we use in the proof of Theorem 1 when
$n=1$ is the one of Section 3, i.e., the
Easton support iteration which adds
a non-reflecting stationary set of ordinals
of cofinality $\g$ to each $V$-measurable
cardinal below $\k$. By Lemma 3,
$V^{P^{\k, 1}} \models ``$No cardinal
$\d < \k$ is measurable''.
Thus, the proof of Theorem 1 for the case
$n=1$ will be complete once we have
shown the following.
\proclaim{Lemma 5}
$V^{P^{\k, 1}} \models ``\k$ is both strongly compact
and $\k^+$ supercompact''.
\endproclaim
\demo{Proof of Lemma 5}
The proof of Lemma 5 is a modification of
the proof of Lemma 4. Let
$\l > 2^{{[\k^+]}^{< \k}} = 2^{\k^+}
= 2^\k = \k^{++}$
be an arbitrary cardinal, and let
$k_1 : V \to M$ be an embedding witnessing the $\l$
supercompactness of $\k$. $\l$ has been chosen large
enough so that
any ultrafilter over $P_\k(\k^+)$
present in $V$ is an element of $M$, so
we may assume by
the remarks in the first paragraph of this section that
$k_2 : M \to N$ is an embedding witnessing the
$\k^+$ supercompactness of $\k$ definable in $M$
so that
$N \models ``\k$ isn't measurable''.
It is easily verifiable
using the embedding definition of $\l$ strong
compactness given in Section 0 that
$j = k_2 \circ k_1$ is so that
$j : V \to N$ is a $\l$ strongly compact embedding
that also witnesses the $\k^+$ supercompactness of $\k$.
We show as in Lemma 4 that $j$ extends to
$j^* : V^{P^{\k, 1}} \to N^{j^*(P^{\k, 1})}$,
thus proving Lemma 5.
To do this, once again write $j(P^{\k, 1})$ as
$P^{\k, 1} \ast \dot Q^\k \ast \dot R^\k$, where
$\dot Q^\k$ is a term for the portion of
$j(P^{\k, 1})$ between $\k$ and $k_2(\k)$ and
$\dot R^\k$ is a term for the rest of
$j(P^{\k, 1})$, i.e., the part above $k_2(\k)$. Note that since
$N \models ``\k$ isn't measurable'',
$\k \not\in {\hbox{\rm field}}(\dot Q^\k)$. Also, since
$M \models ``\k$ is measurable'', by elementarity,
$N \models ``k_2(\k)$ is measurable''.
Thus, as before,
the field of $\dot Q^\k$ is composed of all
$N$-measurable cardinals in the interval
$(\k, k_2(\k)]$
(so $k_2(\k) \in {\hbox{\rm field}}(\dot Q^\k)$),
and the field of $\dot R^\k$ is
composed of all $N$-measurable cardinals in the interval
$(k_2(\k), k_2(k_1(\k)))$.
Let $G_0$ be $V$-generic over $P^{\k, 1}$.
In analogy to Lemma 4, we construct in
$V[G_0]$ an $N[G_0]$-generic object $G_1$ over $Q^\k$
and an $N[G_0][G_1]$-generic object $G_2$ over $R^\k$.
Once again,
since $P^{\k, 1}$ is an Easton support iteration of length
$\k$ with no forcing done at stage $\k$, the construction of
$G_1$ and $G_2$ automatically guarantees that
$j''G_0 \subseteq G_0 \ast G_1 \ast G_2$, meaning that
$j : V \to N$ extends to
$j^* : V[G_0] \to N[G_0][G_1][G_2]$.
To build $G_1$, note that since $k_2$ can be assumed
to be generated by an ultrafiler ${\cal U}$ over
${(P_\k(\k^+))}^M =
{(P_\k(\k^+))}^V$, and since in both $V$ and $M$,
$2^{\k^+} = 2^\k = \k^{++}$,
$|k_2(\k^{++})| = |k_2(2^\k)| =
|\{f : f : P_\k(\k^+) \to \k^{++}$ is a function$\}| =
|{[\k^{++}]}^{\k^+}| = \k^{++}$. Thus, as
$N[G_0] \models ``|Q^\k| = k_2(2^\k)$'',
we can let $\la D_\a : \a < \k^{++} \ra$ enumerate in
$V[G_0]$ the dense open subsets of $Q^\k$ present in
$N[G_0]$.
Since the $\k^+$ closure of $N$ with respect to either
$M$ or $V$ implies the least element of the field of
$Q^\k$ is $> \k^{++}$, the definition of $Q^\k$ as the
Easton support iteration which adds a non-reflecting
stationary set of ordinals of cofinality $\g$ to each
$N[G_0]$-measurable cardinal in the interval
$(\k, k_2(\k)]$ implies that
$N[G_0] \models ``Q^\k$ is $\prec \k^{++}$-strategically
closed''. By the fact the standard arguments
show that forcing with the $\k$-c.c$.$ partial ordering
$P^{\k, 1}$ preserves that $N[G_0]$ remains $\k^+$ closed
with respect to either $M[G_0]$ or $V[G_0]$,
$Q^\k$ is $\prec \k^{++}$-strategically closed in both
$M[G_0]$ and $V[G_0]$.
This means that $G_1$ can now be constructed in either
$M[G_0]$ or $V[G_1]$ in exactly the same manner as
was done in the proof of Lemma 4.
It remains to construct in $V[G_0]$ the desired
$N[G_0][G_1]$-generic object $G_2$ over $R^\k$.
This is done in virtually the same way as
in the proof of Lemma 4.
We first note that as $\l > 2^\k$,
$M \models ``\k$ is measurable''. This means we can
again write
$k_1(P^{\k, 1})$ as $P^{\k, 1} \ast \dot S^\k \ast \dot T^\k$,
where
$\force_{P^{\k, 1}} ``\dot S^\k$ adds a non-reflecting
stationary set of ordinals of cofinality $\g$ to $\k$'', and
$\dot T^\k$ is a term for the rest of $k_1(P^{\k, 1})$.
Since we have assumed
$V \models ``$No cardinal $\d > \k$ is measurable'', the
$\l$ closure of $M$ with respect to $V$ implies
$M \models ``$No cardinal $\d \in (\k, \l]$ is measurable''.
Thus, as before,
the field of $\dot T^\k$ is composed of all
$M$-measurable cardinals in the interval $(\l, k_1(\k))$,
which implies that in $M$,
$\force_{P^{\k, 1} \ast \dot S^\k} ``\dot T^\k$ is
$\prec \l^+$-strategically closed''. Further, since we can
assume $\l$ is regular, $|{[\l]}^{< \k}| = \l$, and
$2^\l = \l^+$ (our ground model $V$ is constructed by
forcing over a model of GCH using a set partial ordering),
and since $k_1$
can be assumed to be generated by an ultrafilter
${\cal U}$ over $P_\k(\l)$,
$|k_1(\l^+)| = |k_1(2^\l)| = |2^{k_1(\l)}| =
|\{f : f : P_\k(\l) \to \l^+$ is a function$\} =
|{[\l^+]}^\l| = \l^+$.
Work until otherwise specified in $M$.
Consider once more the ``term forcing'' partial ordering $T^*$
associated with $\dot T^\k$, i.e., $\tau \in T^*$ iff
$\tau$ is a term in the forcing language with respect to
$P^{\k, 1} \ast \dot S^\k$ and
$\force_{P^{\k, 1} \ast \dot S^\k} ``\tau \in \dot T^\k$'',
ordered by $\tau \ge \sigma$ iff
$\force_{P^{\k, 1} \ast \dot S^\k} ``\tau \ge \sigma$''.
Again, $T^* \in M$. Also, by our earlier arguments, since
$\force_{P^{\k, 1} \ast \dot S^\k} ``\dot T^\k$ is
$\prec \l^+$-strategically closed'', it can easily be verified
that $T^*$ itself is $\prec \l^+$-strategically closed in
$M$ and, since $M^\l \subseteq M$, in $V$ as well. Therefore, as
$\force_{P^{\k, 1} \ast \dot S^\k} ``|\dot T^\k| =
k_1(\l)$ and
$2^{k_1(\l)} = {(k_1(\l))}^+ = k_1(\l^+)$'',
we can as in Lemma 4
assume without loss of generality that in $M$,
$|T^*| = k_1(\l)$.
This means we can let $\la D_\a : \a < \l^+ \ra$
enumerate in $V$ the dense open subsets of $T^*$
present in $M$ and argue as earlier to construct in
$V$ an $M$-generic object $H_2$ over $T^*$.
Note now that since
$N$ can be assumed to be given by an ultrapower
of $M$ via a normal measure ${\cal U} \in M$ over
${(P_\k(\k^+))}^M$,
as in the proof of Lemma 4,
Fact 2 of Section 1.2 of [C]
tells us that $k_2 '' H_2$ generates an
$N$-generic object $G^*_2$ over
$k_2(T^*)$.
By elementariness, $k_2(T^*)$ is the term
forcing whose elements are names for elements of
$k_2(\dot T^\k) = \dot R^\k$ in
$N^{k_2(P^{\k, 1} \ast \dot S^\k)}$.
Therefore, since $G^*_2$ is $N$-generic over
$k_2(T^*)$, and since
$G_0 \ast G_1$ is
$k_2(P^{\k, 1} \ast \dot S^\k)$-generic over $N$,
as in the proof of Lemma 4,
Fact 1 of Section 1.5 of [C]
tells us that for
$G_2 = \{i_{G_0 \ast G_1}(\tau) : \tau \in G^*_2 \}$,
$G_2$ is $N[G_0][G_1]$-generic over $R^\k$.
As $G_0$ is a set of conditions in an Easton support iteration
of length $\k$ in which a direct limit was
taken at $\k$, each condition in $G_0$
has a support which is bounded in $\k$. It follows
once more that
in $V[G_0]$,
$j : V \to N$ extends to
$j^* : V[G_0] \to N[G_0][G_1][G_2]$.
This proves Lemma 5.
\pbf
\hfill $\square$ Lemma 5
Since
$V \models ``|P^{\k, 1}| = \k$'',
$V^{P^{\k, 1}} \models ``2^\k = \k^{++}$''.
This means the proof of Theorem 1 when
$n=1$ is now complete.
\pbf
\hfill $\square$ Theorem 1 ($n=1$)
In conclusion to this section,
we note that the first
author's non-iterative proof of Theorem 1 for
the case $n=1$ [A97a]
used Magidor's notion of iterated Prikry
forcing [Ma] to destroy all measurable cardinals
found below the cardinal $\k$ produced in Theorem 3 of
[AS97b], thereby requiring an initial assumption of a
supercompact limit of supercompact cardinals.
At the Oberwolfach meeting at which this proof was discovered,
Magidor and Woodin independently
of one another told both authors a (non-iterative)
proof of Theorem 1 for the case $n=1$ could be
given using Radin forcing, starting from only one
supercompact cardinal. Neither Woodin's nor Magidor's
proof has been published (and both proofs seem
unlikely to be published anywhere in the foreseeable
future), but our methods here and the methods of [A97a]
provide another non-iterative proof starting from
only one supercompact cardinal for the case $n=1$.
An outline of this proof is as follows:
First, start with a ground model $V$ so that
$V \models ``$GCH + $\k$ is supercompact''.
Next, force using the partial ordering $P^{\k, 0}$
of Lemma 2 to preserve the supercompactness of $\k$,
make $2^\k = \k^{++}$, and construct a $\k^+$
supercompact embedding
$j^* : V^{P^{\k, 0}} \to M^{j^*(P^{\k, 0})}$ so that
$M^{j^*(P^{\k, 0})} \models ``\k$ isn't measurable''.
Finally, force over $V^{P^{\k, 0}}$ using $Q^*$, Magidor's
notion of iterated Prikry forcing of [Ma],
to destroy all measurable cardinals below $\k$.
Magidor's arguments of [Ma] yield
$V^{P^{\k, 0} \ast \dot Q^*} \models ``\k$ is both
strongly compact and the least measurable cardinal'',
and the exact same argument as given in the Lemma of
[A97a] shows
$V^{P^{\k, 0} \ast \dot Q^*} \models ``\k$ is
$\k^+$ supercompact''. Also, since
$\force_{P^{\k, 0}} ``|\dot Q^*| = \k$'',
$V^{P^{\k, 0} \ast \dot Q^*} \models ``2^\k = \k^{++}$''.
The advantage of the non-iterative proof just given and
the earlier non-iterative proofs previously mentioned
is that the large cardinal structure above $\k$ can be
arbitrary in any of these proofs. The proofs
of Lemmas 4 and 5
and the proofs to be given in the next section require
severe restrictions on the large cardinal structure
of the universe. We will comment more upon this in the
concluding remarks of the paper.
\no \S 5 The Case of Arbitrary Finite $n$
In this section, we give a proof of Theorem 1 for
arbitrary finite $n$.
\demo{Proof of Theorem 1}
Let
$V_0 \models ``$ZFC + GCH + $\k_1 < \k_2 < \cdots < \k_n$ are
the first $n$ (for $n \in \omega$) supercompact cardinals +
No cardinal $\l > \k_n$ is measurable''. Let
$P^* \in V_0$ be a partial ordering so that
$V = V^{P^*}_0 \models
``$Each $\k_i$ for $i = 1, \ldots, n$
is indestructible under $\k_i$-directed closed forcing''.
The existence of this sort of generalized version of
Laver's partial ordering of [L] is easy to show and
is found in many places, e.g., [A83], [A98],
or [CFM]. Note that since as in [L],
$P^*$ can be defined as an iteration so that for
$P^*_{\k_i}$ the portion of $P^*$ up through stage $\k_i$,
$|P^*_{\k_i}| = \k_i$, and since also we can assume that
the portion of $P^*$ defined beyond stage $\k_i$ is
at least $\l_i$-directed closed,
where for the rest of this section,
$\l_i$ is the
least inaccessible above $\k_i$,
$V \models ``2^{\k_i} = \k^+_i$ for $i = 1, \ldots, n$''.
Also, by the L\'evy-Solovay results [LS], since
$V_0 \models ``$No cardinal $\l > \k_n$ is measurable'',
$V \models ``$No cardinal $\l > \k_n$ is measurable'' as well.
We take now $V$ as our ground model and let $\k_0 = \omega$.
$P$ will be defined as the cartesian product
$\underset 1 \le i \le n \to{\prod} P^{\k_i}$
where $P^{\k_i}$ for $i = 1, \ldots, n$ can be defined in
two ways, depending upon whether the simpler or more
complicated version of the partial ordering
$P^1_{\d, \l}[S]$
is used. If the simpler version of
$P^1_{\d, \l}[S]$
is used, then $P^{\k_i} = P^{\k_i, 0} \ast
\dot P^{\k_i, 1}$, where $P^{\k_i, 0}$ is defined
as in Section 2, using only those inaccessibles in the interval
$(\k_{i - 1}, \k_i]$ satisfying GCH in its field
and fixing $\k_{i - 1}$ as the cofinality of the
non-reflecting stationary sets added by each
$P^0_{\d, \l}$
(since $V \models ``2^{\k_i} = \k^+_i$'',
reflection shows that unboundedly many cardinals in
$(\k_{i - 1}, \k_i)$ will be in the field of
$P^{\k_i, 0}$),
and $P^{\k_i, 1}$ also adds non-reflecting stationary
sets of ordinals of cofinality $\k_{i - 1}$ to every
$V^{P^{\k_i, 0}}$-measurable cardinal in the interval
$(\k_{i - 1}, \k_i)$.
If the more complicated version of
$P^1_{\d, \l}[S]$
is used, then $P^{\k_i, 0}$ is defined as in Section 2,
using only those inaccessibles in the interval
$(\k_{i - 1}, \k_i]$ in its field satisfying GCH, and
$P^{\k_i, 1}$ is as just defined when the simpler version of
$P^1_{\d, \l}[S]$
is used.
For $i = 1, \ldots, n$, write $P = P_i \times P^{\k_i}
\times P^i$, where $P_i = \underset 1 \le j \le i - 1 \to{\prod}
P^{\k_j}$
and $P^i = \underset i + 1 \le j \le n \to{\prod} P^{\k_j}$.
When the simpler version of
$P^1_{\d, \l}[S]$
is used, it easily follows that $P^i$ is
$\k_i$-directed closed. This is since by the remarks in
the middle of p$.$ 108 of [AS97a], each
$P^1_{\d, \l}[S]$
used in the definition of each $P^{\k_j, 0}$ for
$j = i + 1, \ldots, n$ is at least $\k_i$-directed closed,
so as the cofinalities of the ordinals present in the
non-reflecting stationary sets added by $P^{\k_j, 1}$
for $j = i + 1, \ldots, n$ are at least $\k_i$,
$P^{\k_j}$ for $j = i + 1, \ldots, n$ and
$P^i = \underset i + 1 \le j \le n \to{\prod} P^{\k_j}$
are all $\k_i$-directed closed. Also, when the
more complicated version of
$P^1_{\d, \l}[S]$ is used, it is the case that a
$V$-generic object for $P^i$ is $V$-generic over a
$\k_i$-directed closed partial ordering. This follows
by the argument of Lemma 14 of [AS97b] combined with
the fact that any partial ordering of the form
$P^0_{\d, \l} \ast (P^1_{\d, \l}[ \dot S] \times
P^2_{\d, \l}[\dot S])$
used in the definition of $P^{\k_j, 0}$
for $j = i + 1, \ldots, n$ is
$\k_i$-directed closed.
(See Lemma 4 of [AS97b] for a proof.)
Thus, by the indestructibility properties of $\k_i$
and the fact $P^i$ is $\l_i$-strategically closed,
$V^{P^i} \models ``\k_i$ is supercompact and
$2^{\k_i} = \k^+_i$''.
By the definition of $P^{\k_{i + 1}}$,
$V^{P^i} \models ``$No cardinal $\d \in (\k_i, \k_{i + 1})$
is measurable'', for $i = 1, \ldots, n - 1$.
When $i=n$, we take $(\k_i, \k_{i + 1})$ as all ordinals
$> \k_i$ and $P^i$ as being trivial, so our initial
assumptions on $V$ once more give us
$V^{P^i} \models ``$No cardinal $\d \in (\k_i, \k_{i + 1})$
is measurable''.
Therefore, the arguments of Lemmas 2 - 5 apply to show
$V^{P^i \times P^{\k_i}} \models ``\k_i$ is
$< \k_{i + 1}$ strongly compact, $\k_i$ is
$\k^+_i$ supercompact, $2^{\k_i} = \k^{++}_i$, and
no cardinal $\d \in (\k_{i - 1}, \k_i)$ is measurable'',
taking when $i=n$ ``$< \k_i$ strongly compact'' as meaning
``$\d$ strongly compact for all cardinals $\d > \k_n$'',
i.e., as meaning fully strongly compact.
Since by the definition of $P_i$, $|P_i| < \l_{i - 1}$,
the results of [LS] tell us
$V^{P^i \times P^{\k_i} \times P_i} = V^P \models
``\k_i$ is $< \k_{i + 1}$ strongly compact,
$\k_i$ is $\k^+_i$ supercompact, $2^{\k_i} = \k^{++}_i$,
and no cardinal $\d \in (\k_{i - 1}, \k_i)$ is measurable''.
Therefore, since a result of DiPrisco [DH] tells us that if
$\d$ is $< \g$ strongly compact and $\g$ is $\rho$
strongly compact, then $\d$ is $\rho$ strongly compact,
and since we are working with finitely many cardinals
$\k_1, \ldots, \k_n$, we can apply finitely often the
result of [DH] to infer
$V^P \models ``\k_i$ is strongly compact,
$\k_i$ is $\k^+_i$ supercompact, $2^{\k_i} = \k^{++}_i$,
and no cardinal $\d \in (\k_{i - 1}, \k_i)$ is measurable''.
This completes the proof of Theorem 1 for arbitrary finite $n$.
\pbf
\hfill $\square$ Theorem 1
We note that a simplified version of the
above proof easily yields a proof of
Magidor's unpublished theorem of the
consistency, relative to the existence of
$n \in \omega$ supercompact cardinals,
of the first $n$ measurable cardinals being
the first $n$ strongly compact cardinals.
An outline of the proof is as follows:
We begin by taking $V_0$ and $V$ as just
specified. The definition of each
$P^{\k_i}$, however, is greatly simplified
as the Easton support iteration of partial
orderings which add, to each $V$-measurable
cardinal in the interval
$(\k_{i - 1}, \k_i)$, a non-reflecting
stationary set of ordinals of cofinality
$\k_{i - 1}$.
Here, when $i=1$, $\k_{i - 1}$ is taken
as being $\omega$.
Letting $P$ be the cartesian product
of the $P^{\k_i}$ as above,
noting that each $P^{\k_i}$ is
$\k_{i - 1}$-directed closed,
and reasoning in a similar manner
to what was just done, the
arguments of Lemmas 3 and 4 then show that
for $i = 1, \ldots, n$,
$V^P \models ``\k_i$ is $< \k_{i + 1}$
strongly compact, and no cardinal
$\d \in (\k_{i - 1}, \k_i)$ is measurable''.
We can then infer as before that
$V^P \models ``\k_i$ is strongly compact,
and no cardinal
$\d \in (\k_{i - 1}, \k_i)$ is measurable''.
This yields the desired result.
\no \S 6 Concluding Remarks
In conclusion to this paper, we note that for the moment,
we are restricted to proving Theorem 1, as was Magidor,
only to finite values of $n$. The proof of Theorem 1,
and Magidor's original proof of the consistency of the
first $n \in \omega$ strongly compact cardinals being the
first $n$ measurable cardinals, both heavily use that
the ground model contains only finitely many supercompact
cardinals and no measurable cardinals beyond their
supremum. This is evident in the proof of Lemma 4,
its equivalent form in Magidor's original proof,
and in the proof given in Section 4.
Although it is possible to give alternate proofs,
in both our situation and Magidor's, by composing
supercompact embeddings either with embeddings
generated by normal measures over measurable cardinals
of Mitchell order $0$
in Magidor's case or with a $\k^+$ supercompact
embedding as constructed in Lemma 2 in our case,
the proofs still require the initial assumption
of only finitely many supercompact cardinals with
no measurable cardinals above their supremum.
(See the end of [A95] for a further discussion of this problem.)
Thus, the prime question of the relative consistency of the
first $\omega$ measurable and strongly compact cardinals
coinciding, with or without any additional degrees of
supercompactness, remains open.
\hb\vfill\eject\frenchspacing
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\bye