Reflection implies the SCH
Saharon Shelah \thanks { The author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 242/03). Publication 794. \null\newline I would like to thank Alice Leonhardt for the beautiful typing.} \endthanks
cf
794.tex
% 794, processed by citealice (2006-07-10) on Thu Sep 27 19:00:44 IST 2007

\ifx\shlhetal\undefinedcontrolsequence\let\shlhetal\relax\fi
% To: Saharon Shelah <shelah@math.huji.ac.il>, Andrzej Roslanowski <roslanow@member.ams.org>, "Saharon Shelah's Office" <shlhetal@math.huji.ac.il>, <shani@math.huji.ac.il>
% Subject: sh794
% Date: Thu, 27 Sep 2007 09:59:18 -0400 (EDT)
% From: Alice Leonhardt <leonhard@math.rutgers.edu>
% Mime-Version: 1.0
% Content-Description: three minor revisions
% X-sliced-and-diced-by: 'savemail' 2.0, Oct 2005

\input amstex
% % \input mathdefs
%  *** start including mathdefs.tex *** 
\expandafter\ifx\csname mathdefs.tex\endcsname\relax
  \expandafter\gdef\csname mathdefs.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
  \string\input{mathdefs.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%mathdefs.tex v1.3.2

%%% Changes from v1.0: footnote macros, warning for duplicated tags,
%%%   control sequences \( and \verbatimtags.
%%% From v1.2: \pretags, redefinition of \( using \ifinner, multi-part
%%%   equation numbering, control sequences \[, \references, and
%%%   \resetbracket. 
%%% From v1.3: \rm in \lastpart; write root of multi-part tag to .tgs 

%See file texdefs.doc for documentation.

\catcode`\X=12\catcode`\@=11

%Minor control sequences:
\def\n@wcount{\alloc@0\count\countdef\insc@unt}
\def\n@wwrite{\alloc@7\write\chardef\sixt@@n}
\def\n@wread{\alloc@6\read\chardef\sixt@@n}
\def\r@s@t{\relax}\def\v@idline{\par}\def\@mputate#1/{#1}
\def\l@c@l#1X{\firstpart.#1}\def\gl@b@l#1X{#1}\def\t@d@l#1X{{}}

%Creation of tag families and output of assignments and citations:
\def\crossrefs#1{\ifx\all#1\let\tr@ce=\all\else\def\tr@ce{#1,}\fi
   \n@wwrite\cit@tionsout\openout\cit@tionsout=\jobname.cit 
   \write\cit@tionsout{\tr@ce}\expandafter\setfl@gs\tr@ce,}
\def\setfl@gs#1,{\def\@{#1}\ifx\@\empty\let\next=\relax
   \else\let\next=\setfl@gs\expandafter\xdef
   \csname#1tr@cetrue\endcsname{}\fi\next}
\def\m@ketag#1#2{\expandafter\n@wcount\csname#2tagno\endcsname
     \csname#2tagno\endcsname=0\let\tail=\all\xdef\all{\tail#2,}
   \ifx#1\l@c@l\let\tail=\r@s@t\xdef\r@s@t{\csname#2tagno\endcsname=0\tail}\fi
   \expandafter\gdef\csname#2cite\endcsname##1{\expandafter
     \ifx\csname#2tag##1\endcsname\relax?\else\csname#2tag##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2page\endcsname##1{\expandafter
     \ifx\csname#2page##1\endcsname\relax?\else\csname#2page##1\endcsname\fi
     \expandafter\ifx\csname#2tr@cetrue\endcsname\relax\else
     \write\cit@tionsout{#2tag ##1 cited on page \folio.}\fi}
   \expandafter\gdef\csname#2tag\endcsname##1{\expandafter
      \ifx\csname#2check##1\endcsname\relax
      \expandafter\xdef\csname#2check##1\endcsname{}%
      \else\immediate\write16{Warning: #2tag ##1 used more than once.}\fi
      \multit@g{#1}{#2}##1/X%
      \write\t@gsout{#2tag ##1 assigned number \csname#2tag##1\endcsname\space
      on page \number\count0.}%
   \csname#2tag##1\endcsname}}


\def\multit@g#1#2#3/#4X{\def\t@mp{#4}\ifx\t@mp\empty%
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}%
   \else\expandafter\ifx\csname#2last#3\endcsname\relax
      \expandafter\n@wcount\csname#2last#3\endcsname
      \global\advance\csname#2tagno\endcsname by 1 
      \expandafter\xdef\csname#2tag#3\endcsname
      {#1\number\csname#2tagno\endcsnameX}
      \write\t@gsout{#2tag #3 assigned number \csname#2tag#3\endcsname\space
      on page \number\count0.}\fi
   \global\advance\csname#2last#3\endcsname by 1
   \def\t@mp{\expandafter\xdef\csname#2tag#3/}%
   \expandafter\t@mp\@mputate#4\endcsname
   {\csname#2tag#3\endcsname\lastpart{\csname#2last#3\endcsname}}\fi}
\def\t@gs#1{\def\all{}\m@ketag#1e\m@ketag#1s\m@ketag\t@d@l p
\let\realscite\scite
\let\realstag\stag
   \m@ketag\gl@b@l r \n@wread\t@gsin
   \openin\t@gsin=\jobname.tgs \re@der \closein\t@gsin
   \n@wwrite\t@gsout\openout\t@gsout=\jobname.tgs }
\outer\def\localtags{\t@gs\l@c@l}
\outer\def\globaltags{\t@gs\gl@b@l}
\outer\def\newlocaltag#1{\m@ketag\l@c@l{#1}}
\outer\def\newglobaltag#1{\m@ketag\gl@b@l{#1}}

%Reading in tag information:
\newif\ifpr@ 
\def\m@kecs #1tag #2 assigned number #3 on page #4.%
   {\expandafter\gdef\csname#1tag#2\endcsname{#3}
   \expandafter\gdef\csname#1page#2\endcsname{#4}
   \ifpr@\expandafter\xdef\csname#1check#2\endcsname{}\fi}
\def\re@der{\ifeof\t@gsin\let\next=\relax\else
   \read\t@gsin to\t@gline\ifx\t@gline\v@idline\else
   \expandafter\m@kecs \t@gline\fi\let \next=\re@der\fi\next}
\def\pretags#1{\pr@true\pret@gs#1,,}
\def\pret@gs#1,{\def\@{#1}\ifx\@\empty\let\n@xtfile=\relax
   \else\let\n@xtfile=\pret@gs \openin\t@gsin=#1.tgs \message{#1} \re@der 
   \closein\t@gsin\fi \n@xtfile}

%Sections and subsections; local numbering:
\newcount\sectno\sectno=0\newcount\subsectno\subsectno=0
\newif\ifultr@local \def\ultralocal{\ultr@localtrue}
\def\firstpart{\number\sectno}
\def\lastpart#1{\ifcase#1 \or a\or b\or c\or d\or e\or f\or g\or h\or 
   i\or k\or l\or m\or n\or o\or p\or q\or r\or s\or t\or u\or v\or w\or 
   x\or y\or z \fi}
\def\closeup{\vskip-\bigskipamount}
\def\resetall{\global\advance\sectno by 1\subsectno=0
   \gdef\firstpart{\number\sectno}\r@s@t}
\def\resetsub{\global\advance\subsectno by 1
   \gdef\firstpart{\number\sectno.\number\subsectno}\r@s@t}
\def\newsection#1\par{\resetall\vskip0pt plus.3\vsize\penalty-250
   \vskip0pt plus-.3\vsize\bigskip\bigskip
   \message{#1}\leftline{\bf#1}\nobreak\bigskip}
\def\subsection#1\par{\ifultr@local\resetsub\fi
   \vskip0pt plus.2\vsize\penalty-250\vskip0pt plus-.2\vsize
   \bigskip\smallskip\message{#1}\leftline{\bf#1}\nobreak\medskip}

%jj tags:
% On Andrzej's request:  we want to be able 
% to show tags as in noverbatim, with verbatim in the margin,
% and cites as in verbatim, with nonverbatim in the margin
% mg -- July 2000

\newdimen\marginshift

\newdimen\margindelta
\newdimen\marginmax
\newdimen\marginmin

\def\margininit{       
\marginmax=3 true cm                  % how much room, approximately
				      
\margindelta=0.1 true cm              % distance between entries
\marginmin=0.1true cm                 % where will leftmost entry be
\marginshift=\marginmin
}    % we cannot execute this right now, since 
     % there may be a \magnification coming later in the 
     % main file.   So we call \margininit at the end of 
     % alice2jlem

\def\t@gsjj#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsjj
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\citejj{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\tagjj{##1}}\fi\fi\next}
\newif\ifshowstuffinmargin
\showstuffinmarginfalse
\def\jjtags{\ifx\shlhetal\relax 
      % so this is a public version --> no-op 
  \else
\ifx\shlhetal\undefinedcontrolseq
      % again, this is a public version --> no-op 
\else
\showstuffinmargintrue
\ifx\all\relax\else\expandafter\t@gsjj\all,\fi\fi \fi
}

% \def\zeigentag#1{\def\xxy{#1}\show\xxy\show\realstag\realstag#1\mginpar{\hbox{$\langle$}#1\hbox{$\rangle$}}}
% \def\zeigencite#1{\hbox{$\langle$}#1\hbox{$\rangle$}\mginpar{\realscite#1}}


\def\tagjj#1{\realstag{#1}\oldmginpar{\zeigen{#1}}}
% \def\citejj#1{\zeigen{#1}\mginpar{\rechnen{#1}}}
\def\citejj#1{\rechnen{#1}\mginpar{\zeigen{#1}}}     % modified Sep 02, saharon's suggestion

% \def\rechnen#1{rechnen(\expandafter\ifx\csname stag#1\endcsname\relax ??\else
%                            \csname stag#1\endcsname\fi)endrech }
\def\rechnen#1{\expandafter\ifx\csname stag#1\endcsname\relax ??\else
                           \csname stag#1\endcsname\fi}

% \def\zeigentag#1{//#1--\csname stag#1\endcsname::}

\newdimen\theight



\def\marginfont{\sevenrm}



\def\trymarginbox#1{\setbox0=\hbox{\marginfont\hskip\marginshift #1}%
		\global\marginshift\wd0 
		\global\advance\marginshift\margindelta}


\def \oldmginpar#1{%
\ifvmode\setbox0\hbox to \hsize{\hfill\rlap{\marginfont\quad#1}}%
\ht0 0cm
\dp0 0cm
\box0\vskip-\baselineskip
\else 
             \vadjust{\trymarginbox{#1}%
		\ifdim\marginshift>\marginmax \global\marginshift\marginmin
			\trymarginbox{#1}%
                \fi
             \theight=\ht0
             \advance\theight by \dp0    \advance\theight by \lineskip
             \kern -\theight \vbox to \theight{\rightline{\rlap{\box0}}%
\vss}}\fi}

\newdimen\upordown
\global\upordown=8pt
\font\tinyfont=cmtt8 % scaled 700
\def\mginpar#1{\smash{\hbox to 0cm{\kern-10pt\raise7pt\hbox{\tinyfont #1}\hss}}}
% testing, october 2005, mg
\def\mginpar#1{{\hbox to 0cm{\kern-10pt\raise\upordown\hbox{\tinyfont #1}\hss}}\global\upordown-\upordown}


% \def\mginpar#1{mg-#1-mg }



%Verbatim tags:
\def\t@gsoff#1,{\def\@{#1}\ifx\@\empty\let\next=\relax\else\let\next=\t@gsoff
   \def\@@{p}\ifx\@\@@\else
   \expandafter\gdef\csname#1cite\endcsname##1{\zeigen{##1}}
   \expandafter\gdef\csname#1page\endcsname##1{?}
   \expandafter\gdef\csname#1tag\endcsname##1{\zeigen{##1}}\fi\fi\next}
\def\verbatimtags{\showstuffinmarginfalse
\ifx\all\relax\else\expandafter\t@gsoff\all,\fi}
%%% \def\zeigen#1{zeigen(\hbox{$\langle$}#1\hbox{$\rangle$})endz }
\def\zeigen#1{\hbox{$\scriptstyle\langle$}#1\hbox{$\scriptstyle\rangle$}}

% % \def\margincite#1{\ifshowstuffinmargin\mginpar{\rechnen{#1}}\fi}
%  changed, april 2003, mg: we now have always the verbatim tag in the margin
\def\margincite#1{\ifshowstuffinmargin\mginpar{\zeigen{#1}}\fi}

\def\margintag#1{\ifshowstuffinmargin\oldmginpar{\zeigen{#1}}\fi}

\def\marginplain#1{\ifshowstuffinmargin\mginpar{{#1}}\fi}
\def\marginbf#1{\marginplain{{\bf \ \ #1}}}


%Equation numbering:
\def\(#1){\edef\dot@g{\ifmmode\ifinner(\hbox{\noexpand\etag{#1}})
   \else\noexpand\eqno(\hbox{\noexpand\etag{#1}})\fi
   \else(\noexpand\ecite{#1})\fi}\dot@g}

%Reference numbering:
\newif\ifbr@ck
\def\eat#1{}
\def\[#1]{\br@cktrue[\br@cket#1'X]}
\def\br@cket#1'#2X{\def\temp{#2}\ifx\temp\empty\let\next\eat
   \else\let\next\br@cket\fi
   \ifbr@ck\br@ckfalse\br@ck@t#1,X\else\br@cktrue#1\fi\next#2X}
\def\br@ck@t#1,#2X{\def\temp{#2}\ifx\temp\empty\let\neext\eat
   \else\let\neext\br@ck@t\def\temp{,}\fi
   \def\teemp{#1}\ifx\teemp\empty\else\rcite{#1}\fi\temp\neext#2X}
\def\resetbr@cket{\gdef\[##1]{[\rtag{##1}]}}
\def\references{\resetbr@cket\newsection References\par}

%Footnotes:
\newtoks\symb@ls\newtoks\s@mb@ls\newtoks\p@gelist\n@wcount\ftn@mber
    \ftn@mber=1\newif\ifftn@mbers\ftn@mbersfalse\newif\ifbyp@ge\byp@gefalse
\def\defm@rk{\ifftn@mbers\n@mberm@rk\else\symb@lm@rk\fi}
\def\n@mberm@rk{\xdef\m@rk{{\the\ftn@mber}}%
    \global\advance\ftn@mber by 1 }
\def\rot@te#1{\let\temp=#1\global#1=\expandafter\r@t@te\the\temp,X}
\def\r@t@te#1,#2X{{#2#1}\xdef\m@rk{{#1}}}
\def\b@@st#1{{$^{#1}$}}\def\str@p#1{#1}
\def\symb@lm@rk{\ifbyp@ge\rot@te\p@gelist\ifnum\expandafter\str@p\m@rk=1 
    \s@mb@ls=\symb@ls\fi\write\f@nsout{\number\count0}\fi \rot@te\s@mb@ls}
\def\byp@ge{\byp@getrue\n@wwrite\f@nsin\openin\f@nsin=\jobname.fns 
    \n@wcount\currentp@ge\currentp@ge=0\p@gelist={0}
    \re@dfns\closein\f@nsin\rot@te\p@gelist
    \n@wread\f@nsout\openout\f@nsout=\jobname.fns }
\def\m@kelist#1X#2{{#1,#2}}
\def\re@dfns{\ifeof\f@nsin\let\next=\relax\else\read\f@nsin to \f@nline
    \ifx\f@nline\v@idline\else\let\t@mplist=\p@gelist
    \ifnum\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX0
    \else\currentp@ge=\f@nline
    \global\p@gelist=\expandafter\m@kelist\the\t@mplistX1\fi\fi
    \let\next=\re@dfns\fi\next}
\def\symbols#1{\symb@ls={#1}\s@mb@ls=\symb@ls} 
\def\bigsymbol{\textstyle}
\symbols{\bigsymbol\ast,\dagger,\ddagger,\sharp,\flat,\natural,\star}
\def\ftnumbers{\ftn@mberstrue} \def\ftsymbols{\ftn@mbersfalse}
\def\paginal{\byp@ge} \def\resetftnumbers{\ftn@mber=1}
\def\ftnote#1{\defm@rk\expandafter\expandafter\expandafter\footnote
    \expandafter\b@@st\m@rk{#1}}

%Miscellaneous macros:
\long\def\jump#1\endjump{}
\def\ssum{\mathop{\lower .1em\hbox{$\textstyle\Sigma$}}\nolimits}
\def\down#1{_{{}_{\scriptstyle #1}}}
\def\qed{\nobreak\kern 1em \vrule height .5em width .5em depth 0em}
\def\newneq{\hbox{\rlap{\hbox to 1\wd9{\hss$=$\hss}}\raise .1em 
   \hbox to 1\wd9{\hss$\scriptscriptstyle/$\hss}}}
\def\subsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\subset$}}}}
\def\supsetne{\setbox9 = \hbox{$\subset$}\mathrel{\hbox{\rlap
   {\lower .4em \newneq}\raise .13em \hbox{$\supset$}}}}

%Blackboard bold:
\def\vbar{\mathchoice{\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height6.3ptdepth-.5ptwidth.8pt\kern-.8pt}
   {\vrule height4.1ptdepth-.35ptwidth.6pt\kern-.6pt}
   {\vrule height3.1ptdepth-.25ptwidth.5pt\kern-.5pt}}
\def\f@dge{\mathchoice{}{}{\mkern.5mu}{\mkern.8mu}}
\def\b@c#1#2{{\rm \mkern#2mu\vbar\mkern-#2mu#1}}
\def\b@b#1{{\rm I\mkern-3.5mu #1}}
\def\b@a#1#2{{\rm #1\mkern-#2mu\f@dge #1}}
\def\bb#1{{\count4=`#1 \advance\count4by-64 \ifcase\count4\or\b@a A{11.5}\or
   \b@b B\or\b@c C{5}\or\b@b D\or\b@b E\or\b@b F \or\b@c G{5}\or\b@b H\or
   \b@b I\or\b@c J{3}\or\b@b K\or\b@b L \or\b@b M\or\b@b N\or\b@c O{5} \or
   \b@b P\or\b@c Q{5}\or\b@b R\or\b@a S{8}\or\b@a T{10.5}\or\b@c U{5}\or
   \b@a V{12}\or\b@a W{16.5}\or\b@a X{11}\or\b@a Y{11.7}\or\b@a Z{7.5}\fi}}

\catcode`\X=11 \catcode`\@=12


% Sep 2003, mg:

% % \input 300stuff
%  *** start including 300stuff.tex *** 

% definitions needed to process the 300x papers  (plus 88 etc) 

\let\thischap\jobname



\def\partof#1{\csname returnthe#1part\endcsname}
\def\chapof#1{\csname returnthe#1chap\endcsname}

\def\setchapter#1,#2,#3;{% 
  \expandafter\def\csname returnthe#1part\endcsname{#2}%
  \expandafter\def\csname returnthe#1chap\endcsname{#3}%
}

\setchapter 300a,A,II.A;
\setchapter 300b,A,II.B;
\setchapter 300c,A,II.C;
\setchapter 300d,A,II.D;
\setchapter 300e,A,II.E;
\setchapter 300f,A,II.F;
\setchapter 300g,A,II.G;
\setchapter  E53,B,N;
\setchapter  88r,B,I;
\setchapter  600,B,III;
\setchapter  705,B,IV;
\setchapter  734,B,V;


% \def\cprefix#1{%     \cprefix{300b} generates A.II.  or II.  or "nothing"
%  %\leavevmode\vrule width 2cm height 1cm depth 0cm
% \edef\theotherpart{\partof{#1}}\edef\theotherchap{\chapof{#1}}%
% \ifx\theotherpart\thispart
%    \ifx\theotherchap\thischap % nothing
%     \else % same part, different chap
%      \theotherchap%
%     \fi
%    \else % different part
%      \theotherpart.\theotherchap\fi}


\def\cprefix#1{%     \cprefix{300b} generates II.  or "nothing"
 %\leavevmode\vrule width 2cm height 1cm depth 0cm
\edef\theotherpart{\partof{#1}}\edef\theotherchap{\chapof{#1}}%
\ifx\theotherpart\thispart
   \ifx\theotherchap\thischap % nothing
    \else % same part, different chap
     \theotherchap%
    \fi
   \else % different part
     \theotherchap\fi}


 % \sectioncite[\S4]{300b}  ->  II.\S4   or A.II.\S4
\def\sectioncite[#1]#2{%
     \cprefix{#2}#1}

\def\chaptercite#1{Chapter \cprefix{#1}}

% now define \thispart
\edef\thispart{\partof{\thischap}}
\edef\thischap{\chapof{\thischap}}

\def\lastpage of '#1' is #2.{\expandafter\def\csname lastpage#1\endcsname{#2}}

%  *** end including 300stuff.tex *** 

\def\spuriousreset{}

%  *** end including mathdefs.tex *** 
% % \input citeadd
%  *** start including citeadd.tex *** 
%   citeadd -- a few additions for 
% files from alice that were procesed with "citealice"

\expandafter\ifx\csname citeadd.tex\endcsname\relax
\expandafter\gdef\csname citeadd.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to
\string\input{citeadd.tex} twice.   This does not make sense.} 
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi





\def\sciteu{\sciteerror{undefined}}
\def\sciteuphantom{\complainaboutcitation{undefined}}

\def\scitet{\sciteerror{ambiguous}}
\def\scitetphantom{\complainaboutcitation{ambiguous}}




\def\sciteerror#1#2{{\mathortextbf{\scite{#2}}}\complainaboutcitation{#1}{#2}}
\def\mathortextbf#1{\hbox{\bf #1}}
\def\complainaboutcitation#1#2{%
\vadjust{\line{\llap{---$\!\!>$ }\qquad scite$\{$#2$\}$ #1\hfil}}}





%  *** end including citeadd.tex *** 
\sectno=-1   % start with sect 0
\localtags
\jjtags
\NoBlackBoxes
\define\mr{\medskip\roster}
\define\sn{\smallskip\noindent}
\define\mn{\medskip\noindent}
\define\bn{\bigskip\noindent}
\define\ub{\underbar}
\define\wilog{\text{without loss of generality}}
\define\ermn{\endroster\medskip\noindent}
\define\dbca{\dsize\bigcap}
\define\dbcu{\dsize\bigcup}
\define \nl{\newline}
% (citealice) \define \green{GREEN STARTS HERE}
% (citealice) \define \endgreen{GREEN STOPS HERE}
\magnification=\magstep 1
\documentstyle{amsppt}
% % \input alice2000
%  *** start including alice2000.tex *** 
% This file should be inputted whenever we use amsppt.sty and 
% old tex.  
%  Here we redefine \subjclass (use 1991 instead of 2000, otherwise 
% the following definition comes directly from 
%% 
%%              `amsppt.sty', generated 
%% on <1997/2/2> with the docstrip utility (2.2i).
%% 
%% The original source files were:
%% 
%% amsppt.doc 
%%% ====================================================================
%%% @AMSTeX-style-file{
%%%   filename  = "amsppt.sty",
%%%   version   = "2.1h",
%%%   date      = "1997/02/02",
%%%   time      = "09:27:44 EST",
%%%   checksum  = "56844 3264 16617 137829",
%%%   author    = "American Mathematical Society",
%%%   address   = "PO Box 6248, Providence, RI 02940-6248, USA",
%%%   telephone = "401-455-4080 or (in the USA) 800-321-4AMS",

{    % the braces make the catcode-change local. 
\catcode`@11

\ifx\alicetwothousandloaded@\relax
  \endinput\else\global\let\alicetwothousandloaded@\relax\fi


\gdef\subjclass{\let\savedef@\subjclass
 \def\subjclass##1\endsubjclass{\let\subjclass\savedef@
   \toks@{\def\usualspace{{\rm\enspace}}\eightpoint}%
   \toks@@{##1\unskip.}%
   \edef\thesubjclass@{\the\toks@
     \frills@{{\noexpand\rm2000 {\noexpand\it Mathematics Subject
       Classification}.\noexpand\enspace}}%
     \the\toks@@}}%
  \nofrillscheck\subjclass}
} 

%  *** end including alice2000.tex *** 
% % \input alice2jlem
%  *** start including alice2jlem.tex *** 
%% # Keywords  Input file to be used for texing Alice's files

\expandafter\ifx\csname alice2jlem.tex\endcsname\relax
  \expandafter\xdef\csname alice2jlem.tex\endcsname{\the\catcode`@}
\else \message{Hey!  Apparently you were trying to
\string\input{alice2jlem.tex}  twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi



% % \input bib4plain
%  *** start including bib4plain.tex *** 
\expandafter\ifx\csname bib4plain.tex\endcsname\relax
  \expandafter\gdef\csname bib4plain.tex\endcsname{}
\else \message{Hey!  Apparently you were trying to \string\input
  bib4plain.tex twice.   This does not make sense.}
\errmessage{Please edit your file (probably \jobname.tex) and remove
any duplicate ``\string\input'' lines}\endinput\fi


%  This file should be inputted if you want to use 
%  bibtex fom within plain TeX. 
\def\makeatletter{\catcode`\@11 }      % Not really need for standard
\def\makeatother{\catcode`\@12 }       % bibtex files, but these commands
\def\renewcommand{\newcommand}	       % are used in our literal-unsrt.bst
				       % and other files. 
%  The above lines were written by Martin.Goldstern@tuwien.ac.at
%  The rest is just the file btxmac.tex. 
%  (except for one small change below, regarding undefined citations)

%% @texfile{
%%   author = "Karl Berry and Oren Patashnik",
%%   version = "0.99j",
%%   date = "14 Mar 1992",
%%   filename = "btxmac.tex",
%%   address = "Please use electronic mail",
%%   checksum = "834    4503   33061",
%%   email = "opbibtex@cs.stanford.edu",
%%   codetable = "ISO/ASCII",
%%   supported = "yes",
%%   docstring = "Defines macros that make BibTeX work with plain TeX",
%% }
% BibTeX-for-TeX macros, version 0.99j, for BibTeX 0.99c, TeX 3.0 or later.
% Copyright (C) 1990--92 by Karl Berry and Oren Patashnik; all rights reserved.
% You may copy this file provided: that it's accompanied by the
% "BibTeXing" document, whose text is contained in the file `btxdoc.tex';
% that any documentation you write for these macros also gives a
% reference for "BibTeXing"; and that either you make absolutely no
% changes to your copy, or if you do make changes, (1) you name the file
% something other than `btxmac.tex' and you remove all occurrences of
% `btxmac.tex' from the file, (2) you put, somewhere in the first twenty
% lines of the file, your name, along with an electronic address at which
% others who might use the file may reach you, and (3) you remove each
% occurrence of Oren's name and electronic address from this file.  These
% restrictions help ensure that all standard versions of these macros are
% identical, and that Oren doesn't get deluged with inappropriate e-mail.
%
% This file, btxmac.tex, contains TeX macros that allow BibTeX, a
% bibliography program that was originally designed for use with LaTeX,
% to work with plain TeX.  Please report any bugs (outright goofs,
% improvable macros, misfeatures, or unclear documentation) to Oren
% Patashnik (opbibtex@cs.stanford.edu).  These macros will become frozen
% shortly after BibTeX version 1.00 is released.
%
% AMS-TEX WARNING: We tried very hard, for version .99i of these macros,
% to make them compatible with AmS-TeX.  We succeeded to the extent
% that, if you use one of the standard bibliography styles, you probably
% won't notice any problems with version 0.99i of btxmac.tex.  But
% ultimately we failed, in that the inherent incompatibilities between
% plain TeX and AmS-TeX kept making these macros break, for certain
% inputs or certain styles.  Examples:  (1) AmS-TeX treats at-signs as
% special, in ways that plain TeX and LaTeX don't, so that, for example,
% you can't have any `@' characters in an argument to the \cite command,
% the way you can in TeX or LaTeX; (2) AmS-TeX decided that plain TeX's
% and LaTeX's macron-accent control sequence `\=' should be undefined;
% so you'll need to define `\=' to be `\B' to get the xampl.bib example
% suggested below to work with AmS-TeX; (3) AmS-TeX redefines the tie
% character `~' of plain TeX, and AmS-TeX's `amsppt' style redefines
% plain TeX's `\nobreak' macro, so that if you use an author-date style
% like `apalike' and you have a multiple-author reference for which the
% author-date style automatically produces a citation in the text like
% `(Jones et~al., 1992)' you will throw AmS-TeX's `amsppt' style into
% an infinite loop, exceeding its input stack size.  In practice, such
% incompatibilities surface infrequently; but it is now clear to us that
% it's not worth the effort (perhaps it's not even possible) to make the
% btxmac.tex macros robust when used both with plain TeX and Ams-TeX.
% If the BibTeX/AmS-TeX results attainable with the current btxmac.tex
% macros are sufficient, fine.  But if there's a demand for more robust
% BibTeX/AmS-TeX behavior, then someone who's very familiar with the
% AmS-TeX package should probably make an amsbtxmc.tex version of the
% macros (remembering to follow the copyright restrictions above).
% Until then, if you're an AmS-TeX user, or a LaTeX or plain TeX user
% sharing files with an AmS-TeX user, beware.
% END OF AMS-TEX WARNING.
%
% To use these macros you should be familiar with how BibTeX interacts
% with LaTeX, since BibTeX's interaction with TeX is very similar; that
% interaction is explained in the LaTeX manual.  It also helps to
% have read "BibTeXing", the documentation that accompanies BibTeX.
% Then, if you want, you should redefine any of the macros that begin
% with `\bbl' or `\biblabel' or `\print' that you need to get formatting
% different from the default (the default settings are designed to
% accompany a bibliography style like BibTeX's standard style `plain').
% The macros you might want to change are described briefly a few
% paragraphs hence.  [To get started without reading any documentation,
% try running the nine-line .tex file below through TeX and BibTeX.
% Remember the general scheme: Running (La)TeX writes information on
% the .aux (auxiliary) file; then running BibTeX reads information from
% the .aux, .bst (style), and .bib (database) files, and writes
% information (the bibliography) on a .bbl file; then running (La)TeX
% incorporates the bibliography; then running (La)TeX once more fixes
% the remaining forward references into the bibliography.  Thus, to get
% everything incorporated into your output, you'll have to run (La)TeX,
% BibTeX, (La)TeX, (La)TeX.  (Standup, sitdown, fight, fight, fight.)]
%
% These macros can stand alone or they can be \input into a macro
% package, like Eplain, that is sufficiently compatible with plain TeX.
% To use these macros to format the 0.99 version of the xampl.bib file
% that's distributed with BibTeX (that version of the file has no
% self-identification), you'll need to define \mbox, which is a LaTeX
% command, to be \hbox, as in the example below.
%
% Here's a nine-line plain TeX file for trying out btxmac.tex; of course
% you'll have to remove the comment characters at the beginning of each
% line, and, depending on your system, you might have to take steps so
% that BibTeX can "see" the files xampl.bib and plain.bst (BibTeX will
% give you two empty-field warning messages that you should ignore).
%
%     \def\mbox#1{\leavevmode\hbox{#1}}
%     \input btxmac
%     \noindent This cites Aamport's gnominious article~\cite{article-full}.
%     \medskip
%     \leftline{\bf References}
%     \nocite{*}   % put all database entries into the reference list
%     \bibliography{xampl}   % specify the database files; here, just xampl.bib
%     \bibliographystyle{plain}   % specify plain.bst as the style file
%     \bye
%
%
%   HISTORY
%
% Karl Berry wrote the original version of these macros in 1989 and
% 1990, for use in his `Eplain' package.  Oren Patashnik modified them
% slightly in July 1990, as part of the official BibTeX distribution.
%
%    1-Aug-90  Version 0.99a, not released to the general public.
%   14-Aug-90  0.99b, first general release.
%   26-Aug-90  0.99c, made \@undefinedmessage work with other macro packages.
%    6-Sep-90  0.99d, allowed for general formatting of bibliography labels,
%                     for general formatting of (in-text) citations, and for
%                     changing certain catcodes while reading the .aux file.
%   14-Nov-90  0.99e, changed the way \@setletters works, made some \new...'s
%                     non-outer, and changed the way Eplain reads this file.
%   12-Dec-90  0.99f, made \@resetnumerals change the `,' and `.' catcodes; and
%                     added \biblabelextrahang, \@getoptionalarg, and \bblsc.
%   11-Mar-91  0.99g, made a few minor changes required by the way Eplain reads
%                     this file, but no functional changes.
%   24-Apr-91  0.99h, inhibited the reading and writing of the .aux file if it
%                     isn't used or if the \noauxfile macro is defined, and
%                     removed some .aux-file-opening detritus; printed the
%                     cite-key of undefined citations in \tt font; changed the
%                     catcode of `_' inside \cite; and called \@resetnumerals
%                     from inside a group.
%   29-Feb-92  0.99i, made these macros semi-compatible with AmS-TeX; removed
%                     \@resetnumerals, \@setletters, \@tokstostring, and
%                     friends; changed the way \cite handles catcodes; changed
%                     \@getoptionalarg, and had \bibitem and \newcommand use
%                     it; added \@futurenonspacelet and (to facilitate the use
%                     of multiple reference lists) \bblfilebasename; changed
%                     \biblabelprint to use the new macros \biblabelprecontents
%                     and \biblabelpostcontents, and to, by default, right-
%                     justify numeric labels; and renamed \biblabelextrahang to
%                     the more descriptive \biblabelextraspace.
%   14-Mar-92  0.99j, made 0.99i's use of `\\' local to btxmac.tex.
%
%
% The LaTeX-related commands defined in this file include (a) the four
% commands that a user types (\bibliography, \bibliographystyle, \cite,
% and \nocite); (b) the three commands that BibTeX looks for in the .aux
% file (\bibdata, \bibstyle, and \citation---there is a fourth command
% that BibTeX looks for, but that command is related to LaTeX's \include
% facility, so these macros ignore that command); and (c) a LaTeX
% command (\newcommand) that's written by one of the four standard
% bibliography styles (alpha).  The definitions here are not exactly the
% same as the corresponding LaTeX definitions (those eight LaTeX
% definitions depend on a significant fraction of LaTeX itself).  But
% the only substantial differences are with \newcommand, which here,
% without complaining, lets you redefine a preexisting control sequence
% (in LaTeX, \newcommand won't let you redefine a preexisting command),
% and which here doesn't make the control sequences it defines \long (in
% LaTeX, that happens automatically); there may also be other minor
% differences.  To summarize: Unless you know what you're doing, you
% shouldn't define any control sequences with these eight names:
%
% \bibdata
% \bibliography
% \bibliographystyle
% \bibstyle
% \citation
% \cite
% \newcommand
% \nocite
%
% There are three other commands written by one or more of the four
% standard (plain, abbrv, alpha, unsrt) or four semistandard (acm,
% apalike, ieeetr, siam) bibliography styles; they take effect only
% within the bibliography, and are redefinable, as explained later:
%
% \em
% \newblock
% \sc
%
% There's one control sequence you might want to use (but not redefine)
% in redefining \biblabelprint:
%
% \biblabelwidth
%
% There are fifteen other control sequences (explained later in more detail)
% that the macros of this file will use if you define them---you should
% define them after the \input btxmac command but before the \bibliography
% command.  The first six begin with `\bbl' and affect fonts, spacing,
% perhaps other characteristics of the bibliography, and which .bbl files
% get read; the next five begin with `\biblabel' and determine how labels
% are formatted in the bibliography; and the last four begin with `\print'
% and determine how the in-text citations are formatted:
%
% \bblem
% \bblfilebasename
% \bblhook
% \bblnewblock
% \bblrm
% \bblsc
% \biblabelcontents
% \biblabelprecontents
% \biblabelprint
% \biblabelpostcontents
% \biblabelextraspace
% \printbetweencitations
% \printcitefinish
% \printcitenote
% \printcitestart
%
% If it's defined before the \input btxmac command, the control sequence
% below inhibits the reading and writing of the .aux file(s), and the
% issuing of related warning messages.  Any definition will do.  This
% feature might help when you're working on draft stages of a document:
%
% \noauxfile
%
%
% Here's another control sequence (it's described later) that you
% probably won't want to redefine unless you are writing another macro
% package; if you do redefine it, however, do it before the \input btxmac
% command (and notice that it has an `@' in its name):
%
% \@undefinedmessage
%
% Any other control sequence in this file that might conflict with
% something you've defined will have an `@' in its name, so such conflicts
% are unlikely; but if you're worried about a specific control sequence
% name, do a text search of this file to look for it.
%
%
% So to start things off we turn `@' into a letter (category code 11),
% keeping track of the old category code for future restoration.
% (Simply resetting it to 12 when we leave these macros is
% insufficient.)  The use of `\cite' as a temporary control sequence is
% a kludge, but it's a reasonably simple way to accomplish what we need
% without possibly overwriting something (without an `@' in its name)
% that might already be defined.
%
\edef\cite{\the\catcode`@}%
\catcode`@ = 11
\let\@oldatcatcode = \cite
\chardef\@letter = 11
\chardef\@other = 12
%
%
% Next come some things that will be useful later.
%
% Make an outer definition into an inner one (due to Chris Thompson).
% The arguments should be the control sequence to be defined, and the
% new of the \outer control sequence, as characters; the control
% sequence #1 is defined to be just the same as \csname#2\endcsname, but
% not \outer.  For example, \@innerdef\innernewcount{newcount} would
% define \innernewcount to be a non-outer version of \newcount.
%
\def\@innerdef#1#2{\edef#1{\expandafter\noexpand\csname #2\endcsname}}%
%
% We use \@innerdef to make some of our allocations local, because
% Eplain includes our code inside a conditional.  We put @'s in the
% names to minimize the (already small) chance of conflicts.
%
\@innerdef\@innernewcount{newcount}%
\@innerdef\@innernewdimen{newdimen}%
\@innerdef\@innernewif{newif}%
\@innerdef\@innernewwrite{newwrite}%
%
%
% Swallow one parameter.
%
\def\@gobble#1{}%
%
%
% Use TeX 3.0's \inputlineno to get the line number, for better error
% messages, but if we're using an old version of TeX, don't do anything.
%
\ifx\inputlineno\@undefined
   \let\@linenumber = \empty % Pre-3.0.
\else
   \def\@linenumber{\the\inputlineno:\space}%
\fi
%
%
% The following macro \@futurenonspacelet (from the TeXbook) behaves
% essentially like \futurelet except that it discards any implicit or
% explicit space tokens that intervene before a nonspace is scanned:
%
\def\@futurenonspacelet#1{\def\cs{#1}%
   \afterassignment\@stepone\let\@nexttoken=
}%
\begingroup % The grouping here avoids stepping on an outside use of `\\'.
\def\\{\global\let\@stoken= }%
\\ % now \@stoken is a space token (\\ is a control symbol, so that
   % space after it is seen).
\endgroup
\def\@stepone{\expandafter\futurelet\cs\@steptwo}%
\def\@steptwo{\expandafter\ifx\cs\@stoken\let\@@next=\@stepthree
   \else\let\@@next=\@nexttoken\fi \@@next}%
\def\@stepthree{\afterassignment\@stepone\let\@@next= }%
%
%
% \@getoptionalarg\CS gets an optional argument from the input, enclosed
% in brackets, then expands \CS.  We set \@optionalarg to \empty if we
% don't find one, otherwise to the text of the argument.  This assumes
% the brackets don't have a funny category code.
%
\def\@getoptionalarg#1{%
   \let\@optionaltemp = #1%
   \let\@optionalnext = \relax
   \@futurenonspacelet\@optionalnext\@bracketcheck
}%
%
% The \expandafter's in this macro let us avoid the use of \aftergroup,
% which is somewhat more expensive.
%
\def\@bracketcheck{%
   \ifx [\@optionalnext
      \expandafter\@@getoptionalarg
   \else
      \let\@optionalarg = \empty
      % We can't do the \temp after the \fi, because then the \temp gets
      % in the way of reading the optional argument from the input, if
      % we do have one.
      \expandafter\@optionaltemp
   \fi
}%
%
\def\@@getoptionalarg[#1]{%
   \def\@optionalarg{#1}%
   \@optionaltemp
}%
%
%
% From LaTeX.
%
\def\@nnil{\@nil}%
\def\@fornoop#1\@@#2#3{}%
%
\def\@for#1:=#2\do#3{%
   \edef\@fortmp{#2}%
   \ifx\@fortmp\empty \else
      \expandafter\@forloop#2,\@nil,\@nil\@@#1{#3}%
   \fi
}%
%
\def\@forloop#1,#2,#3\@@#4#5{\def#4{#1}\ifx #4\@nnil \else
       #5\def#4{#2}\ifx #4\@nnil \else#5\@iforloop #3\@@#4{#5}\fi\fi
}%
%
\def\@iforloop#1,#2\@@#3#4{\def#3{#1}\ifx #3\@nnil
       \let\@nextwhile=\@fornoop \else
      #4\relax\let\@nextwhile=\@iforloop\fi\@nextwhile#2\@@#3{#4}%
}%
%
%
% This macro tests if a file \jobname.#1 exists, and sets \if@fileexists
% appropriately.  If an optional argument is given, it is used as the
% root part of the filename instead of \jobname.
%
\@innernewif\if@fileexists
%
\def\@testfileexistence{\@getoptionalarg\@finishtestfileexistence}%
\def\@finishtestfileexistence#1{%
   \begingroup
      \def\extension{#1}%
      \immediate\openin0 =
         \ifx\@optionalarg\empty\jobname\else\@optionalarg\fi
         \ifx\extension\empty \else .#1\fi
         \space
      \ifeof 0
         \global\@fileexistsfalse
      \else
         \global\@fileexiststrue
      \fi
      \immediate\closein0
   \endgroup
}%
%
%
%% [[[start of BibTeX-specific stuff]]]
%
% Now come the four main LaTeX commands and their associated .aux
% commands.  Just as in LaTeX, \bibliographystyle defines the BibTeX
% style name (.bst file, that is), and \bibliography defines the
% database (.bib) file(s).  The corresponding .aux-file commands are
% \bibstyle and \bibdata, which are there only for BibTeX's (but not
% LaTeX's) use.
%
\def\bibliographystyle#1{%
   \@readauxfile
   \@writeaux{\string\bibstyle{#1}}%
}%
\let\bibstyle = \@gobble
%
% As well as writing the \bibdata command to tell BibTeX which .bib
% files to read, we read the .bbl file that BibTeX (or a person,
% conceivably) has produced.  We use \bblfilebasename as the root of the
% filename to read; this defaults to \jobname.
%
\let\bblfilebasename = \jobname
\def\bibliography#1{%
   \@readauxfile
   \@writeaux{\string\bibdata{#1}}%
   \@testfileexistence[\bblfilebasename]{bbl}%
   \if@fileexists
      % We just output a non-discardable item (the `whatsit' with the
      % \bibdata command).  This means that the glue that will be
      % inserted next (\parskip or \baselineskip, most likely) will be a
      % legal breakpoint.  Most likely, this is after some kind of
      % heading, where we don't want to allow a page break.  So:
      \nobreak
      \@readbblfile
   \fi
}%
\let\bibdata = \@gobble
%
% The \nocite{label,label,...} command writes its argument to \@auxfile,
% unless instructed not to, but produces no text in the document.  Both
% the \nocite and \cite commands produce \citation commands in the .aux file.
%
\def\nocite#1{%
   \@readauxfile
   \@writeaux{\string\citation{#1}}%
}%
%
\@innernewif\if@notfirstcitation
%
% \cite[note]{label,label,...} produces the citations for the labels as
% well.  If the optional argument `note' is present, it's added after
% the labels.  Since \cite calls \nocite to do its .aux-file writing,
% \cite doesn't need to call \@readauxfile (\nocite does).
%
\def\cite{\@getoptionalarg\@cite}%
%
% Typeset the citations for the labels in #1, followed by the note, if
% it exists.  To change the citation's format in the text, redefine one
% or more `\print...' macros, whose defaults appear later in this file.
%
\def\@cite#1{%
   % Remember the optional argument, in case one of the macros we call
   % below ends up looking for an optional argument itself.  For
   % example, if a \cite[note] triggers reading the .aux file, then the
   % [note] would be clobbered, since \@testfileexistence looks for an
   % optional arg.
   \let\@citenotetext = \@optionalarg
   % Start printing the text, beginning with a left bracket by default.
   \printcitestart
   % It's complicated, but because \nocite puts a `whatsit' onto the list,
   % \nocite should follow \printcitestart.  It's conceivable, but very
   % unlikely, that this `whatsit' will cause a problem (glue that doesn't
   % disappear when you want it to is the most likely symptom), requiring
   % a change either to \printcitestart or to the label that the .bst file
   % produces.
   \nocite{#1}%
   \@notfirstcitationfalse
   \@for \@citation :=#1\do
   {%
      \expandafter\@onecitation\@citation\@@
   }%
   \ifx\empty\@citenotetext\else
      \printcitenote{\@citenotetext}%
   \fi
   \printcitefinish
}%
%
\newif\ifweareinprivate
\weareinprivatetrue
\ifx\shlhetal\undefinedcontrolseq\weareinprivatefalse\fi
\ifx\shlhetal\relax\weareinprivatefalse\fi
\def\@onecitation#1\@@{%
   \if@notfirstcitation
      \printbetweencitations
   \fi
   %
   \expandafter \ifx \csname\@citelabel{#1}\endcsname \relax
      \if@citewarning
         \message{\@linenumber Undefined citation `#1'.}%
      \fi
      % Give it a dummy definition:
     \ifweareinprivate
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
% Change: marginal remark added, goldstrn@math.huji.ac.il, 
% goldstern@tuwien.ac.at, May 1996 mg
%  !!! change !!!
\strut 
\vadjust{\vskip-\dp\strutbox
\vbox to 0pt{\vss\parindent0cm \leftskip=\hsize 
\advance\leftskip3mm
\advance\hsize 4cm\strut\openup-4pt 
\rightskip 0cm plus 1cm minus 0.5cm ?  #1 ?\strut}}
         {\tt
            \escapechar = -1
            \nobreak\hskip0pt\pfeilsw%\special{ps:100 0 0 setrgbcolor }%
            \expandafter\string\csname#1\endcsname
                     %\special{ps:0 0 0 setrgbcolor }
             \pfeilso
            \nobreak\hskip0pt
         }%
      }%
     \else  % ifweareinprivate = false
      \expandafter\gdef\csname\@citelabel{#1}\endcsname{%
            {\tt\expandafter\string\csname#1\endcsname}
      }%
     \fi  % ifweareinprivate
   \fi
   % Now produce the text, whether it was undefined or not.
   \csname\@citelabel{#1}\endcsname
   \@notfirstcitationtrue
}%
%
% Given a label `foo', the macro `\b@foo' is supposed to
% hold the text that should be produced.
%
\def\@citelabel#1{b@#1}%
%
% So, how does a citation label get defined?  When we read the .bbl file
% (below), a \bibitem writes out a \@citedef command.  And when we read
% the \@citedef, we define \@citelabel{#1}, where #1 is the user's
% label.
%
\def\@citedef#1#2{\expandafter\gdef\csname\@citelabel{#1}\endcsname{#2}}%
%
%
% Reading the .bbl file also produces the typeset bibliography.  Please
% notice, however, that we do not produce the title for the references
% (e.g., `References'), as LaTeX does.  The formatting and spacing of
% that title, whether it should go into the headline, and so on, are all
% things determined by your format.  We cannot know those things in
% advance.  If you wish, you can define \bblhook to produce the title.
% Or just do it before the \bibliography command.
%
\def\@readbblfile{%
   % Define a counter to tell us which item number we are on, unless
   % we've already defined it (because the document has more than one
   % bibliography).
   \ifx\@itemnum\@undefined
      \@innernewcount\@itemnum
   \fi
   %
   \begingroup
      \def\begin##1##2{%
         % ##1 is just `thebibliography'.
         % ##2 is the widest label.
         % We set (new dimen) \biblabelwidth based on the widest label
         \setbox0 = \hbox{\biblabelcontents{##2}}%
         \biblabelwidth = \wd0
      }%
      \def\end##1{}% ##1 is `thebibliography' again.
      %
      % Here we have two possibilities:
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      %
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      %
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         \@writeaux{\string\@citedef{##1}{\csname\@citelabel{##1}\endcsname}}%
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      %
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%
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%
\@innernewdimen\biblabelwidth
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%
% Now come the main macros that are related to the printing of the
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%
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%
% By default the labels are typeset in \bblrm, and enclosed in brackets.
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\def\biblabelcontents#1{{\bblrm [#1]}}%
%
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%
% Emphasis for producing, e.g., titles, is done with \it by default.
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%
% Some styles use a caps-and-small-caps font for author names.  LaTeX
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%
% The major parts of an entry are separated with \bblnewblock.  The
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%
%
% Here are the four default definitions for formatting the in-text
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%
%
% Now comes the stuff for dealing with LaTeX's \newcommand.  As
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%
%% [[[end of BibTeX-specific stuff]]]
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%
% Names of references (arguments given in the \cite and \nocite
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}%
%
% The \@readauxfile macro does all that work the first time it's called.
% Since it's called once for every \cite, \nocite, \bibliography, and
% \bibliographystyle command that the user issues, we need to remember
% whether the work's been done.  It's considered done if we're not to do
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%
\newif\if@auxfiledone
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%
% It's conceivable you'd want to change how other characters are read;
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%
%
% After reading the .aux file, \@readauxfile opens it for writing.
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%
\@innernewwrite\@auxfile
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%
%
% A macro package that uses btxmac.tex might define
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%
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%
% Finally, before leaving we restore @'s old category code.
%
\catcode`@ = \@oldatcatcode



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% # Keywords: Script or Calligraphic (Caligraphic) letters with the RSFS Font

% The story so far:    July 1998 -- Saharon would like to have a
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%  *** end including alice2jlem.tex *** 
\pageheight{8.5truein}
%\pageheight{48.5pc}
\topmatter 
\title{Reflection implies the SCH} \endtitle

\author {Saharon Shelah \thanks {
   The author would like to thank the Israel Science Foundation for
   partial support of this research (Grant no. 242/03). Publication 794. 
\null\newline I would like to thank 
Alice Leonhardt for the beautiful typing.} \endthanks} \endauthor 


\affil{Institute of Mathematics\\
 The Hebrew University\\
 Jerusalem, Israel
 \medskip
 Rutgers University\\
 Mathematics Department\\
 New Brunswick, NJ  USA} \endaffil

\keywords  reflection, stationary sets, Singular Cardinal Hypotheses,
 pcf, set theory \endkeywords

\subjclass  03E04, 03E05  \endsubjclass

\abstract  We prove that, e.g., if $\mu > \text{ cf}(\mu) = \aleph_0$
and $\mu > 2^{\aleph_0}$ and every stationary family of countable
subsets of $\mu^+$ reflect in some subset of $\mu^+$ of cardinality
$\aleph_1$ \ub{then} the SCH for $\mu^+$ holds
(moreover, for $\mu^+$, any scale for
$\mu^+$ has a bad stationary set of cofinality $\aleph_1$).
This answers a question of Foreman and Todor\v cevi\'c who 
gets such conclusion from the simultaneous
reflection of four stationary sets. 
\endabstract
\endtopmatter
\document  
 

\newpage

\head {\S0 Introduction} \endhead  \resetall \sectno=0
 \spuriousreset
\bigskip

In \S1 we prove that the strong hypothesis (pp$(\mu) = \mu^+$ for every
singular $\mu$) hence the SCH (singular cardinal hypothesis, that is
$\lambda^\kappa \le \lambda^+ + 2^\kappa$) holds when:
for every $\lambda \ge \aleph_1$ every stationary ${\Cal S} \subseteq
[\lambda]^{\aleph_0}$ reflect in some $A \in [\lambda]^{\aleph_1}$.

This answers a question of Foreman and Todor\v cevi\'c 
\cite{FoTo} where they
proved that the SCH holds for every $\lambda \ge \aleph_1$ when: every
four stationary ${\Cal S}_\ell \subseteq [\lambda]^{\aleph_0},\ell =
1,2,3,4$ reflect simultaneously in some $A \in
[\lambda]^{\aleph_1}$.  They were probably motivated by Velickovi\'c 
\cite{Ve92a} which used another reflection principle: for every stationary
${\Cal A} \subseteq [\lambda]^{\aleph_0}$ there is $A \in
[\lambda]^{\aleph_1}$ such that ${\Cal A} \cap [A]^{\aleph_0}$ contains
a closed unbounded subset, rather than just a stationary set.

The proof here is self-contained modulo two basic
quotations from \cite{Sh:g}, \cite{Sh:f}; we continued \cite{Sh:e},
\cite{Sh:755} in some respects.  
We prove more in \S1.  In particular if 
$\mu > \text{ cf}(\mu) = \aleph_0$ and pp$(\mu) >
\mu^+$ then some ${\Cal A} \subseteq [\mu^+]^{\aleph_0}$ reflect in no
uncountable $A \in [\mu^+]^{\le \mu}$ and see more in the end.

We thank the referee and Shimoni Garti for not few helpful comments.
\mn
For the reader's convenience let us recall some basic definitions.
\definition{\stag{0.1} Definition}  Assume $\theta$ is regular
uncountable (if $\theta = \sigma^+,[B]^{< \sigma} = [B]^{\le \theta}$,
we can use $[B]^{\le \theta}$, the main case is $B = \lambda$)
\mr
\item "{$(a)$}"  ${\Cal A} \subseteq [B]^{< \theta}$ is closed in
$[B]^{< \theta}$, if for every $\{x_\beta:\beta < \alpha\} \subseteq
{\Cal A}$ where $0 < \alpha < \theta$ and $\beta_1 < \beta_2 < \alpha
\Rightarrow x_{\beta_1} \subseteq x_{\beta_2}$, we have $\dbcu_{\beta
< \alpha} x_\beta \in {\Cal A}$
\sn
\item "{$(b)$}"  ${\Cal A}$ is unbounded in $[B]^{< \theta}$, if for
any $y \in [B]^{< \theta}$ we can find $x \in {\Cal A}$, such that $x
\supseteq y$
\sn
\item "{$(c)$}"  ${\Cal A}$ is a club in $[B]^{< \theta}$, if
${\Cal A} \subseteq [B]^{< \theta}$ and (a)+(b) hold for 
${\Cal A}$
\sn
\item "{$(d)$}"  ${\Cal A}$ is stationary in $[B]^{< \theta}$,
or is a stationary subset of $[B]^{< \theta}$ \ub{when} 
${\Cal A} \subseteq [B]^{<\theta}$ and 
${\Cal A} \cap {\Cal C} \ne \emptyset$ for every club
${\Cal C}$ of $[B]^{< \theta}$
\sn
\item "{$(e)$}"  similarly for $[B]^{\le \theta}$ or
$[B]^\theta$ or consider ${\Cal S} \subseteq [B]^{< \theta}$ as 
a subset of $[B]^{\le \theta}$.
\endroster
\enddefinition
\bigskip

\remark{\stag{0.1A} Remark}  Note: if $B = \theta$ then ${\Cal A}
\subseteq [B]^{< \theta}$ is stationary iff ${\Cal A} \cap \theta$ is
a stationary subset of $\theta$.
\endremark
\bigskip

\definition{\stag{0.2} Definition}  Let ${\Cal A} \subseteq [B_1]^{<
\theta}$ and $B_2 \in [B_1]^\mu$.  We say that ${\Cal A}$ reflects in $B_2$
when ${\Cal A} \cap [B_2]^{< \theta}$ is a stationary subset of
$[B']^{< \theta}$.
\enddefinition
\bigskip

\definition{\stag{0.3} Definition}  Let $\kappa$ be a regular
uncountable cardinal, and assume ${\Cal A}$ is a stationary subset of
$[B]^{< \kappa}$.  We define $\diamondsuit_{\Cal A}$ (i.e., the
diamond principle for ${\Cal A}$) as the following assertion:  
\nl
there exists a
sequence $\langle u_a:a \in {\Cal A}\rangle$, such that $u_a \subseteq a$ for
any $a \in {\Cal A}$, and for every $B' \subseteq B$ the set $\{a \in
{\Cal A}:B' \cap a = u_a\}$ is stationary in $[B']^{< \kappa}$.
\enddefinition
\bigskip

\demo{\stag{0.6} Notation}  1) For regular $\lambda > \kappa$ let
$S^\lambda_\kappa = \{\delta < \lambda$: cf$(\delta) = \kappa\}$.
\nl
2) ${\Cal H}(\lambda)$ is the set of $x$ with transitive closure of
   cardinality $< \lambda$.
\nl
3) $<^*_\lambda$ denotes any well ordering of ${\Cal H}(\lambda)$.
\enddemo
\bn
Let us repeat the definition of the next ideal: (see
\cite{Sh:E12}).
\definition{\stag{0.4} Definition}  For $S \subseteq \lambda$ we say
that $S \in \check I[\lambda]$ iff: there is a club $E$ in $\lambda$,
and a sequence $\langle C_\alpha:\alpha < \lambda\rangle$ such that:
\mr
\widestnumber\item{$(iii)$}
\item "{$(i)$}"  $C_\alpha \subseteq \alpha$ for every $\alpha < \lambda$
\sn
\item "{$(ii)$}"  otp$(C_\alpha) < \alpha$
\sn
\item "{$(iii)$}"  $\beta \in C_\alpha \Rightarrow C_\beta = \beta 
\cap C_\alpha$
\sn
\item "{$(iv)$}"  $\alpha \in E \cap S \Rightarrow \alpha =
\sup(C_\alpha)$.
\endroster
\enddefinition
\bigskip

\proclaim{\stag{0.4.3} Claim}  (By \cite{Sh:420} or see \cite{Sh:E12}).
\nl
1) If $\kappa,\lambda$ are regular and $\lambda > \kappa^+$ \ub{then} there
is a stationary $S \subseteq S^\lambda_\kappa$ 
such that $S \in \check I[\lambda]$.
\nl
2) In \scite{0.4} we can add $\alpha \in E \cap S 
\Rightarrow \text{\rm otp}(C_\alpha) = \,\text{\rm cf}(\alpha)$.
\endproclaim
\bigskip

\definition{\stag{0.5} Definition/Observation}  Let ${\Cal A} \subseteq
[\lambda]^\theta$ be stationary and $\lambda \ge \sigma > \theta$ and
$\sigma$ has uncountable cofinality \ub{then} prj$_\sigma({\Cal A}) 
:= \{\sup(a):a \in {\Cal A}\}$, 
and it is a stationary subset of $\sigma$; if $\sigma = \lambda$ we
may omit it.
\enddefinition
\bigskip

\definition{\stag{0.9} Definition}  Let $f_i$ be a function with domain
$\aleph_0$ to the ordinals, for every 
$i \in I$ where $I$ is a set of ordinals. 
We say that the sequence $\bar f = \langle f_i:i \in
I\rangle$ is free, if we can find a sequence $\bar n = \langle n_i:i
\in I\rangle$ of natural numbers such that:  $(i,j \in I) \wedge (i
<j) \wedge (n_i,n_j \le n < \omega) \Rightarrow f_i(n) < f_j(n)$.  We
say that $\bar f$ is $\mu$-free when for every $J \in [I]^{< \mu}$ the
sequence $\bar f \restriction J$ is free.
\enddefinition
\bigskip

\remark{\stag{0.10} Remark}  If we consider ``$\langle f_\alpha:\alpha
\in S\rangle$ for some stationary $S \subseteq \theta$" when $\theta =
\text{ cf}(\theta) > \aleph_0$, then we can assume (without loss of
generality) that $n_i = n(*)$ for every $i \in S$, as we can decrease $S$.
\endremark
\newpage

\head {\S1 Reflection in $[\mu^+]^{\aleph_0}$ and the strong
hypothesis} 
\endhead  \resetall 
 \spuriousreset
\bigskip

\proclaim{\stag{nr.1} The Main Claim}  Assume
\mr
\item "{$(A)$}"   $\lambda = \mu^+$ and $\mu > {\text {\rm cf\/}}(\mu) =
\aleph_0$ and $\aleph_2 \le \mu_* \le \lambda$ (e.g., $\mu_*
= \aleph_2$ which implies that below always $\theta = \aleph_1$) 
\sn
\item "{$(B)$}"  $\bar \lambda = \langle \lambda_n:n < \omega \rangle$ is an
increasing  sequence of regular cardinals $> \aleph_1$ with limit
$\mu$ and $\lambda = { \text{\rm tcf\/}}(\Pi \lambda_n,
<_{J^{\text{bd}}_\omega})$
\sn
\item "{$(C)$}"  $\bar f = \langle f_\alpha:\alpha < \lambda
\rangle$ is $<_{J^{\text{bd}}_\omega}$-increasing cofinal in 
$(\dsize \prod_{n <  \omega} \lambda_n,<_{J^{\text{bd}}_\omega})$
\sn
\item "{$(D)$}"  the sequence $\bar f$ is $\mu_*$-free or at least for 
every cardinal $\theta$ for which 
$\aleph_1 \le \theta = { \text{\rm cf\/}}(\theta) < \mu_*$ the
following is satisfied: if $\theta \le \sigma < \mu_*,{\Cal A} 
\subseteq [\sigma]^{\aleph_0}$ is stationary (recall \scite{0.1A})
and $\langle
\delta_i:i < \theta \rangle$ is an increasing continuous sequence of
ordinals $< \lambda$ \ub{then} for some stationary subfamily ${\Cal
A}_1$ of ${\Cal A}$ (${\Cal A}_1$ is stationary in 
$[\sigma]^{\aleph_0}$ of course)
letting $R_1 = \text{\rm prj}_\theta({\Cal A}_1)$,
see \scite{0.5} we have $\langle f_{\delta_i}:i \in R_1 \rangle$ is
free.  See \scite{0.9} and by \scite{0.10} we can assume that $i \in R_1
\Rightarrow n_i = n(*)$ so $\langle f_{\delta_i}(n):
i \in R_1 \rangle$ is strictly increasing
for every $n \in [n(*),\omega)$.
\ermn
\ub{Then} some stationary ${\Cal A} \subseteq [\lambda]^{\aleph_0}$
does not reflect in any $A \in [\lambda]^{\aleph_1}$ or even in any
uncountable $A \in [\lambda]^{< \mu_*}$, (see Definition \scite{0.2}). 
\endproclaim  
\bigskip

\remark{\stag{nr.1.3} Remark}  0) From the main claim the result on SCH
should be clear from pcf theory (by translating between the pp,cov
and cardinal arithmetic) but we shall give details (i.e. quotes).
\nl
1) Clause (D) from claim \scite{nr.1} is
related to ``the good set of $\bar f$, gd$(\bar f)$ 
contains $S^\lambda_\theta$ modulo the club filter".  
But the clause (D) is stronger.

Note that the good set gd$(\bar f)$ of $\bar f$ is
$\{\delta < \lambda:\aleph_0 < \text{ cf}(\delta) < \mu$ and for
some increasing sequence $\langle \alpha_i:i < \text{
cf}(\delta)\rangle$ of ordinals with limit $\delta$ and sequence $\bar
n = \langle n_i:i < \text{ cf}(\delta)\rangle$ of natural numbers
we have $i < j < \text{ cf}(\delta) \wedge n_i \le n < \omega 
\wedge n_j \le n < \omega \Rightarrow f_{\alpha_i}(n) < 
f_{\alpha_j}(n)$ (so $\langle \cup\{f_{\alpha_i}(n):i
< \text{ cf}(\delta)$ and $n \ge n_i\}:n < \omega\rangle$ is a
$<_{J^{\text{bd}}_\omega}$-eub of $\bar f \restriction \delta)$.

If we use another ideal $J$ say on $\theta < \mu$, the $n_i$ is
replaced by $s_i \in J$.
\nl
2) Recall that by using the silly square (\cite[II,1.5A,pg.51]{Sh:g}), 
if cf$(\mu) \le \theta < \mu,J$ an 
ideal on $\theta$ (e.g. $\theta = \aleph_0,J =   
J^{\text{bd}}_\omega$) and pp$_J(\mu) > \lambda = 
\text{ cf}(\lambda) > \mu$ \ub{then} we can find a sequence $\langle
\lambda_i:i < \theta \rangle$ of regulars $< \mu$ such that $\mu =
\lim_J\langle \lambda_i:i < \theta\rangle$ and tcf$(\dsize \prod_{i < \theta}
\lambda_i,<_J)=\lambda$ and some $\bar f = \langle f_\alpha:\alpha <
\lambda\rangle$ exemplifying it satisfies gd$(\bar f) = \{\delta <
\lambda:\theta < \text{ cf}(\delta) < \mu\}$ and moreover $\bar f$ is
$\mu^+$-free which here means that for every $u \subseteq \lambda$ of
cardinality $\le \mu$ we can find $\langle s_\alpha:\alpha \in
u\rangle$ such that $s_\alpha \in J$, and for $\alpha < \beta$ from
$u$ we have $\varepsilon \in \theta \backslash (s_\alpha \cup s_\beta)
\Rightarrow f_\alpha(\varepsilon) < f_\beta(\varepsilon)$.  This is
stronger than the demand in clause (D).
\nl
3) Also recall that if $\kappa$ is supercompact, $\mu > \kappa >
\theta = \text{ cf}(\mu)$ and $\langle \lambda_i:i < \theta\rangle$ is
an increasing sequence of regulars with limit $\mu,\langle
f_\alpha:\alpha < \lambda \rangle$ exemplifies $\lambda = \mu^+ 
= \text{ tcf}(\dsize \prod_{i < \theta}
\lambda_i,<_{J^{\text{bd}}_\theta})$ then for unboundedly many
$\kappa' \in \kappa \cap \text{ Reg} \backslash
\theta^+$ the set $S^\lambda_{\kappa'} \backslash \text{gd}(\bar f)$ is
stationary.  This is preserved by e.g. Levy$(\aleph_1,< \kappa)$.
\nl
4) For part of the proof (mainly subclaim \scite{nr.4}) we can weaken
clause (D) of the assumption, e.g. in the end demand ``$\Rightarrow
f_{\delta_i}(n) \ne f_{\delta_j}(n)"$ only.  The weakest version of
clause (D) which suffices there is: for any club $C$ of $\theta$ the
set $\cup\{\text{Rang}(f_\alpha):\alpha \in C\}$ has cardinality
$\theta$. 
\endremark
\bn
Before proving \scite{nr.1} we draw some conclusions.
\demo{\stag{nr.2} Conclusion}  1) Assume $\mu > 2^{\aleph_0}$ then
$\mu^{\aleph_0} = \mu^+$ provided that
\mr
\item "{$(A)_\mu$}"  $\mu > \text{ cf}(\mu) = \aleph_0$
\sn
\item "{$(B)_\mu$}"  every stationary ${\Cal A} \subseteq
[\mu^+]^{\aleph_0}$ reflects in some $A \in [\mu^+]^{\aleph_1}$.
\ermn
2) Assume $\lambda \ge \mu_* \ge \aleph_2$.  We can replace $(B)_\mu$ by
\mr
\item "{$(B)_{\mu,\mu_*}$}"   every stationary ${\Cal A} \subseteq
[\mu^+]^{\aleph_0}$ reflects in some uncountable $A \in [\mu^+]^{< \mu_*}$.
\endroster
\enddemo
\bigskip

\demo{Proof}  1) Easily if $\aleph_1 \le \mu' \le \mu$ then
$(B)_{\mu'}$ holds.  Now if $\mu$ is a counterexample, 
\wilog \, $\mu$ is a minimal
counterexample and then by \cite[IX,\S1]{Sh:g} we have 
pp$(\mu) > \mu^+$, hence there is a sequence $\langle
\lambda^0_n:n < \omega \rangle$ of regular cardinals with limit $\mu$ such that
$\mu^{++} = \text{ tcf}(\dsize \prod_{n < \omega}
\lambda^0_n/J^{\text{bd}}_\omega)$, (see \cite{Sh:g}; more
\cite{Sh:E12} or \cite[6.5]{Sh:430}; e.g. using ``no hole for pp" and
the pcf theorem).
Let $\bar f^0 =
\langle f^0_\alpha:\alpha < \mu^{++} \rangle$ witness this.
Hence by \cite[II,1.5A,p.51]{Sh:g} there is $\bar f$ as required in
\scite{nr.1} even a $\mu^+$-free one and 
also the other assumptions there hold so we can
conclude that there exists ${\Cal A} \subseteq [\mu^+]^{\aleph_0}$
which does not reflect in any $A \in [\mu^+]^{\aleph_1}$,
so we get a contradiction to $(B)_\mu$. 
\nl
2) The same proof. \hfill$\square_{\scite{nr.2}}$
\enddemo
\bigskip

\demo{\stag{nr.3} Conclusion}  1) If for every $\lambda > \aleph_1$, every
stationary ${\Cal A} \subseteq [\lambda]^{\aleph_0}$ reflects in some $A \in
[\lambda]^{\aleph_1}$, \ub{then}
\mr
\item "{$(a)$}"  the strong hypothesis (see \cite{Sh:410}, \cite{Sh:420},
\cite{Sh:E12}) holds, i.e. for every singular $\mu$, pp$(\mu) = \mu^+$
and moreover cf$([\mu]^{\text{cf}(\mu)},\subseteq) = \mu^+$ which follows
\sn
\item "{$(b)$}"  the SCH holds.
\ermn
2) Let $\theta \ge \aleph_0$.  We can restrict ourselves to $\lambda >
\theta^+,A \in [\lambda]^{\theta^+}$ (getting the strong
hypothesis and SCH above $\theta$).
\enddemo
\bigskip

\demo{Proof}  1) As in \scite{nr.2}, by \scite{nr.1} we have
$\mu > \text{cf}(\mu) = \aleph_0
\Rightarrow$ pp$(\mu) = \mu^+$, 
this implies clause (a) (i.e. by
\cite[VIII,\S1]{Sh:g}, $\mu > \text{ cf}(\mu) \Rightarrow$ pp$(\mu) =
\mu^+$).  Hence inductively by 
\cite[IX,1.8,pg.369]{Sh:g}, \cite[1.1]{Sh:430} we have 
$\kappa < \mu \Rightarrow \text{ cf}([\mu]^\kappa,\subseteq) 
= \mu$ if cf$(\mu) > \kappa$ and is
$\mu^+$ if $\mu > \kappa \ge \text{ cf}(\mu)$.  This is a consequence
of the strong hypothesis.)  The SCH follows. 
\nl
2) The same proof.    \hfill$\square_{\scite{nr.3}}$
\enddemo
\bigskip

\demo{Proof of \scite{nr.1}}  Let $M^*$ be 
an algebra with universe $\lambda$ and
countably many functions, e.g. all those definable in $({\Cal
H}(\lambda^+),\in,<^*_{\lambda^+},\bar f)$ and are functions
from $\lambda$ to $\lambda$ or just the functions $\alpha \mapsto
f_\alpha(n),\alpha \mapsto \alpha +1$.
\enddemo
\bigskip

\proclaim{\stag{nr.4} Subclaim}  There are $\bar S,S^*,\bar D$ such that:
\mr
\widestnumber\item{$(*)_3(iii)$}
\item "{$(*)_1\,\,\,\,\,\,\,$}"  $\bar S = \langle S_\varepsilon:
\varepsilon < \omega_1 \rangle$ is 
a sequence of pairwise disjoint stationary subsets of $S^\lambda_{\aleph_0}$ 
\sn
\item "{$(*)_2(i)$}"  $S^* \subseteq S^\lambda_{\aleph_1} = \{\delta <
\lambda:{\text{\rm cf\/}}(\delta) = \aleph_1\}$ is stationary and belongs to 
$\check I[\lambda]$
\sn
\item "{$(ii)$}"  if $\delta \in S^*$ \ub{then} there is an increasing
continuous sequence $\langle \alpha_\varepsilon:\varepsilon < \omega_1
\rangle$ of ordinals with limit $\delta$ such that for some sequence $\bar
\zeta = \langle \zeta_\varepsilon:\varepsilon \in R \rangle$ of
ordinals $< \omega_1$, the
set $R \subseteq \omega_1$ is stationary, $\varepsilon \in R
\Rightarrow \alpha_\varepsilon \in S_{\zeta_\varepsilon}$ and $\bar
\zeta$ is with no repetitions
\sn
\item "{$(*)_3(i)$}"  $\bar D = \langle
(D_{1,\varepsilon},D_{2,\varepsilon}):\varepsilon < \omega_1 \rangle$
\sn
\item "{$(ii)$}"  $D_{\ell,\varepsilon}$ is a filter on $\omega$
containing the filter of cobounded subsets of $\omega$
\sn
\item "{$(iii)$}"  if $R_1 \subseteq \omega_1$ is unbounded and $A \in
\cap \{D_{1,\varepsilon}:\varepsilon \in R_1\}$ \ub{then} for some
$\varepsilon \in R_1$ we have $A \ne \emptyset \,\,\text{\rm mod } 
D_{2,\varepsilon}$
\sn
\item "{$(iv)$}"  for each $\varepsilon < \omega_1$ for some $A$ we have $A \in
D_{1,\varepsilon} \and \omega \backslash A \in D_{2,\varepsilon}$.
\endroster
\endproclaim 
\bigskip

\remark{\stag{nr.4A} Remark}  1) For \scite{nr.4} we can assume
(A), (B), (C) of \scite{nr.1} and weaken clause (D): 
because (inside the proof below) necessarily for any stationary
$S^* \subseteq S^\lambda_{\aleph_1}$, which belongs to $\check I[\lambda]$,
we can restrict the demand in (D) of \scite{nr.1} 
for any $\langle \delta_i:i < \theta \rangle$ with
limit in $S^*$.  See more in \cite{Sh:775}. \nl
2) In Subclaim \scite{nr.4} we can demand
$\zeta_\varepsilon = \varepsilon$ in $(*)_2(ii)$.  See the proof.
\nl
3) If we like to demand that each
$D_{\ell,\varepsilon}$ is an ultrafilter (or just have ``$A \in
D_{2,\varepsilon}"$ in the end of $(*)_3(iii)$ of \scite{nr.4}), use
\cite{Sh:E3}.   
\endremark
\bigskip

\demo{Proof of the subclaim \scite{nr.4}}  How do we choose them?

Let $\langle A_i:i \le \omega_1 \rangle$ be a sequence of infinite
pairwise almost disjoint subsets of $\omega$.  Let $D_{1,i} = \{A
\subseteq \omega:A_i \backslash A$ is finite$\},D_{2,i} = \{A
\subseteq \omega:A_j \backslash A$ is finite for all but finitely many
$j < \omega_1\}$, so $D_{2,i}$ does not depend on $i$.  Clearly
$\langle (D_{1,i},D_{2,i}):i < \omega_1\rangle$ satisfies $(*)_3$.

Recall, \scite{0.4.3}, that by \scite{0.4.3} and the fact that
 $\lambda > \aleph_\omega > \aleph_2$, there is a stationary 
$S^* \subseteq S^\lambda_{\aleph_1}$
from $\check I[\lambda]$, and so every stationary $S' \subseteq S^*$ has the
same properties (i.e. is a stationary subset of $\lambda$ which belongs to
$\check I[\lambda]$ and is included in $S^\lambda_{\aleph_1}$).

Let $N \prec ({\Cal H}((2^\lambda)^+),\in,<^*)$ be of cardinality $\mu$
such that $\mu +1 \subseteq N$ and $\{\bar \lambda,\mu,\bar f\}$ belongs
to $N$.  Let $C^* = \cap\{C:C \in N$ is a club of $\lambda\}$, so
clearly $C^*$ is a club of $\lambda$.
For each $h \in {}^\lambda(\omega_1)$ we can try $\bar S^h =
\langle S^h_\gamma:\gamma < \omega_1\rangle$ where $S^h_\gamma = \{\delta <
\lambda:\text{cf}(\delta) = \aleph_0$ and $h(\delta) = \gamma\}$, 
so it is enough to show that
for some $h \in N$, the sequence $\bar S^h$ is as required.  As $\|N\|
< \lambda$, for this it is enough to show that for every 
$\delta \in S^\lambda_{\aleph_1} \cap C^*$ (or just for every $\delta
\in S^* \cap C^*$,
or just for stationarily many $\delta \in S^* \cap C^*$) the demand holds
for $\bar S^h$ for some $h \in ({}^\lambda(\omega_1)) \cap N$.  That
is, $S^{\bar h}$ satisfies $(*)_1$ and $(*)_2(ii)$ of subclaim \scite{nr.4}.
Given any $\delta \in S^\lambda_{\aleph_1} \cap C^*$ let $\langle
\alpha_\varepsilon:\varepsilon < \omega_1 \rangle$ be an increasing
continuous sequence of ordinals with limit $\delta$, \wilog \,
$\varepsilon < \omega_1 \Rightarrow \text{ cf}(\alpha_\varepsilon) =
\aleph_0$,  and by assumption $(D)$ of
\scite{nr.1} for some \footnote{Note that if we require just that
$\langle f_{\alpha_\varepsilon}(n):\varepsilon \in R\rangle$ is
without repetitions, \ub{then} for some stationary subset $R'$ of $R$ the
sequence $\langle f_{\alpha_\varepsilon}(n):\varepsilon \in R'\rangle$
is increasing.} stationary $R \subseteq \omega_1$ and $n = n(*)
< \omega$, the sequence $\langle
f_{\alpha_\varepsilon}(n):\varepsilon \in R \rangle$ is strictly
increasing, so let its limit be $\beta^*$.  So $\beta^* \le \mu$ and
cf$(\beta^*) = \aleph_1$ but $\mu +1 \subseteq N$ hence $\beta^* \in N$.

Note that
\mr
\item "{$(*)_1$}"  for every $\beta' < \beta^*$ the set 
$\{\alpha \in S^\lambda_{\aleph_0}:f_\alpha(n(*)) \in 
[\beta',\beta^*)\}$ is a stationary subset of $\lambda$.
\ermn
[Why?  So assume that $\beta' < \beta^*$ and that the set $S' =
\{\alpha \in S^\lambda_{\aleph_0}:f_\alpha(n(*)) \in 
[\beta',\beta^*)\}$ is not a
stationary subset of $\lambda$.  As $\beta^* +1 \subseteq N$ and $\bar
f \in N$ clearly $S' \in N$ hence there is a club $C'$ of $\lambda$
disjoint to $S'$ which belongs to $N$.  Clearly acc$(C')$ too is a
club of $\lambda$ which belongs to $N$ hence $C^* \subseteq \text{
acc}(C')$ hence $\delta \in \text{ acc}(C')$.  
So $\delta = \sup(C' \cap \delta)$, so $C' \cap
\delta$ is a club of $\delta$.  Recall that $\{\alpha_\varepsilon:\varepsilon
\in R\}$ is a stationary subset of $\delta$ of order type $\aleph_1$.

Now by the choice of $\beta^*$ for some $\varepsilon(*) \in R$ we have
$\beta' \le f_{\alpha_{\varepsilon(*)}}(n(*))$, 
hence $\varepsilon \in R \backslash
\varepsilon(*) \Rightarrow f_{\alpha_\varepsilon}(n(*)) \in
[\beta',\beta^*)$, so $\delta$ has a stationary subset included in
$S'$ hence disjoint to $C'$, contradiction.]
\mr
\item "{$(*)_2$}"  for every $\beta' < \beta^*$ there is $\beta'' \in
(\beta',\beta^*)$ such that $\{\alpha \in S^\lambda_{\aleph_0}:
f_\alpha(n(*)) \in [\beta',\beta'')\}$ is a stationary subset of $\lambda$.
\ermn
[Why?  Follows from $(*)_1$ as $\aleph_1 < \lambda$.]

As $\beta^* \in N$ we can find an increasing continuous sequence $\langle
\beta_\xi:\xi < \omega_1\rangle \in N$ of ordinals with limit
$\beta^*$.  So by $(*)_2$
\mr
\item "{$(*)_3$}"  for every $\xi_1 < \omega_1$ for some $\xi_2 \in
(\xi_1,\omega_1)$ the set $\{\alpha \in
S^\lambda_{\aleph_0}:f_\alpha(n(*)) \in
[\beta_{\xi_1},\beta_{\xi_2})\}$ is stationary.
\ermn
Hence for some unbounded subset $u$ of $\omega_1$ we have
\mr
\item "{$(*)_4$}"  for every $\xi \in u$ the set
$\{\alpha \in S^\lambda_{\aleph_0}:f_\alpha(n(*)) \in 
[\beta_\xi,\beta_{\xi +1})\}$ is a stationary subset of $\lambda$.
\ermn
If $2^{\aleph_1} \le \mu$ then $u = \omega_1$
recalling we demand $\langle \beta_\xi:\xi < \omega_1\rangle \in N$.

We define $h:\lambda \rightarrow \omega_1$ by $h(\alpha) = \zeta$
\ub{iff} for some $\xi \in u$ we have $\zeta = \text{ otp}
(u \cap f_\alpha(n(*)))$ and/or
$\xi = 0 \and f_\alpha(n(*)) \ge \text{ sup}(u)$.
\nl
Clearly $h \in N$ is as required.  So $\bar S = \bar S^h$ as required 
exists.  But maybe $2^{\aleph_1} > \mu$, then after $(*)_3$ we
continue as follows.  Let $\bar C = \langle C_\delta:\delta \in
S^{\aleph_3}_{\aleph_1}\rangle$ be such that $C_\delta$ is a club of
$\delta$ of order type $\omega_1$ which guess clubs, i.e. for every
club $C$ of $\aleph_3$ for stationarily many $\delta \in
S^{\aleph_3}_{\aleph_1}$ we have $C_\delta \subseteq C$, exists by 
\cite[III]{Sh:g}.  Without loss of generality $\bar C \in N$.

Now let $\delta_* \in \text{ acc}(C^*)$ has cofinality $\aleph_3$.
Again $\delta_* \le \mu$ belongs to $N$ hence some increasing
continuous sequence $\langle \alpha_\varepsilon:\varepsilon <
\aleph_3\rangle \in N$ has limit $\delta_*$.  Now for each
$\varepsilon \in S^{\aleph_3}_{\aleph_1}$ we could choose above
$\delta = \alpha_\varepsilon$ hence for some $n_\varepsilon < \omega$
we have $(\forall \beta' < \alpha_\varepsilon)(\exists \beta'' <
\alpha_\varepsilon)[\beta' < \beta'' \wedge
(\exists^{\text{stat}}\gamma \in S^\lambda_{\aleph_0})(\beta' \le
f_\gamma(n_\varepsilon) < \beta'']$.  So for some $n_* < \omega$ the
set $S' := \{\varepsilon < \aleph_3:\text{cf}(\varepsilon) = \aleph_1$
and $n_\varepsilon = n_*\}$ is a stationary subset of
$S^{\aleph_3}_{\aleph_1}$.  It follows that $(\forall \varepsilon <
\aleph_3)(\exists \zeta < \aleph_3)[\varepsilon < \zeta \wedge
(\exists^{\text{stat}} \gamma \in
S^\lambda_{\aleph_0})(\alpha_\varepsilon \le f_\gamma(n_*) <
\alpha_{\varepsilon +1})]$.

Let $\zeta_\varepsilon$ be the minimal $\zeta$ as required above, so
$C = \{\xi < \aleph_3$: if $\varepsilon < \xi$ then $\zeta_\varepsilon
< \xi$ and $\xi$ is a limit ordinal$\}$ is a club of $\aleph_3$.
Hence for some $\varepsilon(*) \in S^{\aleph_3}_{\aleph_1}$ we have
$C_{\varepsilon(*)} \subseteq C$.  Let $u := \{\alpha_\zeta:\zeta \in
C_{\varepsilon(*)}\}$ so clearly $\langle \alpha_\zeta:\zeta \in
u\rangle$ belongs to $N$.  \hfill$\square_{\scite{nr.4}}$
\enddemo
\bigskip

\remark{\stag{nr.4.8} Remark}  Why can't we, in the proof of
\scite{nr.4}, after $(*)_3$, put the instead assuming $2^{\aleph_1}
\le \mu$ use ``as $N \prec ({\Cal H}(2^\lambda)^+),\in,<^*)$ \wilog \, $u
= w_1"$?

The set $u$ chosen above depends on $\delta$, so if $2^{\aleph_1} \le
\mu$ still $u \in N$, but otherwise the ``\wilog \, $u \in N$"
does not seem to be justified.
\endremark
\bigskip

\demo{Continuation of the proof of \scite{nr.1}}
Let $S := \cup\{S_\varepsilon:\varepsilon < \omega_1\}$.  For $\varepsilon <
\omega_1,\delta \in S_\varepsilon$ let 

$$
\align
{\Cal A}^\varepsilon_\delta = \{a:&a \in [\delta]^{\aleph_0} 
\text{ is } M^* \text{-closed, sup}(a) = \delta, \\
  &\text{otp}(a) \le \varepsilon \text{ and}\\
  &(\forall^{D_{1,\varepsilon}} n)(a \cap \lambda_n \subseteq f_\delta(n)) \\ 
  &\text{and } (\forall^{D_{2,\varepsilon}} n)(a \cap \lambda_n \nsubseteq
f_\delta(n))\}
\endalign
$$

\hskip67pt ${\Cal A}^\varepsilon = 
\cup\{{\Cal A}^\varepsilon_\delta:\delta \in S_\varepsilon\}$
\medskip

\hskip67pt ${\Cal A} = \cup\{{\Cal A}^\varepsilon:\varepsilon < \omega_1\}$.
\mn
So

\hskip67pt ${\Cal A} \subseteq [\lambda]^{\aleph_0}$.
\enddemo
\bn
As the case $\mu_* = \aleph_2$ was the original question and its proof is
simpler we first prove it.
\proclaim{\stag{nr.5} Subclaim}  ${\Cal A}$ does not reflect in any $A
\in [\lambda]^{\aleph_1}$.
\endproclaim
\bigskip

\demo{Proof}  So assume $A \in [\lambda]^{\aleph_1}$, let $\langle
a_i:i < \omega_1 \rangle$ be an increasing continuous sequence of
countable subsets of $A$ with union $A$, and let $R = \{i <
\omega_1:a_i \in {\Cal A}\}$, and assume toward contradiction that $R$
is a stationary subset of $\omega_1$.  As every $a \in {\Cal A}$ is
$M^*$-closed, necessarily $A$ is $M^*$-closed
and so \wilog \, each $a_i$ is $M^*$-closed.
\nl
For each $i \in R$ as $a_i \in {\Cal A}$ by the definition of 
${\Cal A}$ we can find $\varepsilon_i
< \omega_1$ and $\delta_i \in S_{\varepsilon_i}$ such that $a_i \in {\Cal
A}^{\varepsilon_i}_{\delta_i}$ hence by the definition of ${\Cal
A}^{\varepsilon_i}_{\delta_i}$ we have otp$(a_i) \le
\varepsilon_i$.  But as $A = \cup\{a_i:i < \omega_1\}$ with $a_i$ countable
increasing with $i$ and  
$|A| = \aleph_1$, clearly for some club $E$ of $\omega_1$
the sequence $\langle \text{otp}(a_i):i \in E \rangle$ is strictly
increasing, hence $i \in E \Rightarrow \text{ otp}(i \cap E) \le
\text{ otp}(a_i)$ so \wilog \, $i \in E \Rightarrow i \le
\text{ otp}(a_i)$ and \wilog \, $i < j \in E \Rightarrow
\varepsilon_i < j \le \text{ otp}(a_j)$.

Now $j \in E \cap R \Rightarrow j \le \text{ otp}(a_j) 
\le \varepsilon_j$ so $\langle \varepsilon_i:i
\in E \cap R \rangle$ is strictly increasing but $\langle
S_\varepsilon:\varepsilon < \omega_1 \rangle$ are pairwise disjoint and
$\delta_i \in S_{\varepsilon_i}$ so $\langle \delta_i:i \in E \cap R
\rangle$ is without repetitions; but $\delta_i = \sup(a_i)$ and for
$i<j$ from $R \cap E$ we have $a_i \subseteq a_j$ which implies that
$\delta_i = \sup(a_i) \le \sup(a_j) = \delta_j$ so
necessarily $\langle \delta_i:i \in R \cap E \rangle$ is strictly
increasing.

As $\sup(a_i) = \delta_i$ for $i \in R \cap E$, clearly $\sup(A) =
\cup\{\delta_i:i \in E \cap R\}$ and let $\beta_i = \text{ Min}(A
\backslash \delta_i)$ for $i < \omega_1$, it is well defined as $\langle
\delta_j:j \in R \cap E\rangle$ is strictly increasing.  Thinning $E$ \wilog 
\mr
\item "{$\circledast_1$}"  $i < j \in E \cap R \Rightarrow \beta_i <
\delta_j \and \beta_i \in a_j$. 
\ermn
Note that, by the choice of $M^*$,
\mr
\item "{$\circledast_2$}"  $i \in E \cap R \wedge
i < j \in E \cap R \Rightarrow \beta_i \in
a_j \Rightarrow \dsize \bigwedge_n (f_{\beta_i}(n) \in a_j) \Rightarrow
\dsize \bigwedge_n (f_{\beta_i}(n) + 1 \in a_j)$.
\ermn
As $\langle \delta_i:i \in E \cap R \rangle$ is (strictly) increasing 
continuous and $R \cap E$ is a stationary subset of $\omega_1$ 
clearly by clause $(D)$ of the assumption of \scite{nr.1}
we can find a stationary $R_1 \subseteq E \cap R$ and $n(*)$ such that $i
\in R_1 \wedge j \in R_1 \wedge i < j \wedge n(*) \le n < \omega
\Rightarrow f_{\delta_i}(n) < f_{\delta_j}(n)$.

Now if $i \in R_1$, let $\bold j(i) =: \text{ Min}(R_1 \backslash (i+1))$, so
$f_{\delta_i} \le_{J^{\text{bd}}_\omega} f_{\beta_i} <_{J^{\text{bd}}_\omega}
f_{\delta_{\bold j(i)}}$ so for some $m_i < \omega$ we have $n \in
[m_i,\omega) \Rightarrow f_{\delta_i}(n) \le f_{\beta_i}(n) <
f_{\delta_{j(i)}}(n)$.  Clearly for some stationary $R_2 \subseteq
R_1$ we have $i,j \in R_2 \Rightarrow m_i = m_j = m(*)$, so possibly
increasing $n(*)$ \wilog \, $n(*) \ge m(*)$; so we have (where Ch$_a
\in \dsize \prod_{n < \omega} \lambda_n$ is defined by Ch$_a(n) = 
\sup(a \cap \lambda_n)$ for any $a \in [\mu]^{< \lambda_0}$):
\mr
\item "{$\circledast_3$}"  for $i<j$ from $R_2$ we have $\bold j(i) \le j$ and
{\roster
\itemitem{ $(\alpha)$ }  $f_{\delta_i} 
\restriction [n(*),\omega) \le f_{\beta_i} \restriction [n(*),\omega)$
\sn
\itemitem{ $(\beta)$ }  $f_{\beta_i} \restriction [n(*),\omega) <
f_{\delta_{\bold j(i)}} \restriction [n(*),\omega) \le
f_{\delta_j} \restriction [n(*),\omega)$
\sn
\itemitem{ $(\gamma)$ }  $f_{\beta_i} \restriction [n(*),\omega) <
\text{ Ch}_{a_j} \restriction [n(*),\omega)$, by $\circledast_2$.
\endroster}
\ermn
Now by the definition of ${\Cal A}^{\varepsilon_i}_{\delta_i}$ as $a_i
\in {\Cal A}^{\varepsilon_i}_{\delta_i} \subseteq 
{\Cal A}^{\varepsilon_i}$ we have
\mr
\item "{$\circledast_4$}"  if $i \in R_2$ then
{\roster
\itemitem{ $(\alpha)$ }  Ch$_{a_i} \le_{D_{1,\varepsilon_i}}
f_{\delta_i}$
\sn
\itemitem{ $(\beta)$ }  $f_{\delta_i} <_{D_{2,\varepsilon_i}} 
\text{ Ch}_{a_i}$.
\endroster}
\ermn
Let $f^* \in \dsize \prod_{n < \omega} \lambda_n$ be $f^*(n) =
\cup\{f_{\beta_i}(n):i \in R_2\}$ if $n \ge n(*)$ and zero otherwise.  
As $f_{\beta_i}(n) \in a_{j(i)}$ for $i \in R_2$ by $\circledast_3(\gamma)$
clearly $n \ge n(*) \Rightarrow f^*(n) \le \sup\{\text{Ch}_{a_i}(n):i \in
R_2\} = \sup(A \cap \lambda_n) = \text{ Ch}_A(n)$ and by
$\circledast_3(\beta)$ we have $n \ge n(*) \Rightarrow 
\text{ cf}(f^*(n)) = \aleph_1$.  Let
$B_1 =: \{n < \omega:n \ge n(*)$ and $f^*(n) = \sup(A \cap
\lambda_n)\}$ and $B_2 =: [n(*),\omega) \backslash B_1$.
As $\alpha \in A \Rightarrow \alpha +1 \in A$ we have $n \in B_1
\Rightarrow A \cap \lambda_n \subseteq f^*(n) = \sup(A \cap
\lambda_n)$.  Also as by the previous sentence $f^* \restriction
[n(*),\omega) \le \text{ Ch}_A \restriction [n(*),\omega)$ clearly
$n \in B_2 \Rightarrow A \cap \lambda_n \nsubseteq f^*(n)$.  
As $\langle a_i:i \in R_2\rangle$ is increasing with union $A$, 
clearly there is $i(*) \in R_2$ such that: 
$n \in B_2 \Rightarrow a_{i(*)} \cap \lambda_n
\nsubseteq f^*(n)$, so as $i \in R_2 \and \alpha \in a_i \Rightarrow
\alpha +1 \in a_i$ we have $i(*) \le i \in R_2 \Rightarrow \text{ Ch}_{a_i}
\restriction B_2 > f_{\delta_i} \restriction B_2$ hence by clause
$\circledast_4(\alpha)$ we have $i \in R_2 \backslash i(*) 
\Rightarrow B_2 = \emptyset$ mod $D_{1,\varepsilon_i} \Rightarrow 
B_1 \in D_{1,\varepsilon_i}$.  
Also by $\circledast_3$ and the choice of $f^*$ and
$B_1$, for each $n \in B_1$ for some club $E_n$ of $\omega_1$ we
have $i \in E_n \cap R_2 \Rightarrow \sup(a_i \cap \lambda_n) =
\sup\{f_{\beta_j}(n):j \in R_2 \cap i\} = \sup\{f_{\delta_j}(n):j \in
R_2 \cap i\} \subseteq f_{\delta_i}(n)$, hence $R_3 = R_2 \cap \bigcap\{E_n
\backslash i(*):n < \omega\}$ is a stationary subset of 
$\omega_1$.  So $n \in
B_1 \and i \in R_3 \Rightarrow a_i \cap \lambda_n \subseteq
f_{\delta_i}(n)$ hence $i \in R_3 \Rightarrow \text{ Ch}_{a_i}
\restriction B_1 \le f_{\delta_i} \restriction B_1$ hence by
$\circledast_4(\beta)$ we have $i \in R_3 \Rightarrow B_1 
= \emptyset$ mod $D_{2,\varepsilon_i}$
hence $i \in R_3 \Rightarrow B_2 \in D_{2,\varepsilon_i}$.

By the choice of $\langle (D_{1,i},D_{2,i}):i < \omega_1 \rangle$ in
\scite{nr.4} as $B_1 \cup B_2$ is a cofinite subset of $\omega,
B_1 \cap B_2 = \emptyset$
(by the choice of $B_1,B_2$, clearly) 
and $R_3 \subseteq \omega_1$ is stationary we get a contradiction, see
$(*)_3(iii)$ of \scite{nr.4}.  \hfill$\square_{\scite{nr.5}}$ 
\enddemo
\bigskip

\proclaim{\stag{nr.6} Subclaim}  ${\Cal A}$ 
is a stationary subset of $[\lambda]^{\aleph_0}$.
\endproclaim
\bigskip

\remark{Remark}  See \cite{RuSh:117}, \cite[XI,3.5,pg.546]{Sh:f},  
\cite[XV,2.6]{Sh:f}.

We give a proof relying only on \cite[XI,3.5,pg.546]{Sh:f}.
In fact, also if we are interested in Ch$_N = \langle \sup(\theta \cap
N):\aleph_0 < \theta \in N \cap \text{ Reg} \rangle,N \prec ({\Cal
H}(\chi),\in)$ we have full control, e.g., if $\bar S = \langle
S_\theta:\aleph_1 \le \theta \in \text{ Reg } \cap \chi \rangle,S_\theta
\subseteq S^\theta_{\aleph_0}$ stationary we can demand
$\aleph_1 \le \theta = \text{ cf}(\theta) \wedge \theta \in N
\Rightarrow \text{ Ch}_N(\theta) \in S_\theta$ and 
control the order of $f^{{\frak a},\lambda}_{\sup(N \cap \lambda)}$ 
and Ch$_N \restriction {\frak a}$.
\endremark
\bigskip

\demo{Proof}  Let $M^{**}$ be an expansion of $M^*$ by countably many
functions; without loss of generality $M^{**}$ has Skolem functions.

Recall that $S^* \subseteq S^\lambda_{\aleph_1}$ is from \scite{nr.4}
so it belongs to $\check I[\lambda]$ and let
$\bar a = \langle a_\alpha:\alpha < \lambda \rangle$ witness it (see
\scite{0.4}, \scite{0.4.3}) so
otp$(a_\alpha) \le \omega_1$ and $\beta \in a_\alpha \Rightarrow a_\beta =
\beta \cap a_\alpha$, and omitting a non-stationary subset of
$S^*$ we have 
$\delta \in S^* \Rightarrow \text{otp}(a_\delta) = \omega_1 \and \delta =
\sup(a_\delta)$.

Let

$$
\align
T^* = \{\eta:&\eta \text{ is a finite sequence of ordinals, } \eta(2n) <
\lambda \text { and} \\
  &\eta(2n+1) < \lambda_n\}.
\endalign
$$
\mn
Let $\lambda_\eta = \lambda$ if $\ell g(\eta)$ is even and
$\lambda_\eta = \lambda_n$ if $\ell g(\eta) = 2n+1$ and let $\bold I_\eta$
be the non-stationary ideal on $\lambda_\eta$ for $\eta \in T^*$, so
$(T^*,\bar{\bold I})$ is well defined where $\bar{\bold I} := \langle \bold
I_\eta:\eta \in T^*\rangle$.

For $\eta \in T^*$, let 
$M_\eta$ be the $M^{**}$-closure of $\{\eta(\ell):\ell < \ell
g(\eta)\}$ so each $M_\eta$ is countable and $\eta \triangleleft \nu
\in T^* \Rightarrow M_\eta \subseteq M_\nu$ and for $\eta \in \text{
lim}(T^*) = \{\eta \in {}^\omega \lambda:\eta \restriction n \in T^*$
for every $n < \omega\}$ let 
$M_\eta = \cup\{M_{\eta \restriction n}:n < \omega\}$,
so it is enough to prove that $M_\eta \in {\Cal A}$ for some $\eta \in
\text{ lim}(T^*)$, more exactly $|M_\eta| \in {\Cal A}$ recall
$M_\eta \subseteq M^{**} \Leftrightarrow M_\eta \prec M^{**}$ as
$M^{**}$ has Skolem functions.  
Let $\bar M = \langle M_\eta:\eta \in T^*\rangle$ and
we can find a subtree $T \subseteq T^*$ such that
\mr
\item "{$\boxtimes$}"   $(T^*,\bold I) \le (T,\bold I)$ and for some
$\varepsilon^* < \omega_1$ we have $\eta \in \text{ lim}(T) \Rightarrow
\text{ otp}(M_\eta) = \varepsilon^*$ \nl
(recalling $(T^*,\bold I) \le (T,\bold I)$ means $T \subseteq
T^*,(\forall \eta \in T^*)(\forall \ell < \ell g(\eta))(\eta
\restriction \ell \in T^*),<> \in T^*$ and $(\forall \eta \in
T)(\{\alpha < \lambda_\eta:\eta \char 94 \langle \alpha\rangle \in
T^*\} \ne \emptyset$ mod $\bold I_\eta$, i.e. is stationary)). 
\ermn
Why?  As lim$(T^*) = \cup\{\bold B_\varepsilon:\varepsilon <
\omega_1\}$, see $\boxtimes_4$ below, and by $\boxtimes_1$ below each $\bold
B_\varepsilon$ is a Borel subset of lim$(T^*)$ and note that
$\boxtimes$ says the $(\exists \varepsilon < \omega_1)(\exists T)[(T^*,\bold I)
\le (T,\bold I) \cap \text{ lim}(T) \subseteq \bold B_\varepsilon)$.
The existence of such $\varepsilon$ is, e.g.,  
\cite[XI;3.5,p.546]{Sh:f}; the reader may ask to
justify the sets being Borel, so let $u_\eta$ be the universe of
$M_\eta$, a countable set of ordinals.

So we use
\mr
\item "{$\boxtimes_1$}"  for any $\varepsilon < \omega_1$ the set
$\bold B_\varepsilon = \{\eta \in \lim(T):\text{otp}(u_\eta) 
= \varepsilon\}$ is a Borel set.
\ermn
[Why?  Without loss of generality $u_\eta \ne \emptyset$ and let
$\langle \alpha_{\eta,n}:n < \omega\rangle$ enumerate the members of
$u_\eta$ and for $n_1,n_2 < \omega$ and $m_1,m_2 <\omega$ let 
$\bold B_{n_1,n_2,m_1,m_2} := \{\eta \in \lim(T^*):\alpha_{\eta
\restriction n_1,m_1} < \alpha_{\eta \restriction n_2,m_2}\}$.

Clearly
\mr
\item "{$\boxtimes_2$}"  $\bold B_{n_1,n_2,m_1,m_2}$ is an open subset of
lim$(T^*)$
\sn
\item "{$\boxtimes_3$}"  there is a $\Bbb L_{\omega_1,\omega}$ sentence
$\psi_\varepsilon$ in the vocabulary consisting of
$\{p_{n_1,n_2,m_1,m_2,\ell}:n_1,n_2,m_1,m_2 < \omega\}$
such that: the $p$'s are propositional variables 
(i.e. $0$-place predicates) and
if $\langle \alpha_{n,m}:n,m < \omega\rangle$ is a sequence
of ordinals and $p_{n_1,n_2,m_1,m_2}$ is assigned the truth value of
$\alpha_{n_1,m_1} < \alpha_{n_2,m_2}$ then $\gamma = \text{
otp}\{\alpha_{n,m}:n,m <\omega\}$ iff $\psi_\varepsilon$ is assigned the truth
value true
\sn
\item "{$\boxtimes_4$}"  lim$(T^*) = \cup\{\bold B_\varepsilon:
\varepsilon < \omega_1\}$.
\ermn
[Why?  As otp$(M_\eta \cap \lambda) < \|M_\eta\|^+ = \aleph_1$.
Together $\boxtimes$ should be clear.]

Note that for every $\eta \in T^*$ of length $2n+2$ we have $\eta
\trianglelefteq \nu \in T^* \Rightarrow \bold I_\nu$ is $\lambda^+_n$-complete.
As we can shrink $T$ further by \cite[XI,3.5,pg.346]{Sh:f} without loss
of generality
\mr
\item "{$\circledast$}"  for every $n < \omega$ and $\eta \in T \cap
{}^{2n+2}\lambda$ for some $\alpha = \alpha_\eta < \lambda_n$ we have:
if $\eta \triangleleft \nu \in \lim(T)$ then $\alpha_\eta =
\sup(\lambda_n \cap M_\nu)$.
\ermn
[Why?  As above applied to each $T' = \{\rho \in 
{}^{\omega >}\lambda:\eta \char 94 \rho \in T\}$.]

Let $\chi = (2^\lambda)^+$ and  
$N^*_\alpha \prec {\frak B} = ({\Cal H}(\chi),\in,<^*_\chi)$ for
$\alpha < \lambda$ be
increasing continuous, $\|N^*_\alpha\| = \mu,\alpha \subseteq
N^*_\alpha,\langle N^*_\beta:\beta \le \alpha \rangle \in N^*_{\alpha
+1}$ and $(T,\bold I,\bar M,\bar a,\bar f,\bar \lambda,\mu) \in
N^*_\alpha$, clearly possible and $E = \{\delta < \lambda:N^*_\delta
\cap \lambda = \delta\}$ is a club of $\lambda$, hence we can find
$\delta(*) \in S^* \cap E$, so $a_{\delta(*)}$ is well defined.  
Let $\bar N^* = \langle N^*_\alpha:\alpha < \lambda\rangle$.
Let
$C_{\delta(*)}$ be the closure of $a_{\delta(*)}$ as a subset of
$\delta(*)$ in the order topology and 
let $\langle \alpha_\varepsilon:\varepsilon < \omega_1
\rangle$ list $C_{\delta(*)}$ in increasing order, so is increasing continuous.

We define $N_\varepsilon$ by induction on $\varepsilon < \omega_1$ by:
\mr
\item "{$(*)_0$}"  $N_\varepsilon$ is the Skolem hull in ${\frak B}$
of
\nl
$\{\alpha_\zeta:\zeta < \varepsilon\} \cup \{\langle N_\xi:\xi < \zeta
\rangle,\bar N^* \restriction \zeta:\zeta < \varepsilon\} \cup 
\{(T,\bold I,\bar M,\bar a,\bar f,\bar \lambda,\mu)\}$.
\ermn
Let
\mr
\item "{$(*)_1$}"  $g_\varepsilon \in \dsize \prod_{n < \omega}
\lambda_n$ be defined by $g_\varepsilon(n) = \text{ sup}(N_\varepsilon
\cap \lambda_n)$.
\ermn
Clearly
\mr
\item "{$(*)_2$}"   $(a) \quad \langle N_\zeta:\zeta \le \varepsilon\rangle \in
N^*_{\delta(*)}$ and even $\in N_\xi$ for every $\xi \in [\varepsilon
+1,\omega_1)$
\sn
\item "{${{}}$}"  $(b) \quad C_{\delta(*)} \cap (\alpha_\varepsilon +1)$ and
$a_{\alpha_\varepsilon}$ belongs to $N_\xi$ for $\xi \in [\varepsilon
+1,\omega_1)$. 
\ermn
[Why?  For clause (a), $\langle N_\zeta:\zeta \le \varepsilon\rangle$
appear in the set whose Skolem Hull is $N_\xi$.  For clause (b)
because $\bar a \in N^*_{\delta(*)}$ and $\alpha \in a_{\delta(*)}
\Rightarrow a_\alpha = a_{\delta(*)} \cap \alpha$ and $C_{\delta(*)}
\cap (\alpha_\varepsilon +1) =$ the closure of $a_{\alpha_{\varepsilon
+1}} \cap (\alpha_\varepsilon +1)$.]
\mn
Let $e = \{\varepsilon < \omega_1:\varepsilon$ is a limit ordinal and
$N_\varepsilon \cap \omega_1 = \varepsilon\}$.  So
\mr
\item "{$(*)_3$}"  $(a) \quad e$ is a club of $\omega_1$,
\sn
\item "{${{}}$}"  $(b) \quad$ if $\varepsilon \in e$ then
sup$(N_\varepsilon \cap \lambda) = \alpha_\varepsilon =
N^*_{\alpha_\varepsilon} \cap \lambda,N_\varepsilon \subseteq
N^*_{\alpha_\varepsilon}$ and $\varepsilon < \zeta < \omega_1 \Rightarrow
N_\varepsilon \in N_\zeta$
\ermn
hence
\mr
\item "{$(*)_4$}"  if $\varepsilon +2 < \zeta \in e$ then
$g_\varepsilon,g_{\varepsilon +1} \in N_{\varepsilon +2} \prec N_\zeta$.
\ermn
Now $\bar f$ is increasing and cofinal in $\dsize \prod_{n < \omega} \lambda_n$
hence
\mr
\item "{$(*)_5$}"   if $\varepsilon < \zeta \in e$ then $g_\varepsilon
<_{J^{\text{bd}}_\omega} f_{\alpha_\zeta}$ and $f_{\alpha_\varepsilon}
<_{J^{\text{bd}}_\omega} g_\zeta$.
\ermn
Also easily
\mr
\item "{$(*)_6$}"  if $\varepsilon < \zeta \in e$ then $g_\varepsilon
< g_\zeta$.
\ermn
For $n < \omega,\varepsilon < \omega_1$ let $N_{\varepsilon,n+1}$ be
the Skolem hull inside ${\frak B}$ of $N_\varepsilon \cup \lambda_n$
and let $N_{\varepsilon,0} = N_\varepsilon$.  Easily
\mr
\item "{$(*)_7$}"  if $n \le m < \omega$ and $\varepsilon < \omega_1$ then
$g_\varepsilon(m) =\sup(N_{\varepsilon,n} \cap \lambda_m)$.
\ermn
Recall that $\varepsilon^*$ is the order type of $M_\eta \cap \lambda$
for every $\eta \in \text{ lim}(T)$.
\nl
Choose $\varepsilon \in \text{ acc}(e)$ such that $\varepsilon >
\varepsilon^*,\alpha_\varepsilon \in S_\zeta$ for some $\zeta \in
[\varepsilon,\omega_1)$ (possible by subclaim \scite{nr.4} particularly
clause $(*)_2(ii)$) and choose $\varepsilon_k \in
e \cap \varepsilon$ for $k < \omega$ such that $\varepsilon_k <
\varepsilon_{k+1} < \varepsilon = \cup\{\varepsilon_\ell:\ell < \omega\}$.
We also choose $n_k$ by induction on $k < \omega$ such that
\mr
\item "{$(*)_8$}"  $(a) \quad n_\ell < n_k < \omega$ for $\ell < k$
\sn
\item "{${{}}$}"   $(b) \quad g_{\varepsilon_{k+1}} 
\restriction [n_k,\omega) < f_{\alpha_\varepsilon} \restriction [n_k,\omega)$.
\ermn
[Why is this choice possible?  By $(*)_5$.]
\mn
Stipulate $n_{-1} = 0$.

Let $B_1 \in D_{1,\zeta}$ be such that $B_2 = \omega \backslash B_1
\in D_{2,\zeta}$, exists by clause $(*)_3(iv)$ of subclaim \scite{nr.4}.

Now we choose $\eta_n$ by induction on $n < \omega$ such that
\mr
\item "{$\boxdot$}"  $(a) \quad \eta_n \in T$ and $\ell g(\eta_n) = n$
\sn
\item "{${{}}$}"  $(b) \quad m < n \Rightarrow \eta_m \triangleleft
\eta_n$
\sn
\item "{${{}}$}"  $(c) \quad$ if $n \in [n_{k-1},n_k)$ then
$\eta_{2n},\eta_{2n+1} \in N_{\varepsilon_k,n}$
\sn
\item "{${{}}$}"  $(d) \quad$ if $n \in [n_{k-1},n_k)$ then
$\eta_{2n+1}(2n) = \text{ Min}\{\alpha < \lambda:\eta_{2n} \char 94 \langle
\alpha \rangle \in T$ and 
\nl

\hskip25pt $\alpha \ge \alpha_{\varepsilon_{k-1}}$ if $k>0\}$
\sn
\item "{${{}}$}"  $(e) \quad$ if $n \in [n_{k-1},n_k)$ and $n \in B_1$
then $\eta_{2n+2}(2n+1) = \text{ Min}\{\alpha < \lambda_n$:
\nl

\hskip25pt $\eta_{2n+1} \char 94 \langle \alpha \rangle \in T\}$
\sn
\item "{${{}}$}"  $(f) \quad$ if $n \in [n_{k-1},n_k)$ and $n \in B_2$
then $\eta_{2n+2}(2n+1) = \text{ Min}\{\alpha < \lambda_n$:
\nl

\hskip25pt $\eta_{2n+1} \char 94 \langle \alpha \rangle \in T$ and $\alpha >
f_{\alpha_\varepsilon}(n)\}$.
\ermn
No problem to carry the induction.
\mn
[Clearly if $\eta_n$ is well defined then $\eta_{n+1}(n)$ is well
defined (by clause (c) or (d) or (e) according to the case; hence
$\eta_{n+1} \in T \cap {}^{n+1}\lambda$ is well defined by why clause
(c) holds, i.e. assume $n \in [n_{k-1},n_k)$, why
$\eta_{2n},\eta_{2n+1} \in N_{\varepsilon_{k,n}}$?
\bn
\ub{Case 1}:  If $n=0$, then $\eta_{2n} = <> \in N_{\varepsilon_k,n}$
trivially.
\bn
\ub{Case 2}:  $\eta_{2n}$ is O.K. hence $\in N_{\varepsilon_{k,n}}$
and show $\eta_{2n+1} \in N_{\varepsilon_{k,n}}$.
\nl
[Why?  Because $N_{\varepsilon_k,n} \prec {\frak B}$, if $k=0$ as
$\eta_{2n+2}(2)$ is defined from $\eta_{2n}$ and $T$ both of which
belongs to $N_{\varepsilon_k,n}$.  If $k>0$ we have to check that also
$\alpha_{\varepsilon_{k-1}} \in N_{\varepsilon_k,n}$ which holds by $(*)_0$.
\bn
\ub{Case 3}:  $\eta_{2n+1}$ is O.K. so $\in N_{\varepsilon_k,n}$
and we have to show $\eta_{2n+2} \in N_{\varepsilon_k,n+1}$.

As $\eta_{2n+2}(n) < \lambda_n \subseteq N_{\varepsilon_k,n+1}$ this
should be clear.]  

Let $\eta = \cup\{\eta_n:n < \omega\}$.  Clearly $\eta \in \text{
lim}(T)$ hence $u =: |M_\eta| \in [\lambda]^{\aleph_0}$ and $M_\eta
\subseteq M^{**}$, hence it is enough to prove that $u \in {\Cal A}$.

Now
\mr
\item "{$\circledast_1$}"  sup$(u) \le \alpha_\varepsilon$
\nl
[Why?  As $\eta_n$ belongs to the Skolem hull of $N_\varepsilon \cup
\mu \subseteq N^*_{\alpha_\varepsilon}$ hence $M_{\eta_n}
\subseteq N_\varepsilon \subseteq N^*_{\alpha_\varepsilon}$ 
and $N^*_{\alpha_\varepsilon} \cap
\lambda = \alpha_\varepsilon$ as $\alpha_\varepsilon \in E$.]
\sn
\item "{$\circledast_2$}"  sup$(u) \ge \alpha_{\varepsilon_n}$, for
every $n < \omega$
\nl
[by clause (d) of $\boxdot$]
\sn
\item "{$\circledast_3$}"  sup$(u) = \alpha_\varepsilon$
\nl
[Why?  By $\circledast_1 + \circledast_2$]
\sn
\item "{$\circledast_4$}"  $\alpha_\varepsilon \in S_\zeta$ and $\zeta
\ge \varepsilon > \varepsilon^* = \text{otp}(u)$
\nl
[Why?  By the choice of $\varepsilon$]
\sn
\item "{$\circledast_5$}"  if $n \ge n_0,n > 0$ and $n \in B_1$ then $u
\cap \lambda_n \subseteq f_{\alpha_\varepsilon}(n)$
\nl
[Why?  By the choice of $\eta_{2n+2}(2n+1)$, i.e., let $k$ be such
that $n \in [n_{k-1},n_k)$, so $\eta_{2n+1} \in N_{\varepsilon_k,n}$
by clause (c) and by clause (e) of $\boxdot$ we have 
$\eta_{2n+2}(2n+1) \in \lambda_n \cap 
N_{\varepsilon_k,n}$ hence by $\otimes$ above, as $\eta \in \text{
lim}(T)$ we have $\alpha_{\eta \restriction (2n+2)} =
\alpha_{\eta_{2n+2}} = \sup(u \cap \lambda_n)$ and as $\bar M \in
N_{\varepsilon_k,n}$ we have $\alpha_{\eta_{2n+2}} \in
N_{\varepsilon_k,n}$ so sup$(u \cap \lambda_n) = \alpha_{\eta_{2n+2}}  
 < \sup(N_{\varepsilon_k,n} \cap \lambda_n)$ but the latter is equal to
sup$(N_{\varepsilon_k} \cap \lambda_n)$ by $(*)_7$ which is equal to 
$g_{\varepsilon_k}(n)$
which is $< f_{\alpha_\varepsilon}(n)$ by $(*)_8$, as required.]
\sn
\item "{$\circledast_6$}"  if $n \ge n_1$ and $n \in B_2$ then $u \cap
\lambda_n \nsubseteq f_{\alpha_\varepsilon}(n)$
\nl
[Why?  By the choice of $\eta_{2n+2}(2n+1)$.]
\ermn
So we are done.  \hfill$\square_{\scite{nr.6}}$
\enddemo
\bn
This (i.e., \scite{nr.5} + \scite{nr.6}) is 
enough for proving \scite{nr.1} in the case $\mu_* = \aleph_2$.  In
general we should replace \scite{nr.5} by the following claim.
\proclaim{\stag{nr.7} Claim}  The family ${\Cal A}$ does not reflect in any
uncountable $A \in [\lambda]^{< \mu_*}$.
\endproclaim
\bigskip

\demo{Proof}  Assume $A$ is a counterexample.

Trivially 
\mr
\item "{$\circledast_0$}"  $A$ is $M^*$-closed.
\ermn
For $a \in {\Cal A}$ let $(\delta(a),\varepsilon(a))$ be such that $a \in
{\Cal A}^{\varepsilon(a)}_{\delta(a)}$ hence $\delta(a) = \sup(a)$, otp$(a)
\le \varepsilon(a)$.  Let ${\Cal A}^- = {\Cal A} \cap
[A]^{\aleph_0}$ and let $\Gamma = \{\delta(a):a \in {\Cal A}^-\}$.
Of course, $\Gamma \ne \emptyset$.  Assume that $\delta_n \in \Gamma$
for $n < \omega$ so let $\delta_n = \delta(a_n)$ where $a_n \in {\Cal A}$ 
so necessarily $\delta_n \in S_{\varepsilon(a_n)}$.  As $A$ is uncountable
we can find a countable $b$ such that $a_n \subseteq b \subseteq A$ and
$\varepsilon(a_n) < \text{ otp}(b)$ for every $n < \omega$
and as ${\Cal A}^- \subseteq [A]^{\aleph_0}$ is stationary 
we can find $c$ such that $b
\subseteq c \in {\Cal A}^-$; so $\varepsilon(c) \ge \text{ otp}(c) \ge \text{
otp}(b) > \varepsilon(a_n) \and \delta_n \in S_{\varepsilon(a_n)} 
\and \delta(a_n) =
\delta_n \le \sup(a_n) \le \sup(c) = \delta(c)$ for each $n < \omega$.  So if
$\delta(a_n) = \delta_n = \delta(c),n < \omega$ necessarily $\varepsilon(a_n) 
= \varepsilon(c)$ contradiction so $\delta_n \ne \delta(c)$; hence
$\delta(c) > \delta(a_n)$ and, of course, $\delta(c) \in \Gamma$ so $n
< \omega \Rightarrow \delta_n < \delta(c) \in 
\Gamma$.  As $\delta_n$ for $n < \omega$ were
any members of $\Gamma$, clearly $\Gamma$ has no
last element, and let $\delta^* = \sup(\Gamma)$.  Similarly cf$(\delta^*)
= \aleph_0$ is impossible, so clearly cf$(\delta^*) > \aleph_0$ and
let $\theta = \text{ cf}(\delta^*)$ so $\theta \le |A| < \mu_*$ and
$\theta$ is a regular uncountable cardinal. \nl
As $a \in {\Cal A}^\varepsilon_\delta \Rightarrow \sup(a) = \delta$ and
${\Cal A}^- \subseteq [A]^{\aleph_0}$ is stationary 
clearly $A \subseteq \delta^* = \sup(A) = \sup(\Gamma)$.  
Let $\langle \delta_i:i < \theta \rangle$ 
be increasing continuous with limit $\delta^*$ and if
$\delta_i \in S_\varepsilon$ then we let $\varepsilon_i = \varepsilon$.

For $i < \theta$ let $\beta_i = \text{ Min}(A \backslash \delta_i)$,
so $\delta_i \le \beta_i < \delta^*,\beta_i \in A$ and $i < j < \theta
\Rightarrow \beta_i \le \beta_j$.
But $i < \theta \Rightarrow \beta_i < \delta^* \Rightarrow (\exists j)(i
< j < \theta \wedge \beta_i < \delta_j)$ so 
for some club $E_0$ of $\theta$ we
have $i < j \in E_0 \Rightarrow \beta_i < \delta_j \le \beta_j$; as we
can replace $\langle \delta_i:i < \theta \rangle$ by $\langle \delta_i:i
\in E_0 \rangle$ \wilog \, $\beta_i < \delta_{i+1}$ hence $\langle
\beta_i:i < \theta \rangle$ is strictly increasing. 
\nl
Let $A^- := \{\beta_i:i < \theta\}$ and let $H:[\theta]^{\aleph_0}
\rightarrow \theta$ be $H(b) = \text{ sup}\{i:\beta_i \in b\}$ and let
$J := \{R \subseteq \theta$: the family $\{b \in {\Cal A}^-:H(b) \in
R\} = \{b \in {\Cal A}^-:\sup(\{i < \theta:\beta_i \in b\}) \in
R\}$ is not a stationary subset of $[A^-]^{\aleph_0}\}$. 
\nl
Clearly
\mr
\item "{$\circledast_1$}"   $J$ is an $\aleph_1$-complete ideal on $\theta$
extending the non-stationary ideal and $\theta \notin J$ by the
definition of the ideal
\sn
\item "{$\circledast_2$}"   if $B \in J^+$ (i.e., $B \in {\Cal
P}(\theta) \backslash J$) \ub{then} $\{a \in {\Cal A}^-:H(a) \in B\}$
is a stationary subset of $[\theta]^{\aleph_0}$.
\ermn
By clause (D) of the assumption of \scite{nr.1}, for some stationary
$R_1 \in J^+$ and $n_i < \omega$ for $i \in R_1$ we have 
\mr
\item "{$\circledast_3$}"   if $i < j$ are from $R_1$
and $n \ge n_i,n_j$ (but $n < \omega$) then $f_{\beta_i}(n) <
f_{\beta_j}(n)$.
\ermn
Recall that 
\mr
\item "{$\circledast_4$}"  $i < j \in R_1 \Rightarrow \beta_i <
\delta_j$. 
\ermn
Now if $i \in R_1$, let $j(i) = \text{ Min}(R_1 \backslash (i+1))$, so
$f_{\delta_i} \le_{J^{\text{bd}}_\omega} f_{\beta_i} <_{J^{\text{bd}}_\omega}
f_{\delta_{\bold j(i)}}$ hence for some $m_i < \omega$ we have $n \in
[m_i,\omega) \Rightarrow f_{\delta_i}(n) \le f_{\beta_i}(n) <
f_{\delta_{\bold j(i)}}(n)$.  Clearly for some $n(*)$ satisfying
$\lambda_{n(*)} > \theta$ and $R_2 \subseteq R_1$ from $J^+$ we have 
$i \in R_2 \Rightarrow n_i,m_i \le n(*)$, so
\mr
\item "{$\circledast_5$}"  for $i<j$ in $R_2$ we have
{\roster
\itemitem{ $(\alpha)$ }  $f_{\delta_i} 
\restriction [n(*),\omega) \le f_{\beta_i}[n(*),\omega)$
\sn
\itemitem{ $(\beta)$ }  $f_{\beta_i} \restriction [n(*),\omega) <
f_{\delta_j} \restriction [n(*),\omega)$.
\endroster}
\ermn
Let $f^* \in \dsize \prod_{n < \omega} \lambda_n$ be defined by
$f^*(n) = \cup \{f_{\delta_i}(n):i \in R_2\}$ if $n \ge n(*)$ and zero
otherwise.  Clearly $f^*(n) \le \sup(A \cap \lambda_n)$ for $n <
\omega$.

Let ${\Cal A}' = \{a \in {\Cal A}^-:(\forall i < \theta)(i \in a
\equiv \beta_i \in a \equiv \delta_i \in \beta_i)
\sup\{i \in R_2:\beta_i \in a\} =
\sup\{i:\beta_i \in a\} = \sup(a \cap \theta) \in R_2$ and 
sup$(A \cap \lambda_n) > f^*(n)
\Rightarrow a \cap \lambda_n \nsubseteq f^*(n)\}$.  As $R_2 \in J^+$ 
clearly ${\Cal A}'$ is a stationary subset of $[A]^{\aleph_0}$. 
\nl
Let $R_3 = \{i \in R_2:i = \sup(i \cap R_2)\}$ so $R_3 \subseteq
R_2,R_2 \backslash R_3$ is a non-stationary subset of $\theta$ (hence
belongs to $J$) and $a \in {\Cal A}' \Rightarrow \sup(a) 
\in \{\delta_i:i \in R_3\}$. \nl
Let

$$
\align
{\Cal A}^* = \biggl\{ a \in [A]^{\aleph_0}:&(a) \quad \beta_{\text{min}(R_2)}
\in a \text{ and } a \text{ is } M^*\text{-closed} \\
  &(b) \quad \text{if } i \in R_2 \and j =
\text{ Min}(R_2 \backslash (i+1)) \text{ \ub{then} } 
[a \nsubseteq \delta_i \Rightarrow a \nsubseteq \delta_j] \text{ and} \\
 &\qquad \,n \in [n(*),\omega) \and a \cap \lambda_n \nsubseteq
f_{\delta_i}(n) \Rightarrow a \cap 
\lambda_n \backslash f_{\delta_j}(n) \ne \emptyset \\
  &(c) \quad \text{if } i < \theta \and n \in 
[n(*),\omega) \text{ then } (\exists \gamma)(\beta_i \le \gamma \in a)
\equiv \\
  &\qquad \, (\exists j)(i < j < \theta
\and \beta_j \in a) \equiv (\exists \gamma)(f_{\beta_i}(n) \le \gamma
\in a \cap f^*(n)) \text{ and} \\
  &(d) \quad \text{if } A \cap \lambda_n \nsubseteq f^*(n) \text{ then
} a \cap \lambda_n \nsubseteq f^*(n) \\
  &\qquad \, \text{ but } (\forall \gamma \in a)(\gamma +1 \in a) \\
  &\qquad \, \text{hence sup } (a \cap \lambda_n) > f^*(n) \biggr\}.
\endalign
$$
\mn
Clearly ${\Cal A}^*$ is a club of 
$[A]^{\aleph_0}$ (recall that $A$ is $M^*$-closed).  But if $a
\in {\Cal A}^* \cap {\Cal A}'$, then for some limit ordinal $i \in R_3
\subseteq  \theta$ we have $a \subseteq \sup(a) = \delta_i$ and 
$n \in [n(*),\omega) \Rightarrow \sup(a \cap f^*(n)) = \sup(a \cap \cup
\{f_{\delta_j}(n):j \in R_2\})$.
\nl
Let 

$$
B_1 = \{n:n(*) \le n < \omega \text{ and } A \cap \lambda_n \subseteq
f^*(n) = \sup(A \cap \lambda_n)\}.
$$

$$
B_2 = \{n:n(*) \le n < \omega \text{ and } f^*(n) < \sup(A \cap
\lambda_n)\}.
$$
\mn
Clearly $B_1,B_2$ are disjoint with union $[n(*),\omega)$ 
recalling $\alpha \in A \Rightarrow \alpha + 1
\in A$ by $\circledast_0$.

By the definition of ${\Cal A}'$, for every $a \in{\Cal A}' \cap {\Cal
A}^*$, we have
\mr
\item "{$\circledast_6$}"  $n \in B_2 \Rightarrow 
{ \text{\rm Ch\/}}_a(n) \ge f^*(n) > f_{\delta(a)}(n)$
\sn
\item "{$\circledast_7$}"  $n \in B_1 \Rightarrow 
{ \text{\rm Ch\/}}_a(n) =
\cup\{f_{\beta_\varepsilon}(n):\varepsilon \in R_2 \cap \delta(a)\}
\le f_{\delta(a)}(n)$.
\ermn
But this contradicts the observation below.
\enddemo
\bigskip

\demo{\stag{nr.8} Observation}  If $B \subseteq \omega$, \ub{then} for
some $\varepsilon < \omega_1$ we have:

if $a \in {\Cal A}$ is $M^*$-closed and 
$\{n < \omega:\sup(a \cap \lambda_n) \le
f_{\sup(a)}(n)\} = B \text{ mod } J^{\text{bd}}_\omega$, \ub{then} otp$(a) <
\varepsilon$.
\enddemo
\bigskip

\demo{Proof}  Read the definition of ${\Cal A}$ 
(and ${\Cal A}^\varepsilon,{\Cal A}^\varepsilon_\delta$) and subclaim
\scite{nr.4} particularly $(*)_3$.
\nl
${{}}$  \hfill$\square_{\scite{nr.8}},
\square_{\scite{nr.7}},\square_{\scite{nr.1}}$
\enddemo
\bigskip

\remark{Remark}  Clearly 
\scite{nr.8} shows that we have much freeness in the choice of
${\Cal A}^\varepsilon_\delta$'s.
\endremark
\bn
We can get somewhat more, as in \cite{Sh:e}
\proclaim{\stag{nr.10} Claim}  In Claim \scite{nr.1} we can add to the
conclusion
\mr
\item "{$(*)$}"  ${\Cal A}$ satisfies the diamond,
i.e. $\diamondsuit_{\Cal A}$.
\endroster
\endproclaim
\bigskip

\demo{Proof}  In \scite{nr.4} we can add
\mr
\item "{$(*)_5$}"  $\{2n+1:n < \omega\} = \emptyset$ mod
$D_{\ell,\varepsilon}$ for $\ell < 2,\varepsilon < \omega_1$. 
\ermn
This is easy:  replace $D_{\ell,\varepsilon}$ by
$D'_{\ell,\varepsilon} = \{A \subseteq \omega:\{n:2n \in A\} \in
D_{\ell,\varepsilon}\}$.  We can fix a countable vocabulary $\tau$ and
for $\zeta < \omega_1$ choose a function $F_\zeta$ from 
${\Cal P}(\omega)$ onto $\{N:N$ is a $\tau$-model with universe $\zeta\}$ such
that $F_\zeta(A) = F_\zeta(B)$ if $A=B$ mod finite. 
\bn
\ub{Case 1}:  $\mu > 2^{\aleph_0}$.

Lastly, for $a \in {\Cal A}$ let $\delta_a,\varepsilon_a$ be such that $a
\in {\Cal A}^{\varepsilon_a}_{\delta_a}$ and let $A_a = \{n:\sup(a \cap
\lambda_{2n+1}) < f_{\delta_a}(2n)\}$, and let $N_a$ be the $\tau$-model with
universe $a$ such that the one-to-one order preserving function from
$\zeta$ onto $a$ is an isomorphism from $F_\zeta(N)$ onto $N$.
Note that in the proof of ``${\Cal A} \subseteq [\lambda]^{\aleph_0}$
is stationary", i.e. of \scite{nr.6}, given a $\tau$-model $M$ with
universe $\lambda$ \wilog \, $\lambda_0 > 2^{\aleph_0}$ and so can
demand that the isomorphism type of $M_\eta$ is the same for all $\eta
\in \text{ lim}(T)$ and, of course, $M \in M_\eta$.  Hence the
isomorphic type of $M \restriction u_\eta$ is the same for all $\eta
\in \text{ lim}(T)$ where $u_\eta$ is the universe of  $M_\eta$.  Now
in the choice for $B_1$ we can add the demand
$F_{\varepsilon^*}(\{n:2n+1 \in B_1\})$ is isomorphic to $M
\restriction u_\eta$ for every $\eta \in \text{ lim}(T)$.
\nl
Now check.   
\bn
\ub{Case 2}:  $\mu \le 2^{\aleph_0}$.

Similarly letting $\{2n +1:n < \omega\}$ be the disjoint union of
$\langle B^*_n:n <\omega\rangle$, each $B_n$ infinite.  We use
$A_a \cap B^*_n$ to code model with universe $\subseteq \zeta$ for
some $\zeta < \omega_1$, by a function $\bold F_n$.  We then
let $N_a$ be the model with universe $a$ sucht hat the order
preserving function from $a$ onto a countable ordinal $\zeta$ is an
isomorphism from $N_a$ onto $\cup\{\bold F_n(A_a \cap B^*_n):n <
\omega\}$ when the union is a $\tau$-model with universe $\zeta$.

Now we cannot demand them all $M_\eta,\eta \in \text{ lim}(T)$ has the
same isomorphism type but only the same order type.  The rest should
be clear.  \hfill$\square_{\scite{nr.10}}$
\enddemo
\bn
We can also generalize
\proclaim{\stag{nr.11} Claim}  We can weaken the assumption of
\scite{nr.1} as follows
\mr
\item "{$(a)$}"  $\lambda = \,\text{\rm cf}(\lambda) > \mu$ instead
$\lambda = \mu^+$ (still necessarily $\mu_* \le \mu$)
\sn
\item "{$(b)$}"   replace $J^{\text{bd}}_\omega$ by an ideal $J$ on $\omega$
containing the finite subsets, $\lambda_n = \,\text{\rm cf}(\lambda_n) >
\aleph_1,\mu = \,\text{\rm lim}_J \langle \lambda_n:n < \omega \rangle$ but
not necessarily $n < \omega \Rightarrow \lambda_n < \lambda_{n+1}$ and
add ${\Cal P}(\omega)/J$ is infinite (hence uncountable).
\endroster
\endproclaim
\bigskip

\demo{Proof}  In \scite{nr.4} in $(*)_3$ we choose $\langle
A_\varepsilon:\varepsilon < \omega_1 \rangle$, a sequence of subsets
of $\omega$ such that $\langle A_\varepsilon/J:\varepsilon < \omega_1
\rangle$ are pairwise distinct.  This implies some changes and waiving
$\lambda_n < \lambda_{n+1}$ requires some changes in \scite{nr.6}, 
in particular for each $n$ using $\langle \bold B_\alpha:\alpha \in
S^{\lambda_n}_{\aleph_0}\rangle$ with $\bold B_\delta = \{\eta \in
\text{ lim}(T^*):a \cap \lambda_n \subseteq \alpha\}$ and the
partition theorem \cite[XI,3.7,pg.549]{Sh:f}.
\nl
${{}}$    \hfill$\square_{\scite{nr.11}}$ 
\enddemo
\newpage


% PRIVATE excursion starts here
% back from PRIVATE part number 1/1

\nocite{ignore-this-bibtex-warning} 
%% you may want to move the following lines up a bit
\newpage
    
REFERENCES.  
\bibliographystyle{lit-plain}
\bibliography{lista,listb,listx,listf,liste}

\enddocument %%

   




794.bbl
\def\germ{\frak} \def\scr{\cal} \ifx\documentclass\undefinedcs
  \def\bf{\fam\bffam\tenbf}\def\rm{\fam0\tenrm}\fi % f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{=}{\B} % doublef**k-amstex!!
  \defaultdefine{mathfrak}{\frak} \defaultdefine{mathbb}{\bf}
  \defaultdefine{mathcal}{\cal}
  \defaultdefine{beth}{BETH}\defaultdefine{cal}{\bf} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt>
  \def\implies{\Rightarrow}\def\mathfrak{\bf}\def\germ{\frak} \def\scr{\cal}
  \ifx\documentclass\undefinedcs
  \def\bf{\fam\bffam\tenbf}\def\rm{\fam0\tenrm}\fi % f**k-amstex!
  \def\defaultdefine#1#2{\expandafter\ifx\csname#1\endcsname\relax
  \expandafter\def\csname#1\endcsname{#2}\fi} \defaultdefine{Bbb}{\bf}
  \defaultdefine{frak}{\bf} \defaultdefine{=}{\B} % doublef**k-amstex!!
  \defaultdefine{mathfrak}{\frak} \defaultdefine{mathbb}{\bf}
  \defaultdefine{mathcal}{\cal}
  \defaultdefine{beth}{BETH}\defaultdefine{cal}{\bf} \def\bbfI{{\Bbb I}}
  \def\mbox{\hbox} \def\text{\hbox} \def\om{\omega} \def\Cal#1{{\bf #1}}
  \def\pcf{pcf} \defaultdefine{cf}{cf} \defaultdefine{reals}{{\Bbb R}}
  \defaultdefine{real}{{\Bbb R}} \def\restriction{{|}} \def\club{CLUB}
  \def\w{\omega} \def\exist{\exists} \def\se{{\germ se}} \def\bb{{\bf b}}
  \def\equivalence{\equiv} \let\lt< \let\gt>
\begin{thebibliography}{RuSh 117}
\makeatletter \renewcommand{\@biblabel}[1]{[#1]} \makeatother
\def\eprintfootnotetext{References of the form {\tt math.XX/$\cdots$}
 refer to the {\tt xxx.lanl.gov} archive}
\ifx\documentstyle\undefinedcontrolsequence
   \def\anyfootnote{\footnote{*}}
   \else\def\anyfootnote{\footnote}\fi
\def\eprintfn{\ifEprint\anyfootnote{\eprintfootnotetext}\fi\Eprintfalse }
\newif\ifEprint  \Eprinttrue

\bibitem[FoTo]{FoTo}Matthew Foreman and Stevo Todor\v{c}evi\'{c}.
\newblock {A new L\"owenheim-Skolem theorem}.
\newblock {\em Transactions of the American Mathematical Society}, {\bf
  357}:1693--1715, 2005.

\bibitem[RuSh 117]{RuSh:117}Matatyahu Rubin and Saharon Shelah.
\newblock {Combinatorial problems on trees: partitions, $\Delta$-systems and
  large free subtrees}.
\newblock {\em {Annals of Pure and Applied Logic}}, {\bf 33}:43--81, 1987.

\bibitem[Sh:E3]{Sh:E3}S.~Shelah.
\newblock {On some problems in general topology}.
\newblock In {\em Set Theory, Boise ID, 1992--1994}, volume 192 of {\em
  Conteporary Mathematics}, pages 91--101.

\bibitem[Sh:E12]{Sh:E12}Saharon Shelah.
\newblock {Analytical Guide and Corrections to \cite{Sh:g}.}
\newblock math.LO/9906022.

\bibitem[Sh:e]{Sh:e}Saharon Shelah.
\newblock {\em {Non--structure theory}}, accepted.
\newblock {Oxford University Press}.

\bibitem[Sh 420]{Sh:420}Saharon Shelah.
\newblock {Advances in Cardinal Arithmetic}.
\newblock In {\em {Finite and Infinite Combinatorics in Sets and Logic}}, pages
  355--383. Kluwer Academic Publishers, 1993.
\newblock N.W. Sauer et al (eds.). 0708.1979.

\bibitem[Sh 410]{Sh:410}Saharon Shelah.
\newblock {More on Cardinal Arithmetic}.
\newblock {\em {Archive for Mathematical Logic}}, {\bf 32}:399--428, 1993.
\newblock math.LO/0406550.

\bibitem[Sh:g]{Sh:g}Saharon Shelah.
\newblock {\em {Cardinal Arithmetic}}, volume~29 of {\em {Oxford Logic
  Guides}}.
\newblock {Oxford University Press}, 1994.

\bibitem[Sh 430]{Sh:430}Saharon Shelah.
\newblock {Further cardinal arithmetic}.
\newblock {\em {Israel Journal of Mathematics}}, {\bf 95}:61--114, 1996.
\newblock math.LO/9610226.

\bibitem[Sh:f]{Sh:f}Saharon Shelah.
\newblock {\em {Proper and improper forcing}}.
\newblock {Perspectives in Mathematical Logic}. {Springer}, 1998.

\bibitem[Sh 755]{Sh:755}Saharon Shelah.
\newblock {Weak Diamond}.
\newblock {\em Mathematica Japonica}, {\bf 55}:531--538, 2002.
\newblock math.LO/0107207.

\bibitem[Sh 775]{Sh:775}Saharon Shelah.
\newblock {Middle Diamond}.
\newblock {\em Archive for Mathematical Logic}, {\bf 44}:527--560, 2005.
\newblock math.LO/0212249.

\bibitem[Ve92a]{Ve92a}Boban Veli{\v c}kovi\'c.
\newblock {Forcing axioms and stationary sets}.
\newblock {\em Advances in Mathematics}, {\bf 94}:256--284, 1992.

\end{thebibliography}
