\documentstyle[12pt,fleqn,leqno]{article} % %--------------------------------------------------------------------------- % coding on: 01.02.96 % extended/modified on : 17.03.96/26.03.96/28.03.96/03.04.96/04.04.96 % 05.04.96,26.04.96, 29.04.96, 23.05.96, 24.05.96 % 97.05.05: abstract added. %--------------------------------------------------------------------------- \def\lastrevision{April 29, 1996, revised on May 24, 1996} \title{{\LARGE\bf More set-theory\protect\\ {\it around}\protect\\ the weak Freese-Nation property}\\} \author{Saka\'e Fuchino$^{0}$, Lajos Soukup$^{1}$} \date{{\large\lastrevision}\\[2\baselineskip] \fbox{\normalsize\it\phantom{$A^A_A$}to appear in `Fundamenta Mathematicae'% \phantom{$A^A_A$}}} %--------------------------------------------------------------------------- \typeout{ \jobname.tex} \typeout{.......... (c) Sakae Fuchino, Lajos Soukup} \typeout{.......... \lastrevision} %-------- %Switches %-------- \newif\ifwrong \newif\iftesting % if set label names are printed in the margin. %\testingtrue %%% \newif\ifcommented % if set also comments which are to be eliminated in the final % version will be printed. %\commentedtrue %%% %----- %Tools %----- \let\nc\newcommand \let\rnc\renewcommand \let\nth\newtheorem \let\nenv\newenvironment % \iftesting \typeout{ } \typeout{* Compiling for test print. Label names are printed in the margin.} \typeout{* ............ to suppress this, turn off the switch: ''testing''.} \typeout{ } \let\Label\label% \def\label#1{\mbox{}\marginpar{{\tiny #1}}\Label{#1}}% \fi \ifcommented \typeout{* Some extra remarks, proofs, comments etc. which are not going to} \typeout{* be included in the final version, are also printed. ............} \typeout{* .......... to suppress this, turn off the switch: ''commented''.} \fi \nenv{comments}{\ifcommented\begin{footnotesize}}% {\ifcommented\end{footnotesize}\fi} % comments put if 'commented' is set % comments are to be given by \begin{comments} ... \fi\end{comments} % at the moment I cannot create a better structure without extra outer '\fi' \nc{\?}{\iftesting% \marginpar{$\emptyset\emptyset\emptyset$}% \fi \typeout{Something is still incomplete here.} \errmessage{................................}} % %----------- %Page format %----------- %\setlength{\voffset}{0cm} \ifx\kanjiskip\otankonasu\setlength{\topmargin}{-1.6cm}% if not jlatex \else\setlength{\topmargin}{-1.2cm}\fi \setlength{\oddsidemargin}{0.6cm} \setlength{\textwidth}{14.8cm} \setlength{\textheight}{23cm} \setlength{\footskip}{1.4cm} \rnc{\baselinestretch}{1.17} % \nth{Thm}{{\bf Theorem}}%[section] \nth{Cor}[Thm]{{\bf Corollary}} \nth{Lemma}[Thm]{{\bf Lemma}} \nth{Prop}[Thm]{{\bf Proposition}} \nth{Claim}{{\bf Claim}}[Thm] \nth{Subclaim}{{\bf Subclaim}}[Claim] \nth{Problem}{{\bf Problem}} \nth{Ex}[Thm]{{\bf Example}} \nth{Fact}[Thm]{{\bf Fact}} % \nc{\Thmof}[1]{{Theorem \ref{#1}}} \nc{\Corof}[1]{{Corollary \ref{#1}}} \nc{\Propof}[1]{{Proposition \ref{#1}}} \nc{\Lemmaof}[1]{{Lemma \ref{#1}}} \nc{\Factof}[1]{{Fact \ref{#1}}} \nc{\Claimof}[1]{{Claim \ref{#1}}} \nc{\Exof}[1]{{Example \ref{#1}}} \nc{\Problemof}[1]{{Problem \ref{#1}}} % \nc{\Thmabove}{{Theorem \number\theThm}} \nc{\Corabove}{{Corollary \number\theThm}} \nc{\Propabove}{{Proposition \number\theThm}} \nc{\Lemmaabove}{{Lemma \number\theThm}} \nc{\Factabove}{{Fact \number\theThm}} \nc{\Claimabove}{{Claim \number\theClaim}} \nc{\Exabove}{{Example \number\theThm}} % \nc{\prf}{{\bf Proof\ \ }} \nc{\prfof}[1]{{\bf Proof of #1\ \ }} \nc{\prfofClaim}{\raisebox{-.4ex}{\Large $\vdash$\ \ }} % \newsavebox{\qedbox}\sbox{\qedbox}{% QED symbol (by U.Fuchs modified by S.F.) {\unitlength=0.034ex \begin{picture}(40,60) \put(0,0){\framebox(30,44)[cc]{}} \put(30,-7){\rule{7\unitlength}{44\unitlength}} \put(10,-7){\rule{27\unitlength}{7\unitlength}} \end{picture}}} \nc{\qed}{\nolinebreak\mbox{}\nolinebreak\hfill% \ifvmode\mbox{}\hfill\fi\usebox{\qedbox}} \nc{\qedskip}{\medskip\smallskip} \nc{\smallqed}% {\mbox{}\smallskip\hfill\raisebox{-.4ex}{\Large $\dashv$}\\} \nc{\qedof}[1]% {\nopagebreak\mbox{}\nolinebreak\hspace*{\fill}{\usebox{\qedbox}{~(#1)}}} \nc{\Qedof}[1]% {\nopagebreak\mbox{}\nolinebreak\hspace*{\fill}{\usebox{\qedbox}% {~(#1~\number\theThm)}}} \nc{\qedofThm}{\Qedof{Theorem}} \nc{\qedofCor}{\Qedof{Corollary}} \nc{\qedofProp}{\Qedof{Proposition}} \nc{\qedofLemma}{\Qedof{Lemma}} \nc{\qedofFact}{\Qedof{Fact}} % \nc{\qedofClaim}% {\nopagebreak\mbox{}\nolinebreak\hfill\raisebox{-.4ex}{\Large $\dashv$ }\nolinebreak% \mbox{~(Claim~\number\theClaim)}} % \nc{\qedofSubclaim}% {\mbox{}\hfill\raisebox{-.4ex}{\Large $\dashv$ }\nolinebreak% \mbox{~(Subclaim~\number\theSubclaim)}} % \nc{\noindentafterqed}{\bigskip\\} \nc{\indentafterqed}{\bigskip\\\hspace*{\parindent}} % \nc{\xitem}[1]{\item[$(\mbox{\it #1})$]} \nc{\xitemof}[1]{{$(\mbox{\it #1})$}} \def\assert#1{\makebox[5ex][l]{\rm #1)}\ignorespaces} \def\lassert#1{\llap{\makebox[5ex][l]{\rm #1)}}\ignorespaces} \def\assertof#1{{\rm #1)}} % \nenv{assertion}[1]{\begin{trivlist} \newbox\assertbox \dimen255=\textwidth \setbox\assertbox=\hbox{\hspace*{\parindent}{#1}\hspace{\labelsep}} \advance\dimen255 by -1\wd\assertbox \advance\dimen255 by -1ex \item[]\unhbox\assertbox\hfill \begin{minipage}[t]{\dimen255}}% {\end{minipage}\end{trivlist}} % \nenv{subassertion}[1]{\begin{trivlist} \newbox\assertbox \dimen255=\textwidth \setbox\assertbox=\hbox{\hspace*{2\parindent}{#1}\hspace{\labelsep}} \advance\dimen255 by -1\wd\assertbox \advance\dimen255 by -1ex \item[]\unhbox\assertbox\hfill \begin{minipage}[t]{\dimen255}}% {\end{minipage}\end{trivlist}} % \nc{\implies}{$\,\Rightarrow\,$} \nc{\equivto}{$\,\Leftrightarrow\,$} % \nenv{formula}[1]{\\[\abovedisplayskip]${#1}$\hfill$}% {$\hfill\mbox{}\\[\belowdisplayskip]} % \nc{\restr}% {\mathop{\hspace{0.01ex}|\hspace*{-0.02ex}{\grave{}}\hspace{0.4ex}}} \nc{\upper}{\mathop{\hspace{0.1ex}\uparrow\hspace{0.2ex}}} \nc{\subsetnoneq}% {\mathrel{\raisebox{-0.8ex}{$\stackrel{\subset}{\scriptstyle\,\not=\,}$}}} \nc{\Linf}[1]{{\cal L}_{\infty{#1}}} \nc{\powersetof}[1]{{\cal P}(#1)} \nc{\psof}[1]{{\cal P}(#1)} \nc{\cardof}[1]{{\mathopen{\mid}{#1}\mathclose{\mid}}} \nc{\generatedby}[1]{{\langle#1\rangle}} \nc{\devides}{^{^{\,}}\mid^{^{\,}}} \nc{\setof}[2]{\{\,#1\,:\,#2\,\}} \nc{\smallsetof}[1]{\{\,#1\,\}} \nc{\orderedsetof}[2]{\langle\,#1\,:\,#2\,\rangle} \nc{\tuple}[1]{\langle\,#1\,\rangle} \nc{\continuum}{2^{\aleph_0}} \nc{\reals}{\mbox{$\rm I_{\!\!}R$}} \nc{\rationals}{% \rlap{\hbox{\hspace*{0.35ex}\raise0.35ex\hbox{$\scriptstyle\wr$}}}% \hbox{\rm Q}} \nc{\integers}{Z\hspace*{-1.12ex}Z} \nc{\unitint}{\mbox{$\rm I_{\!\!}I$}} \nc{\lvalue}{\lbrack\!\lbrack} \nc{\rvalue}{\rbrack\!\rbrack} \nc{\Bvalueof}[2]{\lvalue\,#1\,\rvalue^{(#2)}} \nc{\forces}[2]{\,\|\hspace{-.35ex}\mbox{\sf--}_{\,#1\,}% \mbox{\rm``}\,#2\,\mbox{\rm''}} \let\Models\models \def\models#1{\Models\mbox{\rm``}\,#1\,\mbox{\rm''}} \nc{\xmbox}[1]{ ${\rm #1}$ } % \nc{\st}{such that} \nc{\wolog}{without loss of generality} \nc{\Wolog}{Without loss of generality} \nc{\wrt}{with respect to} \nc{\Tfae}{The following are equivalent} \nc{\tfae}{the following are equivalent} % \nc{\Ba}{Boolean algebra} \nc{\Bas}{Boolean algebras} \nc{\cBa}{complete Boolean algebra} \nc{\cBas}{complete Boolean algebras} \nc{\po}{partial ordering} \nc{\pos}{partial orderings} \nc{\Pos}{Partial orderings} % \nc{\Fr}{{\rm Fr}\,} \nc{\cof}{\mathop{\rm cof}} \nc{\cf}{\mathop{\rm cf}} \nc{\pcf}{\mathop{\rm pcf}} \nc{\Sub}{{\rm Sub}} \nc{\pr}{{\rm pr}} \nc{\proj}{{\rm proj}} \nc{\wproj}{{\rm wproj}} \nc{\rc}{{\rm rc}} \nc{\cm}{{cm}} \nc{\reg}{{\rm reg}} \nc{\Card}{{\rm Card}} \nc{\Ord}{{\rm Ord}} \nc{\Fn}{{\rm Fn}} \nc{\dom}{{\rm dom}} \nc{\range}{{\rm range}} \nc{\Lim}{{\rm Lim}} \nc{\otp}{{\rm otp}} % \nc{\cohenalg}[1]% {\mbox{$\,\raisebox{0.05ex}{\small$\wr$}\!\!_{_{\!\!}}\mbox{\rm C}_{#1}$}} \nc{\cantorsp}[1]{\mbox{}^{#1^{\mbox{}\!}}2} \nc{\fnsp}[2]{\mbox{}^{#1^{\mbox{}\!}}#2} \nc{\mapping}[3]{#1:#2\rightarrow #3} \nc{\pfeil}{\mathrel{{\rightarrow\,}\llap{$\rightarrow$}}} % \nc{\ccc}{{\mbox{\rm ccc}}} \nc{\ZFC}{{\rm ZFC}} \nc{\CH}{{\rm CH}} \nc{\GCH}{{\rm GCH}} \nc{\MA}{{\rm MA}} \nc{\MM}{{\rm MM}} \nc{\MAp}{${\rm MA}^+(\sigma${\it-closed\/}$)$} \nc{\stick}% {\mbox{{\hspace{0.4ex}$\mbox{\raisebox{1ex}{$\bullet$}}% \hspace*{-0.94ex}|$\hspace{0.6ex}}}} % \nc{\Subrc}[1]{\Sub_\rc^{\aleph_0}(#1)} \nc{\dpr}{\overline{\pr}} % \nc{\calA}{{\cal A}} \nc{\calB}{{\cal B}} \nc{\calC}{{\cal C}} \nc{\calD}{{\cal D}} \nc{\calF}{{\cal F}} \nc{\calG}{{\cal G}} \nc{\calH}{{\cal H}} \nc{\calM}{{\cal M}} \nc{\calN}{{\cal N}} \nc{\calO}{{\cal O}} \nc{\calP}{{\cal P}} \nc{\calS}{{\cal S}} \nc{\calT}{{\cal T}} % \nc{\dota}{{\dot{a}}} \nc{\dotb}{{\dot{b}}} \nc{\dotc}{{\dot{c}}} \nc{\dotf}{{\dot{f}}} \nc{\dotg}{{\dot{g}}} \nc{\dotm}{{\dot{m}}} \nc{\dotr}{{\dot{r}}} \nc{\dotx}{{\dot{x}}} \nc{\doty}{{\dot{y}}} \nc{\dotB}{{\dot{B}}} \nc{\dotP}{{\dot{P}}} \nc{\dotQ}{{\dot{Q}}} \nc{\dotR}{{\dot{R}}} \nc{\dotS}{{\dot{S}}} \nc{\dotU}{{\dot{U}}} \nc{\dotX}{{\dot{X}}} \nc{\dotY}{{\dot{Y}}} % \nc{\undercirc}[1]{% \setbox255=\hbox{$#1$} \dimen255=0.9ex \advance\dimen255 by \dp255 \rlap{\hbox to\wd255{\hss{\lower\dimen255\hbox{$\scriptstyle\circ$}}\hss}} \box255 } \nc{\sundercirc}[1]{% \setbox255=\hbox{$\scriptstyle#1$} \dimen255=0.8ex \advance\dimen255 by \dp255 \rlap{\hbox to\wd255{\hss{\lower\dimen255 \hbox{$\scriptscriptstyle\circ$}}\hss}} \box255 } \nc{\multiundercirc}[1]{% \mathchoice{\undercirc{#1}}{\undercirc{#1}}{\sundercirc{#1}}{\sundercirc{#1}}} \nc{\acirc}{\multiundercirc{a}} % \nc{\Dag}{^\dagger} % \nc{\deq}{\mathrel{\sqsubseteq}} \nc{\Deq}{\mathrel{{\sqsubseteq}\hspace{-1.5ex}{\sqsubseteq}}} \nc{\notDeq}{\mathrel{{\sqsubseteq}\hspace{-1.5ex}{\not\sqsubseteq}}} % \nc{\kappagame}[2]{\calG^{#1}(#2)}% \nc{\kappalambdagame}[3]{\calG^{#1}_{#2}(#3)}% %------------ \hyphenation% %------------ {al-gebra al-gebras sub-order-ing co-ab-so-lute theo-rem} \begin{document} \maketitle \footnotetext{\ Institut f\"ur Mathematik II, Freie Universit\"at Berlin, 14195 Berlin, Germany. e-mail: {\tt fuchino@math.fu-berlin.de} (present address: Department of Computer Sciences, Kitami Institute of Technology, Kitami Hokkaido 090 Japan. e-mail: {\tt fuchino@info.kitami-it.ac.jp}). $^1$ Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary. e-mail: {\tt soukup@math-inst.hu}. The second author was supported by the Hungarian National Foundation for Scientific Research grant no.\ 16391.} \begin{abstract} We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (\Thmof{correction}) of partial orderings with $\kappa$-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang's conjecture for $\aleph_\omega$, we can find a counter-example to the characterization (\Thmof{ccchang}). We then show that, in the model obtained by adding Cohen reals, a lot of ccc complete Boolean algebras of cardinality $\leq\lambda$ have the $\aleph_1$-Freese-Nation property provided that $\mu^{\aleph_0}=\mu$ holds for every regular uncountable $\mu<\lambda$ and the very weak square principle holds for each cardinal $\aleph_0<\mu<\lambda$ of cofinality $\omega$ (\Thmof{cohen-model-rev}). Finally we prove that there is no $\aleph_2$-Lusin gap if $\powersetof{\omega}$ has the $\aleph_1$-Freese Nation property (\Thmof{lusin-gap}). \end{abstract} \section{Introduction} For a regular $\kappa$, a \po\ $P$ is said to have the $\kappa$ Freese-Nation property (the $\kappa$-FN for short) if there is a mapping ($\kappa$-FN mapping) $\mapping{f}{P}{[P]^{<\kappa}}$ \st\ for any $p$, $q\in P$ if $p\leq q$ then there is $r\in f(p)\cap f(q)$ \st\ $p\leq r\leq q$. Freese and Nation \cite{freese-nation} used the $\aleph_0$-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness. L.\ Heindorf gave a negative answer to the question showing that the \Bas\ with the $\aleph_0$-FN are exactly those which are openly generated. It is known that the class of openly generated \Bas\ contains projective \Bas\ as a proper subclass (see \cite{heindorf-shapiro} --- openly generated \Bas\ are called `rc-filtered' there). Heindorf and Shapiro \cite{heindorf-shapiro} then studied the $\aleph_1$-FN which they called the weak Freese-Nation property and proved some elementary properties of the \Bas\ with this property. \Pos\ with the $\kappa$-FN for arbitrary regular $\kappa$ were further studied in Fuchino, Koppelberg and Shelah \cite{WFN}. Koppelberg \cite{koppelberg} gives some nice applications of the $\aleph_1$-FN. In the following we shall quote some elementary facts from \cite{WFN} which we need later. First of all, it can be readily seen that every small \po\ has the $\kappa$-FN: \begin{Lemma}{\rm (\cite{WFN})} \label{baka} Every \po\ $P$ of cardinality $\leq\kappa$ has the $\kappa$-FN.\qed \end{Lemma} For a \po\ $P$ and a sub-ordering $Q\subseteq P$, we say that $Q$ is a $\kappa$-sub-ordering of $P$ and denote it with $Q\leq_\kappa P$ if, for every $p\in P$, the set $\setof{q\in Q}{q\leq p}$ has a cofinal subset of cardinality $<\kappa$ and the set $\setof{q\in Q}{q\geq p}$ has a coinitial subset of cardinality $<\kappa$. \begin{Lemma}{\rm (\cite{WFN})} \label{conti-chain}Suppose that $\delta$ is a limit ordinal and $(P_\alpha)_{\alpha\leq\delta}$ a continuously increasing chain of \pos\ \st\ $P_\alpha\leq_\kappa P_\delta$ for all $\alpha<\delta$. If $P_\alpha$ has the $\kappa$-FN for every $\alpha<\delta$, then $P_\delta$ also has the $\kappa$-FN.\qed \end{Lemma} For application of \Lemmaof{conti-chain}, it is enough to have $P_\alpha\leq_\kappa P_\delta$ and the $\kappa$-FN of $P_\alpha$ for every $\alpha<\delta$ \st\ either $\alpha$ is a successor or of cofinality $\geq\kappa$: $P_\alpha\leq_\kappa P_\delta$ for $\alpha<\delta$ of cofinality $<\kappa$ follows from this since such $P_\alpha$ can be represented as the union of $<\kappa$ many $\kappa$-sub-orderings of $P_\delta$. Hence by inductive application of \Lemmaof{conti-chain}, we can show that $P_\alpha$ satisfies the $\kappa$-FN for every $\alpha\leq\delta$. Similarly, if $\delta$ is a cardinal $>\kappa$, then it is enough to have $P_\alpha<_\kappa P_\delta$ and the $\kappa$-FN of $P_\alpha$ for every limit $\alpha<\delta$ of cofinality $\geq\kappa$. %% \begin{Prop}{\rm(\cite{WFN})} \label{characterization} For a regular $\kappa$ and a \po\ $P$, \tfae:\smallskip\\ \assert{1} $P$ has the $\kappa$-FN;\smallskip\\ \assert{2} For some, or equivalently, any sufficiently large $\chi$, if $M\prec\calH_\chi=(\calH_\chi,\in)$ is \st\ $P\in M$, $\kappa\subseteq M$ and $\cardof{M}=\kappa$ then $P\cap M\leq_\kappa P$ holds;\smallskip\\ \assert{3} $\setof{C\in[P]^{\kappa}}{C\leq_\kappa P}$ contains a club set.\qed \end{Prop} Though \Propof{characterization},\assertof{2} is quite useful to show that a \po\ has the $\kappa$-FN, sometimes it is quite difficult to check \Propof{characterization},\,\assertof{2} as in the case of the $\aleph_1$-FN of $P(\omega)$ or $[\kappa]^{<\omega}$: in these cases it is independent if \Propof{characterization},\,\assertof{2} holds. Applications like \Corof{appl-of-char} in mind, we could think of another possible variant of \Propof{characterization},\,\assertof{2} in terms of the following weakening of the notion of internal approachability from \cite{MM}: for a regular $\kappa$ and a sufficiently large $\chi$, we shall call an elementary submodel $M$ of $\calH_\chi$ {\em $V_{\kappa}$-like} if, either $\kappa=\aleph_0$ and $M$ is countable, or there is an increasing sequence $(M_\alpha)_{\alpha<\kappa}$ of elementary submodels of $M$ of cardinality less than $\kappa$ \st\ $M_\alpha\in M_{\alpha+1}$ for all $\alpha<\kappa$ and $M=\bigcup_{\alpha<\kappa}M_\alpha$. %% If $M$ is %% $V_{\kappa}$-like, we say that a sequence %% $(M_\alpha)_{\alpha<\kappa}$ as above witnesses the %% $V_{\kappa}$-likeness of %% $M$. In \cite{WFN}, a characterization of the $\kappa$-FN using $V_\kappa$-like elementary submodels in place of elementary submodels in \Propof{characterization},\,\assertof{2} was discussed. Unfortunately it appeared that some consequences of $\neg0^\#$ are necessary for the characterization (see ``Added in Proof'' in \cite{WFN}). In this paper, we introduce a weakening of the very weak square principle from \cite{foreman-magidor} --- the principle $\Box^{***}_{\kappa,\mu}$. In section \ref{jensen-matrix} we show the equivalence of $\Box^{***}_{\kappa,\mu}$ with the existence of a matrix $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\cf(\mu)}$ --- which we called (weak) $(\kappa,\mu)$-dominating matrix --- of elementary submodels of $\calH(\chi)$ for sufficiently large $\chi$ with certain properties. This fact is used in section \ref{xchar} to show that $\Box^{***}_{\kappa,\lambda}$ together with a very weak version of the Singular Cardinals Hypothesis yields the characterization of \pos\ with the $\kappa$-FN in terms of $V_\kappa$-like elementary submodels (\Thmof{correction}). ZFC or even ZFC $+$ GCH is not enough for this characterization: in section \ref{chang}, we show that, under Chang's conjecture for $\aleph_\omega$, there is a counter-example to the characterization. Together with \Thmof{correction}, this counter-example also shows that $\Box^{***}_{\aleph_1,\aleph_\omega}$ is not a theorem in ZFC $+$ GCH. One of the most natural questions concerning the $\kappa$-FN would be if $(\powersetof{\omega},\subseteq)$ has the $\aleph_1$-FN. It is easy to see that $(\powersetof{\omega},\subseteq)$ has the $\aleph_1$-FN iff $(\powersetof{\omega}/fin,\subseteq^*)$ does (see \cite{WFN}). See also Koppelberg \cite{koppelberg} for some consequences of the $\aleph_1$-FN of $\powersetof{\omega}/fin$. By \Lemmaof{baka}, $\powersetof{\omega}$ has the $\aleph_1$-FN under CH. In section \ref{cohen}, we show that $\powersetof{\omega}$ and a lot of other ccc \cBas\ still have the $\aleph_1$-FN in a model obtained by adding arbitrary number of Cohen reals to a model of, say, $V=L$. On the other hand it can happen very easily that $\powersetof{\omega}$ does not have the $\aleph_1$-FN. In \cite{WFN}, it was shown that this is the case when ${\bf b}>\aleph_1$ or, more generally, if there is a $\subseteq^*$-sequence of elements of $\psof{\omega}$ of order-type $\geq\omega_2$. In section \ref{lusin}, we show that the existence of an $\aleph_2$-Lusin gap can be another reason for failure of the $\aleph_1$-FN. At the moment the authors do not know if there are yet other reasons for failure of the $\aleph_1$-FN of $\powersetof{\omega}$: \begin{Problem} Suppose that $\powersetof{\omega}$ does not have any increasing chain of length $\geq\aleph_2$ \wrt\ $\subseteq^*$ and that there is no $\aleph_2$-Lusin gap. Does it follow that $\powersetof{\omega}$ has the $\aleph_1$-FN\,? \end{Problem} \par %! notation ... %! notation $(C)'$, cf Our notation is fairly standard. The following are possible deviations from the standard: for $C\subseteq\kappa$, we denote with $(C)'$ the set of limit points of $C$ other than $\kappa$. For an ordinal $\alpha$, $\Lim(\alpha)=\setof{\beta<\alpha}{\beta\xmbox{ is a limit ordinal}}$. For a \po\ $P$, $\cf(P)=\min\setof{\cardof{X}}{X\subseteq P,X\mbox{ is cofinal in }P}$. $[\lambda]^{<\kappa}=\setof{X\subseteq\lambda}{\cardof{X}<\kappa}$ is often seen as the \po\ $([\lambda]^{<\kappa},\subseteq)$. If $Q$ is a sub-ordering of a \po\ $P$ and $p\in P$ then $Q\upper p=\setof{q\in Q}{q\geq p}$ and $Q\restr p=\setof{q\in Q}{q\leq p}$. Let $I$ be an infinite set. Adopting the notation of \cite{kunen}, we denote with $\Fn(I,2)$ the standard \po\ for adding $\cardof{I}$-many Cohen reals, i.e. $\Fn(I, 2)$ is the \po\ $\setof{p}{p\xmbox{ is a mapping with } \dom(p)\in[I]^{<\aleph_0}\xmbox{ and }\range(p)\subseteq 2}$ with the inverse inclusion. \section{Very weak square and dominating matrix}\label{jensen-matrix} For a cardinal $\mu$, the {\em weak square principle for $\mu$ } (notation: $\Box^*_\mu$) is the statement: there is a sequence $(\calC_\alpha)_{\alpha\in\Lim(\mu^+)}$ \st\ for every $\alpha\in\Lim(\mu^+)$ \begin{assertion}{}\mbox{}% \lassert{w1} $\calC_\alpha\subseteq\powersetof{\alpha}$ and $\cardof{\calC_\alpha}\leq\mu$;\smallskip\\ \lassert{w2} every $C\in\calC_\alpha$ is club in $\alpha$ and if $\cf(\alpha)<\mu$ then $\otp(C)<\mu$;\smallskip\\ \lassert{w3} for every $C\in\calC_\alpha$ and $\delta\in(C)'$, we have $C\cap\delta\in\calC_\delta$. \end{assertion} Clearly we have $\Box_\mu\rightarrow\,\Box^*_\mu$. Jensen \cite{jensen} proved that $\Box^*_\mu$ is equivalent to the existence of a special Aronszajn tree on $\mu^+$. Ben-David and Magidor \cite{ben-david-magidor} showed that the weak square principle for a singular $\mu$ is actually weaker than the square principle: they constructed a model of $\Box^*_{\aleph_\omega}$ and $\neg\Box_{\aleph_\omega}$ starting from a model with a supercompact cardinal. Foreman and Magidor considered in \cite{foreman-magidor} the following principle which is, e.g.\ under GCH, a weakening of $\Box^*$ principle: for a cardinal $\mu$, {\em the very weak square principle for $\mu$} holds if there is a sequence\ $(C_\alpha)_{\alpha<\mu^+}$ and a club $D\subseteq\mu^+$ \st\ for every $\alpha\in D$ \begin{assertion}{}\mbox{}% \lassert{v1} $C_\alpha\subseteq\alpha$, $C_\alpha$ is unbounded in $\alpha$;\smallskip\\ \lassert{v2} for all bounded $x\in[C_\alpha]^{<\omega_1}$, there is $\beta<\alpha$ \st\ $x=C_\beta$. \end{assertion} In this paper, we shall use the following yet weaker variant of the very weak square principle. For a regular cardinal $\kappa$ and $\mu>\kappa$, let $\Box^{***}_{\kappa,\mu}$ be the following assertion: there exists a sequence $(C_\alpha)_{\alpha<\mu^+}$ and a club set $D\subseteq\mu^+$ \st\ for $\alpha\in D$ with $\cf(\alpha)\geq\kappa$ \begin{assertion}{}\mbox{}% \lassert{y1} $C_\alpha\subseteq\alpha$, $C_\alpha$ is unbounded in $\alpha$;\smallskip\\ %% \lassert{y2} $[\alpha]^{<\kappa}\cap\setof{C_{\alpha'}}{\alpha'<\alpha}$ dominates $[C_\alpha]^{<\kappa}$ (\wrt\ $\subseteq$). \end{assertion} Since \assertof{y2} remains valid when $C_\alpha$'s for $\alpha\in D$ are shrunk, we may replace \assertof{y1} by \begin{assertion}{}\mbox{}% \lassert{y1'} $C_\alpha\subseteq\alpha$, $C_\alpha$ is unbounded in $\alpha$ and $\otp(C_\alpha)=\cf(\alpha)$. \end{assertion} A corresponding remark holds also for the sequence of the very weak square principle. \begin{Lemma} \assert{a} The very weak square principle for $\mu$ implies $\Box^{***}_{\omega_1,\mu}$. \smallskip\\ \assert{b} For a singular cardinal $\mu$ and a regular $\kappa$ \st\ $\cf([\lambda]^{<\kappa},\subseteq)\leq\mu$ for every $\lambda<\mu$, $\Box^*_\mu$ implies $\Box^{***}_{\kappa,\mu}$. \end{Lemma} \prf \assertof{a} is clear. For \assertof{b}, let $(\calC_\alpha)_{\alpha\in\Lim(\mu^+)}$ be a weak square sequence. Let $\calC_\alpha=\setof{C_{\alpha,\beta}}{\beta<\mu}$ for every $\alpha\in\Lim(\mu^+)$. \iffalse \Wolog, we may assume that $C_{\alpha,0}$ is as the $C$ in \assertof{w3}. \fi By shrinking $C_{\alpha,\beta}$'s if necessary, we may assume that $\cardof{C_{\alpha,\beta}}<\mu$ for every $\alpha\in\Lim(\mu^+)$ and $\beta<\mu$. Note that we need here the assumption that $mu$ be singular. For $\alpha\in\Lim(\mu^+)$ and $\beta<\mu$, let $X_{\alpha,\beta}$ be a cofinal subset of $[C_{\alpha,\beta}]^{<\kappa}$ of cardinality $\leq\mu$ and let $\setof{C_\alpha}{\alpha\in\mu^+\setminus\Lim(\mu^+)}$ be an enumeration of $\bigcup\setof{X_{\alpha,\beta}}{\alpha\in\Lim(\mu^+),\beta<\mu}$. For each $\alpha\in\Lim(\mu^+)$, let $C_\alpha=C_{\alpha,0}$. Let $\mapping{F}{\mu^+}{\mu^+}$ be defined by $F(\xi)= \min\setof{\gamma<\mu^+}{% X_{\xi,\beta}\subseteq\setof{C_\alpha}{\alpha<\gamma} \xmbox{ for every }\beta<\mu}$. Let $D\subseteq\mu^+$ be a club set closed \wrt\ $F$. Then $(C_\alpha)_{\alpha<\mu^+}$ and $D$ are as in the definition of $\Box^{***}_{\kappa,\mu}$. To see that \assertof{y2} is satisfied, let $\alpha\in D$ be \st\ $\cf(\alpha)\geq\kappa$ and $x\in[C_\alpha]^{<\kappa}$. By definition we have $C_\alpha=C_{\alpha,0}$. Hence there are $\alpha'\in\alpha\cap\Lim(\mu^+)$ and $\beta<\lambda$ \st\ $x\in[C_{\alpha',\beta}]^{<\kappa}$. Since $\alpha$ is closed \wrt\ $F$, there is some $\gamma<\alpha$ \st\ $x\subseteq C_\gamma\in [C_{\alpha',\beta}]^{<\kappa}$. \qedofLemma \qedskip\par $\Box^{***}_{\kappa,\mu}$ has some influence on cardinal arithmetic of cardinals below $\mu$: \begin{Lemma}\label{card-arith} Suppose that $\kappa$ is regular and $\mu$ is \st\ $\cf(\mu)<\kappa$. If $\Box^{***}_{\kappa,\mu}$ holds, then we have $\cf([\lambda]^{<\kappa},\subseteq)<\mu$ for every $\lambda<\mu$. \end{Lemma} \prf Let $(C_\alpha)_{\alpha<\mu^+}$, and $D\subseteq\mu^+$ be witnesses of $\Box^{***}_{\kappa,\mu}$. For $\lambda<\mu$, let $\delta\in D$ be \st\ $\cf(\delta)\geq\lambda+\kappa$. Then $\setof{C_\alpha}{\alpha<\delta}\cap[\delta]^{<\kappa}$ is cofinal in $[C_\delta]^{<\kappa}$. Since $\cardof{\delta}\leq\mu$ it follows that $\cf([C_\delta]^{<\kappa},\subseteq)\leq\mu$. As the order type of $C_\delta$ is at least $\lambda$, it follows that $\nu=\cf([\lambda]^{<\kappa},\subseteq)\leq\mu$. But $\cf(\nu)\geq\kappa$. Hence $\nu<\mu$. \qedofLemma\qedskip\\ %% Suppose now that $\kappa$ is a regular cardinal and $\mu>\kappa$ is \st\ $\cf(\mu)<\kappa$. Let $\mu^*=\cf(\mu)$. For a sufficiently large $\chi$ and $x\in\calH(\chi)$, let us call a sequence $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ a {\em$(\kappa,\mu)$-dominating matrix over $x$} --- or just {\em dominating matrix over $x$} if it is clear from the context which $\kappa$ and $\mu$ are meant --- if the following conditions hold: \begin{assertion}{}\mbox{}% \lassert{j1} $M_{\alpha,\beta}\prec\calH(\chi)$, $x\in M_{\alpha,\beta}$, $\kappa+1\subseteq M_{\alpha,\beta}$ and $\cardof{M_{\alpha,\beta}}<\mu$ for all $\alpha<\mu^+$ and $\beta<\mu^*$\,;\smallskip\\ %% \lassert{j2} $(M_{\alpha,\beta})_{\beta<\mu^*}$ is an increasing sequence for each $\alpha<\mu^+$\,;\smallskip\\ \lassert{j3} if $\alpha<\mu^+$ is \st\ $\cf(\alpha)\geq\kappa$, then there is $\beta^*<\mu^*$ \st, for every $\beta^*\leq\beta<\mu^*$, $[M_{\alpha,\beta}]^{<\kappa}\cap M_{\alpha,\beta}$ is cofinal in $([M_{\alpha,\beta}]^{<\kappa},\subseteq)$\,; \end{assertion} For $\alpha<\mu^+$, let $M_\alpha=\bigcup_{\beta<\mu^*}M_{\alpha,\beta}$. By \assertof{j1} and \assertof{j2}, we have $M_\alpha\prec\calH(\chi)$. %% By \assertof{j3}, $(M_\alpha)_{\alpha<\mu^+}$ is increasing. \begin{assertion}{}\mbox{}% \lassert{j4} $(M_\alpha)_{\alpha<\mu^+}$ is continuously increasing and $\mu^+\subseteq\bigcup_{\alpha<\mu^+}M_\alpha$. \end{assertion} \iffalse The choice of the term ``Jensen matrix'' was suggested by \cite{foreman-magidor} in which (under GCH) a matrix of subsets of $\mu^+$ having some properties similar to those of the sequence $(\mu^+\cap M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ for $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ as above is called a Jensen sequence. \else Foreman and Magidor \cite{foreman-magidor} called a sequence of subsets of $\mu^+$ having some properties similar to those of the sequence $(\mu^+\cap M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ for $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ as above Jensen matrix. Our definition of dominating matrices and some ideas in the proofs in this section are inspired by their paper. \fi Note that, in the case of $2^{<\kappa}=\kappa$, \assertof{j3} can be replaced by the following seemingly stronger property: \begin{assertion}{}\mbox{}% \lassert{j3'} if $\alpha<\mu^+$ is \st\ $\cf(\alpha)\geq\kappa$, then there is $\beta^*<\mu^*$ \st, for every $\beta^*\leq\beta<\mu^*$, $[M_{\alpha,\beta}]^{<\kappa}\subseteq M_{\alpha,\beta}$. \end{assertion} This is simply because of the following observation: \begin{Lemma}\label{easy} Suppose that $2^{<\kappa}=\kappa$ and $M$ is an elementary submodel of $\calH(\chi)$ for some sufficiently large $\chi$ and $\kappa\subseteq M$. If $[M]^{<\kappa}\cap M$ is cofinal in $[M]^{<\kappa}$, then we have $[M]^{<\kappa}\subseteq M$. \end{Lemma} \prf Let $x\in[M]^{<\kappa}$. We show that $x\in M$. By assumption there is $y\in [M]^{<\kappa}\cap M$ \st\ $x\subseteq y$. Let $\eta=\cardof{\psof{y}}$. Then $\eta\in M$ and $\eta\leq\kappa$. Let $(y_\alpha)_{\alpha<\eta}\in M$ be an enumeration of $\psof{y}$. Then there is an $\alpha_0<\eta$ \st\ $x=y_{\alpha_0}$. But since $\kappa\subseteq M$, we have $\alpha_0\in M$ and $y_{\alpha_0}\in M$ as well. \qedofLemma\qedskip \\ Note also that, if $M\prec\calH(\chi)$ is $V_\kappa$-like, then $[M]^{<\kappa}\cap M$ is cofinal in $[M]^{<\kappa}$. Hence, under $2^{<\kappa}=\kappa$, $M\prec\calH(\chi)$ is $V_\kappa$-like if and only if $\cardof{M}=\kappa$ and $[M]^{<\kappa}\subseteq M$. In the following theorem, we show that $\Box^{***}_{\kappa,\mu}$ together with a very weak version of the Singular Cardinals Hypothesis below $\mu$ implies the existence of a dominating matrix: \begin{Thm}\label{Jensen-exists} Suppose that $\kappa$ is a regular cardinal and $\mu>\kappa$ is \st\ $\cf(\mu)<\kappa$. If we have $\cf([\lambda]^{<\kappa},\subseteq)=\lambda$ for cofinally many $\lambda<\mu$ and $\Box^{***}_{\kappa,\mu}$ holds, then, for any sufficiently large $\chi$ and $x\in\calH(\chi)$, there is a $(\kappa,\mu)$-dominating matrix over $x$. \end{Thm} \prf Let $\mu^*=\cf(\mu)$ and $(\mu_\beta)_{\beta<\mu^*}$ be an increasing sequence of cardinals below $\mu$ \st\ $\mu_0>\mu^*$, $\sup\setof{\mu_\beta}{\beta<\mu^*}=\mu$ and $\cf([\mu_\beta]^{<\kappa},\subseteq)=\mu_\beta$ for every $\beta<\mu^*$. Let $(C_\alpha)_{\alpha\in\mu^+}$ and $D\subseteq\mu^+$ be as in the definition of $\Box^{***}_{\kappa,\mu}$. \Wolog, we may assume that $\cardof{C_\alpha}\leq\cf(\alpha)$ for all $\alpha<\mu^+$. We may also assume that $\alpha>\mu$ for every $\alpha\in D$. % introduction に持ってゆく In the following, we fix a well ordering $\unlhd$ on $\calH(\chi)$ and, when we talk about $\calH(\chi)$ as a structure, we mean $\calH(\chi)=(\calH(\chi),\in,\unlhd)$. $X\subseteq\calH(\chi)$ as a substructure of $\calH(\chi)$ is thus the structure $(X,\in\cap X^2,\unlhd\cap X^2)$ --- for notational simplicity we shall denote such a structure simply by $(X,\in,\unlhd)$. Let $N\in\calH(\chi)$ be an elementary substructure of $\calH(\chi)$ \st\ $N$ contains every thing needed below --- in particular, we let $\mu^+\subseteq N$ and $x$, $(C_{\alpha})_{\alpha<\mu^+}$, $D$, $(\mu_\alpha)_{\alpha<\mu^*}\in N$. Let $(N_\xi)_{\xi<\kappa}$ be an increasing sequence of elementary submodels of $\calH(\chi)$ \st\ \begin{assertion}{}\mbox{}% \lassert{0} $N_0=N$\,; \\ \lassert{1} $N_\xi\in \calH(\chi)$ for every $\xi<\kappa$ and \\ \lassert{2} $(N_\eta)_{\eta\leq\xi}\in N_{\xi+1}$ for every $\xi<\kappa$. \end{assertion} Now, for each $\xi<\kappa$, let \[ \calN_\xi =(N_\xi,\in,\unlhd,R_\xi,\kappa, \mu,\mu^*, D, (C_{\alpha})_{\alpha<\mu^+}, (\mu_\alpha)_{\alpha<\mu^*},x,\eta)_{\eta<\xi} \]\noindent where $R_\xi$ is the relation $\setof{(\eta, N_\eta)}{\eta<\xi}$. For $X\subseteq\mu^+$, let us denote with $sk_\xi(X)$ the Skolem-hull of $X$ \wrt\ the built-in Skolem functions of $\calN_\xi$ (induced from $\unlhd$). For $\xi<\xi'<\kappa$, $\calN_\xi$ is an element of $\calN_{\xi'}$ by \assertof{2} and the Skolem functions of $\calN_\xi$ are also elements of $\calN_{\xi'}$. In particular, we have $sk_\xi(X)\subseteq sk_{\xi'}(X)$. It follows that $sk(X)=\bigcup_{\xi<\kappa}sk_\xi(X)$ is an elementary submodel of $\calH(\chi)$. Note also that, if $X\subseteq\mu^+$ is an element of $sk_{\xi'}(Y)$ then, since $sk_\xi(X)$ is definable in $sk_{\xi'}(Y)$, we have $sk_\xi(X)\in sk_{\xi'}(Y)$. For the proof of the theorem, it is clearly enough to construct $M_{\alpha,\beta}$ with \assertof{j1} --- \assertof{j4} for every $\alpha$ in the club set $D$ and for every $\beta<\mu^*$. Let \[ M_{\alpha,\beta}=sk(\mu_\beta\cup C_{\alpha}) \]\noindent for $\alpha\in D$ and $\beta<\mu^*$. We show that $(M_{\alpha,\beta})_{\alpha\in D,\beta<\mu^*}$ is as desired. It is clear that \assertof{j1} and \assertof{j2} hold. We need the following claim to show the other properties: \begin{Claim}\label{claim-alpha} $M_\alpha=sk(\alpha)$ for every $\alpha\in D$. \end{Claim} \prfofClaim ``$\subseteq$'' is clear since $\mu_\beta\cup C_{\alpha}\subseteq\alpha$ for every $\alpha\in D$ and $\beta<\mu^*$. For ``$\supseteq$'', it is enough to show that $\alpha\subseteq M_\alpha$. Let $\gamma<\alpha$. By \assertof{y1}, there is $\gamma_1\in C_{\alpha}$ \st\ $\gamma<\gamma_1$. Let $f\in M_{\alpha,0}$ be a surjection from $\mu$ to $\gamma_1$, and let $\delta<\mu$ be \st\ $f(\delta)=\gamma$. Then we have $\gamma=f(\delta)\in M_{\alpha,\beta^*}\subseteq M_\alpha$ for $\beta^*<\mu^*$ \st\ $\delta<\mu_{\beta^*}$. \qedofClaim% claim-alpha \smallskip For \assertof{j3}, suppose that $\alpha\in D$ and $\cf(\alpha)\geq\kappa$. Let $\beta^*$ be \st\ $\cardof{C_\alpha}<\mu_{\beta^*}$ and $[\alpha]^{<\kappa}\cap \setof{C_{\alpha'}}{\alpha'<\alpha,\,\alpha'\in M_{\alpha,\beta^*}}$ dominates $[C_\alpha]^{<\kappa}$. The last property is possible by \assertof{y2}, \Claimabove\ and $\mu^*<\kappa$. We show that this $\beta^*$ is as needed in \assertof{j3}. Let $\beta<\mu^*$ be \st\ $\beta^*\leq\beta$ and suppose that $x\in[M_{\alpha,\beta}]^{<\kappa}$. Then there are $u\in[\mu_\beta]^{<\kappa}$ and $v\in[C_{\alpha}]^{<\kappa}$ \st\ $x\subseteq sk(u\cup v)$. Since $\mu_\beta\in M_{\alpha,\beta}$ and $\cf([\mu_\beta]^{<\kappa},\subseteq)=\mu_\beta$, there is $X\in M_{\alpha,\beta}$ \st\ $X\subseteq[\mu_\beta]^{<\kappa}$, $\cardof{X}=\mu_\beta$ and $X$ is cofinal in $([\mu_\beta]^{<\kappa},\subseteq)$. Since $\mu_\beta\subseteq M_{\alpha,\beta}$, it follows that $X\subseteq M_{\alpha,\beta}$. Hence there is $u'\in M_{\alpha,\beta}\cap[\mu_\beta]^{<\kappa}$ \st\ $u\subseteq u'$. On the other hand, by definition of $\beta^*$, $\cf(\alpha)\geq\kappa$, there is $\alpha'\in \alpha\cap M_{\alpha,\beta}$ \st\ $C_{\alpha'}\in [\alpha]^{<\kappa}$ and $v\subseteq C_{\alpha'}$. Thus, we have $x\subseteq sk(u\cup v)\subseteq sk(u'\cup C_{\alpha'})$. By regularity of $\kappa$, there is $\xi<\kappa$ \st\ $x\subseteq sk_\xi(u'\cup C_{\alpha'})$. But $sk_\xi(u'\cup C_{\alpha'})\in M_{\alpha,\beta}$ and $\cardof{sk_\xi(u'\cup C_{\alpha'})}<\kappa$. \assertof{j4} follows immediately from \Claimof{claim-alpha}. \qedofThm % Jensen-exists \qedskip\\ Note that, in the proof above, the sequence $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ satisfies also: \begin{assertion}{}\mbox{}% \lassert{j5} for $\alpha<\alpha'<\mu^+$ and $\beta<\mu^*$, there is $\beta'<\mu^*$ \st\ $M_{\alpha,\beta}\subseteq M_{\alpha',\beta'}$. \end{assertion} [\,Suppose that $\alpha$, $\alpha'\in D$ are \st\ $\alpha<\alpha'$ and $\beta<\mu^*$. By \Claimof{claim-alpha}, there is $\beta'<\mu^*$ \st\ $\beta<\beta'$, $\alpha\in M_{\alpha',\beta'}$ and $\otp(C_{\alpha})\leq\mu_{\beta'}$. Then we have $C_{\alpha}\in M_{\alpha',\beta'}$ and $C_{\alpha}\subseteq M_{\alpha',\beta'}$. Also $\mu_\beta\in M_{\alpha',\beta'}$ and $\mu_\beta\subseteq\mu_{\beta'}\subseteq M_{\alpha',\beta'}$. Hence it follows that $M_{\alpha,\beta}=sk(\mu_\beta\cup C_{\alpha}) \subseteq M_{\alpha',\beta'}$\,.] Conversely, the existence of a $(\kappa,\mu)$-dominating matrix (over some/any $x$) implies $\Box^{***}_{\kappa,\mu}$: %%% \begin{Thm}\label{converse} Suppose that $\kappa$ is a regular cardinal and $\mu>\kappa$ is \st\ $\cf(\mu)<\kappa$. If there exists a $(\kappa,\mu)$-dominating matrix, then $\Box^{***}_{\kappa,\mu}$ holds. \end{Thm} \prf Let $\mu^*=\cf(\mu)$ and $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ be a $(\kappa,\mu)$-dominating matrix. Let $M_\alpha=\bigcup_{\beta<\mu^*}M_{\alpha,\beta}$ for each $\alpha<\mu^+$. For \[ \textstyle X= \bigcup\setof{[M_{\alpha,\beta}]^{<\kappa}\cap M_{\alpha,\beta}}{% \alpha<\mu^+,\,\beta<\mu^*}, \]\noindent let $\setof{C_{\alpha+1}}{\alpha<\mu^+}$ be an enumeration of $X$. Let \[ D= \setof{\alpha<\mu^+}{% \begin{array}[t]{@{}l} M_\alpha\cap\mu^+=\alpha,\\ \setof{C_{\alpha'+1}}{\alpha'<\alpha} \supseteq [M_{\alpha^\dagger,\beta}]^{<\kappa}\cap M_{\alpha^\dagger,\beta}\\ \mbox{for every }\alpha^\dagger<\alpha,\,\beta<\mu^*\,\,}. \end{array} \]\noindent By \assertof{j4}, $D$ is a club subset of $\mu^+$. For $\alpha\in D$ with $\cf(\alpha)\geq\kappa$, let $C_\alpha=M_{\alpha,\beta_\alpha}\cap \alpha$ where $\beta_\alpha<\mu^+$ be \st\ $M_{\alpha,\beta_\alpha}\cap\alpha$ is cofinal in $\alpha$ (this is possible as $M_\alpha\cap \mu^+=\alpha$ and $\mu^*<\kappa$) and that $[M_{\alpha,\beta_\alpha}]^{<\kappa}\cap M_{\alpha,\beta_\alpha}$ is cofinal in $[M_{\alpha,\beta_\alpha}]^{<\kappa}$ (possible by \assertof{j3}\,). For $\alpha\in\Lim(\mu^+)\setminus\setof{\alpha\in D}{\cf(\alpha)\geq\kappa}$, let $C_\alpha$ be anything, say $C_\alpha=\emptyset$. We claim that $(C_\alpha)_{\alpha<\mu^+}$ and $D$ as above satisfy the conditions in the definition of $\Box^{***}_{\kappa,\mu}$: \assertof{y1} is clear by definition of $C_\alpha$'s. To show \assertof{y2}, let $\alpha\in D$ be \st\ $\cf(\alpha)\geq\kappa$ and $x\in[C_\alpha]^{<\kappa}$. Then, by the choice of $\beta_\alpha$, there is $y\in[\alpha]^{<\kappa}\cap M_{\alpha,\beta_\alpha}$ \st\ $x\subseteq y$. By \assertof{j4}, there are $\alpha^\dagger<\alpha$ and $\beta^\dagger<\mu^*$ \st\ $y\in M_{\alpha^\dagger,\beta^\dagger}$. By definition of $D$, it follows that $y=C_{\alpha'+1}$ for some $\alpha'<\alpha$. This shows that $[\alpha]^{<\kappa}\cap\setof{C_{\alpha'}}{\alpha'<\alpha}$ dominates $[C_\alpha]^{<\kappa}$. \qedofThm\qedskip \\ Thus, by \Thmof{Jensen-exists} and \Thmof{converse}, if $\cf([\lambda]^{<\kappa},\subseteq)=\lambda$ for cofinally many $\lambda<\mu$, $\Box^{***}_{\kappa,\mu}$ is equivalent to the existence of a $(\kappa,\mu)$-dominating matrix. The assumption ``$\cf([\lambda]^{<\kappa},\subseteq)=\lambda$ for cofinally many $\lambda<\mu$'' cannot be removed from this equivalence theorem since $\Box^{***}_{\kappa,\mu}$ implies this. However, using the following weakening of the notion of dominating matrix, we obtain a characterization of $\Box^{***}_{\kappa,\mu}$ in ZFC: for a regular cardinal $\kappa$ and $\mu>\kappa$ \st\ $\mu^*=\cf(\mu)<\kappa$, let us call a matrix $(M_{\alpha,\beta})_{\alpha<\mu^+,\beta<\mu^*}$ of elementary submodels of $\calH(\chi)$ for a sufficiently large $\chi$, {\em a weak $(\kappa,\mu)$-dominating matrix over $x$}, if it satisfies \assertof{j1}, \assertof{j2}, \assertof{j4} for $M_\alpha=\bigcup_{\beta<\mu^*}M_{\alpha,\beta}$, $\alpha<\mu^+$, and \begin{assertion}{}\mbox{}% \lassert{j3$^-$\relax} if $\alpha<\mu^+$ is \st\ $\cf(\alpha)\geq\kappa$, then there is $\beta^*<\mu^*$ \st, for every $\beta^*\leq\beta<\mu^*$, $[M_{\alpha,\beta}]^{<\kappa}\cap M_{\alpha}$ is cofinal in $([M_{\alpha,\beta}]^{<\kappa},\subseteq)$\,. \end{assertion} Since $\mu^*<\kappa$, the condition above is equivalent to: \begin{assertion}{}\mbox{}% \lassert{j3$^*$\relax} if $\alpha<\mu^+$ is \st\ $\cf(\alpha)\geq\kappa$, then there is $\beta^*<\mu^*$ \st, for every $\beta^*\leq\beta<\mu^*$, there is $\beta'<\mu^*$ \st\ $[M_{\alpha,\beta}]^{<\kappa}\cap M_{\alpha,\beta'}$ is cofinal in $([M_{\alpha,\beta}]^{<\kappa},\subseteq)$\,. \end{assertion} \begin{Thm} Suppose that $\kappa$ is a regular cardinal and $\mu>\kappa$ is \st\ $\mu^*=\cf(\mu)<\kappa$. Then $\Box^{***}_{\kappa,\mu}$ holds if and only if there is a weak $(\kappa,\mu)$-dominating matrix over some/any $x$. \end{Thm} \prf For the forward direction the proof is almost the same as that of \Thmof{Jensen-exists}. We let $(\mu_\beta)_{\beta<\mu^*}$ here merely an increasing sequence of regular cardinals with the limit $\mu$. Then $(M_{\alpha,\beta})_{\alpha\in D,\beta<\mu^*}$ is constructed just as in the proof of \Thmof{Jensen-exists}. \Lemmaof{card-arith} is then used to see that \assertof{j3$^-$\relax} is satisfied by this matrix. For the converse, just the same proof as that of \Thmof{converse} will do. \qedofThm\qedskip\\ Existence of a (weak) dominating-matrix is not a theorem in ZFC: we show in section \ref{chang} that the Chang's conjecture for $\aleph_\omega$ together with $2^{\aleph_\omega}=\aleph_{\omega+1}$ implies that there is no $(\aleph_n,\aleph_\omega)$-dominating matrix for any $n\geq 1$. \section{A characterization of the $\kappa$-Freese-Nation \protect\\property}\label{xchar} The following game over a \po\ $P$ was considered in \cite{WFN,pogame}. Let $\kappagame{\kappa}{P}$ be the following game played by Players I and II: in a play in $\kappagame{\kappa}{P}$, Players I and II choose subsets $X_\alpha$ and $Y_\alpha$ of $P$ of cardinality less than $\kappa$ alternately for $\alpha<\kappa$ \st \[ X_0\subseteq Y_0\subseteq X_1\subseteq Y_1\subseteq\cdots\subseteq X_\alpha \subseteq Y_\alpha\subseteq\cdots\subseteq X_\beta\subseteq Y_\beta\subseteq\cdots \]\noindent for $\alpha\leq\beta<\kappa$. Thus a play in $\kappagame{\kappa}{P}$ looks like \[ \begin{array}{l@{}l@{}@{}l@{}@{}l@{}l} \mbox{\it Player I}\ &:\ \ &X_0,\ &X_1,\ \ldots,\ &X_\alpha,\ \ldots\medskip\\ \mbox{\it Player II}\ &:\ \ &Y_0,\ &Y_1,\ \ldots,\ &Y_\alpha,\ \ldots \end{array} \]\noindent where $\alpha<\kappa$. Player II wins the play if $\bigcup_{\alpha<\kappa}X_\alpha=\bigcup_{\alpha<\kappa}Y_\alpha$ is a $\kappa$-sub-ordering of $P$. Let us call a strategy $\tau$ for Player II {\em simple} if, in $\tau$, each $Y_\alpha$ is decided from the information of the set $X_\alpha\subseteq P$ alone (i.e.\ also independent of $\alpha$). Another notion we need here is the following generalization of $V_\kappa$-likeness. Let $\kappa$ be regular and $\chi$ be sufficiently large. For $\calD\subseteq\setof{M\prec\calH(\chi)}{\cardof{M}<\kappa}$, we say that $M\in[\calH(\chi)]^\kappa$ is {\em$\calD$-approachable} if there is an increasing sequence $(D_\alpha)_{\alpha<\kappa}$ of elements of $\calD$ \st\medskip\\ \assert{a} $D_\alpha\cup\smallsetof{D_\alpha}\subseteq D_{\alpha+1}$ for every $\alpha<\kappa$; and \\ \assert{b} $M=\bigcup_{\alpha<\kappa}D_\alpha$. \medskip\\ Clearly $M\prec\calH(\chi)$ is $V_\kappa$-like if and only if $M$ is $\calD$-approachable for $\calD=\setof{M\prec\calH(\chi)}{\cardof{M}<\kappa}$. A slightly weaker version of the following theorem was announced in \cite{WFN}: \begin{Thm} \label{correction} Let $\kappa$ be a regular uncountable cardinal and $\kappa\leq\lambda$. Suppose that \smallskip\\ \assert{i} $([\mu]^{<\kappa},\subseteq)$ has a cofinal subset of cardinality $\mu$ for every $\mu$ \st\ $\kappa<\mu<\lambda$ and $\cf(\mu)\geq\kappa$\,; and\\ \assert{ii} $\Box^{***}_{\kappa,\mu}$ holds for every $\mu$ \st\ $\kappa\leq\mu<\lambda$ and $\cf(\mu)<\kappa$.\smallskip\\ Then, for a \po\ $P$ of cardinality $\leq\lambda$, \tfae:\medskip\\ \assert{1} $P$ has the $\kappa$-FN\,;\smallskip\\ \assert{2} Player II has a simple winning strategy in $\kappagame{\kappa}{P}$\,;\smallskip\\ \assert{3} for some, or equivalently any sufficiently large $\chi$, and any $V_\kappa$-like $M\prec\calH(\chi)$ with $P$, $\kappa\in M$, we have $P\cap M\leq_\kappa P$\,;\smallskip\\ \assert{4} for some, or equivalently any sufficiently large $\chi$, there is $\calD\subseteq[\calH(\chi)]^{<\kappa}$ \st\ $\calD$ is cofinal in $[\calH(\chi)]^{<\kappa}$ and for any $\calD$-approachable $M\subseteq\calH(\chi)$, we have $P\cap M\leq_\kappa P$. \end{Thm} Note that $\neg 0^\#$ implies the conditions \assertof{i} and \assertof{ii}. Also note that, for every $\lambda<\kappa^{+\omega}$, the condition \assertof{i} holds in ZFC. Hence the characterization above holds for partial orderings of cardinality $\leq\kappa^{+\omega}$ without any additional assumptions. \medskip\\ \prf A proof of \assertof{1}\implies\assertof{2}\implies\assertof{3} is given in \cite{WFN}. For \assertof{3}\implies\assertof{4}, suppose that $P$ satisfies \assertof{3}. Then $P$ together with $\calD=\setof{M\prec\calH(\chi)}{\cardof{M}<\kappa,\,P,\kappa\in M}$ satisfies \assertof{4}. The proof of \assertof{4}\implies\assertof{1} is done by induction on $\nu=\cardof{P}\leq\lambda$. If $\nu\leq\kappa$, then $P$ has the $\kappa$-FN by \Lemmaof{baka}. For $\nu>\kappa$, let $P$ and $\calD$ be as in \assertof{4} and assume that \assertof{4}\implies\assertof{1} holds for every \po\ of cardinality $<\nu$. We need the following claims: \begin{Claim}\label{claim-1} Let $\chi^*$ be sufficiently large above $\chi$. Suppose that $M$ is an elementary submodel of $\calH(\chi^*)$ \st\ $P$, $\calH(\chi)$, $\calD\in M$, $\kappa+1\subseteq M$ and $[M]^{<\kappa}\cap M$ is cofinal in $[M]^{<\kappa}$ \wrt\ $\subseteq$. Then we have $P\cap M\leq_\kappa P$. \end{Claim} \prfofClaim Suppose not. then there is $b\in P$ \st\ either \medskip\\ \assert{a} $(P\cap M)\restr b$ has no cofinal subset of cardinality $<\kappa$\,; or \\ \assert{b} $(P\cap M)\uparrow b$ has no coinitial subset of cardinality $<\kappa$. \medskip\\ To be definite, let us assume that we have the case \assertof{a} --- for the case \assertof{b}, just the same argument will do. We can construct an increasing sequence $(N_\alpha)_{\alpha<\kappa}$ of elements of $\calD$ \st\medskip\\ \assert{c} $N_\alpha\in M$ and $\cardof{N_\alpha}<\kappa$ for $\alpha<\kappa$ (since $\kappa+1\subseteq M$, it follows that $N_\alpha\subseteq M$)\,;\\ \assert{d} $N_\alpha\in N_{\alpha+1}$ for every $\alpha<\kappa$\,;\\ \assert{e} $(P\cap N_\alpha)\restr b$ is not cofinal in $(P\cap N_{\alpha+1})\restr b$ for every $\alpha<\kappa$.\medskip\\ %% Then $N=\bigcup_{\alpha<\kappa}N_\alpha$ is $\calD$-approachable elementary submodel of $\calH(\chi)$ by \assertof{c} and \assertof{d}. Hence, by \assertof{4}, we have $P\cap M\leq_\kappa P$. But, by \assertof{e}, $(P\cap N)\restr b$ has no cofinal subset of cardinality $<\kappa$. This is a contradiction. To see that the construction of $N_\gamma$ is possible at a limit $\gamma<\kappa$, assume that $N_\alpha$, $\alpha<\gamma$ have been constructed in accordance with \assertof{c},\assertof{d} and \assertof{e}. Let $N'=\bigcup_{\alpha<\gamma}N_\alpha$. By \assertof{c}, we have $N'\subseteq M$ and $\cardof{N'}<\kappa$. Since $[M]^{<\kappa}\cap M$ is cofinal in $[M]^{<\kappa}$, there is some $N''\in M$ \st\ $N'\subseteq N''\subseteq \calH(\chi)$ and $\cardof{N''}<\kappa$. Hence by elementarity of $M$ and by $\calD\in M$ there is $N'''\in M\cap\calD$ \st\ $N''\subseteq N'''$. Clearly, we may let $N_\gamma=N'''$. For the construction at a successor step, assume that $N_\alpha$ have been chosen in accordance with \assertof{c},\assertof{d} and \assertof{e}. By assumption, there is $c\in (P\cap M)\restr b$ \st\ there is no $c'\in (P\cap N_\alpha)\restr b$ with $c\leq c'$. By elementarity of $M$ there is $N^*\in M\cap\calD$ \st\ $N_\alpha\cup\smallsetof{N_\alpha,c}\subseteq N^*$ and $\cardof{N^*}<\kappa$. Then $N_{\alpha+1}=N^*$ is as desired. \qedofClaim \begin{Claim}\label{claim-2} If $Q\leq_\kappa P$, then for every $\calD$-approachable $M\prec\calH(\chi)$ with $Q\in M$ we have $Q\cap M\leq_\kappa Q$. In particular, such $Q$ also satisfies the condition \assertof{4}. \end{Claim} \prfofClaim Let $M\prec\calH(\chi)$ be $\calD$-approachable with $Q\in M$. By assumption, we have $P\cap M\leq_\kappa P$. By elementarity of $M$ and since $Q\in M$, we have $Q\cap M\leq_\kappa P\cap M$. It follows that $Q\cap M\leq_\kappa P$ and hence $Q\cap M\leq_\kappa Q$. Now, let $\calD_0=\setof{M\in\calD}{Q\in M}$. Then it is clear that $Q$ satisfies the condition \assertof{4} with $\calD_0$ in place of $\calD$. \qedofClaim \medskip\\ Now we are ready to prove the induction steps.\\ {\bf Case I\,:} $\nu$ is a limit cardinal or $\nu=\mu^+$ with $\cf(\mu)\geq\kappa$. Let $\nu^*=\cf\nu$. Then, by \assertof{i}, we can find an increasing sequence $(M_\alpha)_{\alpha<\nu^*}$ of elementary submodels of $\calH(\chi^*)$ \st, for every $\alpha<\nu^*$, $\cardof{M_\alpha}<\nu$ and $M_\alpha$ satisfies the conditions in \Claimof{claim-1}; and $P\subseteq\bigcup_{\alpha<\nu^*}M_\alpha$. By \Claimof{claim-1}, $P\cap M_\alpha\leq_\kappa P$ for every $\alpha<\nu^*$. For $\alpha<\nu^*$ let \[ P_\alpha=\left\{\, \begin{array}{@{}l@{}l} P\cap M_\alpha,\quad&\mbox{if }\alpha\mbox{ is a successor},\\[\jot] P\cap(\bigcup_{\beta<\alpha}M_\beta),\quad&\mbox{otherwise}. \end{array}\right. \]\noindent Then $(P_\alpha)_{\alpha<\nu^*}$ is a continuously increasing sequence of sub-orderings of $P$ \st\ $\cardof{P_\alpha}<\nu$ for every $\alpha<\nu^*$ and $P=\bigcup_{\alpha<\nu^*}P_\alpha$. We have also $P_\alpha\leq_\kappa P$ for every $\alpha<\nu^*$: for a successor $\alpha<\nu^*$ this is clear. If a limit $\alpha<\nu^*$ has cofinality $<\kappa$ then $P_\alpha$ can be represented as the union of an increasing sequence of $<\kappa$ many $\kappa$-sub-ordering of $P$ and hence $P_\alpha\leq_\kappa P$. If $\alpha<\nu^*$ is a limit with cofinality $\geq\kappa$, then $P_\alpha=P\cap M$ where $M=\bigcup_{\beta<\alpha}M_\beta$. Now it is clear that $M$ satisfies the conditions in \Claimof{claim-1}. Hence we again obtain that $P_\alpha=P\cap M\leq_\kappa P$. Now, by \Claimof{claim-2}, each of $P_\alpha$, $\alpha<\nu^*$ satisfies the condition \assertof{4} and hence, by induction hypothesis, has the $\kappa$-FN. Thus, by \Lemmaof{conti-chain}, $P$ also has the $\kappa$-FN. \medskip\\ {\bf Case II\,:} $\nu=\mu^+$ with $\cf(\mu)<\kappa$. Let $\mu^*=\cf(\mu)$. \Wolog\ we may assume that the underlying set of $P$ is $\nu$. By \Thmof{Jensen-exists}, there is a $(\kappa,\mu)$-dominating matrix $(M_{\alpha,\beta})_{\alpha<\nu,\beta<\mu^*}$ over $(P,\calH(\chi))$. For $\alpha<\nu$ and $\beta<\mu^*$, let $P_{\alpha,\beta}=P\cap M_{\alpha,\beta}$ and $P_\alpha=\bigcup_{\beta<\mu^*}P_{\alpha,\beta}$. By \assertof{j4}, the sequence $(P_\alpha)_{\alpha<\nu}$ is continuously increasing and $\bigcup_{\alpha<\nu}P_\alpha=P$. $\cardof{P_\alpha}\leq\mu$ for every $\alpha<\nu$ by \assertof{j1}. \begin{Claim}\label{claim-4} $P_\alpha\leq_\kappa P$ for every $\alpha<\nu$. \end{Claim} \prfofClaim For $\alpha<\nu$ \st\ $\cf(\alpha)\geq\kappa$, we have $P_{\alpha,\beta}\leq_\kappa P$ for every sufficiently large $\beta<\mu^*$ by \assertof{j3} and \Claimof{claim-1}. Since $\mu^*<\kappa$, it follows that $P_\alpha\leq_\kappa P$. If $\cf(\alpha)<\kappa$, then, by the argument above, we have $P_{\alpha'}\leq_\kappa P$ for every $\alpha'<\alpha$ with $\cf(\alpha')\geq\kappa$. Since $P_\alpha$ can be represented as the union of $<\kappa$ many of such $P_{\alpha'}$'s, it follows again that $P_\alpha\leq_\kappa P$. \qedofClaim \medskip\\ Now, by \Claimof{claim-2}, each of $P_\alpha$, $\alpha<\nu$ satisfies the condition of \assertof{4}. Hence, by induction hypothesis, they have the $\kappa$-FN. By \Lemmaof{conti-chain}, it follows that $P$ also has the $\kappa$-FN.\qedofThm \qedskip\\ \begin{Cor}\label{appl-of-char} Suppose that $\kappa$ and $\lambda$ satisfy \assertof{i}, \assertof{ii} in \Thmof{correction} and $2^{<\kappa}=\kappa$. Then:\smallskip\\ \assert{a} Every $\kappa$-cc \cBa\ of cardinality $\leq\lambda$ has the $\kappa$-FN. \smallskip\\ \assert{b} For any $\mu$ \st\ $\mu^{<\kappa}\leq\lambda$, the \po\ $([\mu]^{<\kappa},\subseteq)$ has the $\kappa$-FN. \end{Cor} \prf Let $\chi$ be sufficiently large. For \assertof{a}, let $B$ be a $\kappa$-cc \cBa. We show that $B$ satisfies \assertof{3} in \Thmof{correction}. Let $M\prec\calH(\chi)$ be $V_\kappa$-like with $B$, $\kappa\in M$. By $2^{<\kappa}=\kappa$, \Lemmaof{easy} and by the remark after the lemma, we have $[M]^{<\kappa}\subseteq M$. Hence $B\cap M$ is a complete subalgebra of $B$. It follows that $B\cap M\leq_\kappa M$. For \assertof{b}, let $M\prec\calH(\chi)$ be $V_\kappa$-like with $\lambda$, $\kappa\in M$. Then as above we have $[M]^{<\kappa}\subseteq M$. Hence, letting $X=\lambda\cap M$, we have $[\lambda]^{<\kappa}\cap M=[X]^{<\kappa} \leq_\kappa [\lambda]^{<\kappa}$. \qedofCor \section{Chang's Conjecture for $\aleph_\omega$}\label{chang} Recall that $(\kappa,\lambda)\pfeil(\mu,\nu)$ is the following assertion: \begin{assertion}{}\it For any structure ${\cal A}=(A,U,\ldots)$ of countable signature with $\cardof{A}=\kappa$, $U\subseteq A$ and $\cardof{U}=\lambda$, there is an elementary substructure ${\cal A}'=(A',U',\ldots)$ of $\cal A$ \st\ $\cardof{A'}=\mu$ and $\cardof{U'}=\nu$. \end{assertion} In \cite{levinski}, a model of ZFC $+$ GCH $+$ Chang's Conjecture for $\aleph_\omega$, i.e.\ $(\aleph_{\omega+1},\aleph_\omega)\pfeil(\aleph_1,\aleph_0)$, is constructed starting from a model with a cardinal having a property slightly stronger than huge. The following theorem together with \Corof{appl-of-char} shows that the $\aleph_1$-FN of the \po\ $([\aleph_\omega]^{\aleph_0},\subseteq)$ is independent from ZFC (or even from ZFC $+$ GCH). \begin{Thm}\label{ccchang} Suppose that $(\aleph_\omega)^{\aleph_0}=\aleph_{\omega+1}$ and $(\aleph_{\omega+1},\aleph_\omega)\pfeil(\aleph_1,\aleph_0)$. Then $([\aleph_\omega]^{\aleph_0},\subseteq)$ does not have the $\aleph_1$-FN. \end{Thm} \prf \nc{\alo}{{\aleph_{\omega}}} \nc{\aloo}{{\aleph_{\omega+1}}} Assume to the contrary that there is an $\aleph_1$-FN mapping $\mapping{F}{[\alo]^{\aleph_0}}% {[[\alo]^{\aleph_0}]^{\aleph_0}}$. Let us fix an enumeration $(b_\alpha)_{\alpha<\aloo}$ of $[\alo]^{\aleph_0}$ and consider the structure: \[ \calA=(\aloo, \alo,\leq, E,f,g,h), \]\noindent where\medskip\\ \assert{1} $\leq$ is the canonical ordering on $\aloo$;\smallskip\\ \assert{2} $E=\setof{(\alpha,\beta)}{\alpha\in\alo,\beta\in\aloo,\alpha\in b_\beta}$; \smallskip\\ \assert{3} $\mapping{f}{\aloo\times\aloo}{\aloo}$ is \st, for each $\alpha\in\aloo$, \[ F(b_\alpha)=\setof{b_{f(\alpha,n)}}{n\in\omega}; \]\noindent \assert{4} $\mapping{g}{\aloo\times\aloo}{\alo}$ is \st, for each $\alpha\in\aloo$, $g(\alpha,\cdot)\restr\alpha$ is an injective mapping from $\alpha$ to $\alo$; and \smallskip\\ \assert{5} $\mapping{h}{\aloo\times\aloo\times\aloo}{\omega+1}$ is \st\ for each $\alpha$, $\beta\in\aleph_{\omega+1}$, $h(\alpha,\beta,\cdot)\restr(b_\alpha\cap b_\beta)$ is injective.\medskip\\ %% Note that, by \assertof{5} and since $\omega$ is definable in $\calA$, we can express ``$b_\alpha\cap b_\beta$ is finite'' as a formula $\varphi(\alpha,\beta)$ in the language of $\calA$. Now, applying the Chang's conjecture for $\calA$ with $A=\aleph_{\omega+1}$ and $U=\aleph_\omega$, we obtain elementary substructure $\calA'=(A',U',\leq',E',f',g',h')$ of $\calA$ \st\ $\cardof{A'}=\aleph_1$ and $\cardof{U'}=\aleph_0$. \begin{Claim}\label{claim=1} $otp(A')=\omega_1$. \end{Claim} \prfofClaim By \assertof{4} and elementarity of $\calA'$, every initial segment of $A'$ can be mapped into $U'$ injectively and hence countable. Since $\cardof{A'}=\aleph_1$, it follows that $otp(A')=\omega_1$. \qedofClaim \begin{Claim}\label{claim=2} For any $\alpha<\aloo$, there is $\gamma<\aloo$ \st\ $b_\beta\cap b_\gamma$ is finite for every $\beta<\alpha$. \end{Claim} \prfofClaim Since $\cardof{\alpha}\leq\alo$, we can find a partition $(I_n)_{n\in\omega}$ of $\alpha$ \st\ $\cardof{I_n}<\alo$ for every $n<\omega$. For $n<\omega$, let $\eta_n=\min(\alo\setminus\bigcup\setof{b_\xi}{\xi\in\bigcup_{m\leq n}I_m})$. Let $z=\setof{\eta_n}{n\in\omega}$ and $\gamma<\aloo$ be \st\ $b_\gamma=z$. For any $\beta<\alpha$, if $\beta\in I_{m_0}$ for some $m_0<\omega$, then we have $b_\beta\cap b_\gamma\subseteq\setof{\eta_n}{n\omega$ and every sufficiently large $n<\omega$. By induction hypothesis, it follows that $\forces{P}{\dotB_{\alpha,n}\xmbox{ has the }\aleph_1\xmbox{-FN}}$ for such $\alpha$ and $n$. By \Lemmaof{conti-chain} (applied in $V^P$) it follows that $\forces{P}{\dotB_\alpha\xmbox{ has the }\aleph_1\xmbox{-FN}}$ for every $\alpha<\nu$ with $\cf(\alpha)>\omega$. Hence again by \Lemmaof{conti-chain} and the remark after that (applied in $V^P$) we obtain that $\forces{P}{\dotB\xmbox{ has the }\aleph_1\xmbox{-FN}}$. \qedofThm \begin{Cor} Suppose that $V=L$ holds and let $G$ be $V$-generic over $P=\Fn(\tau,2)$ for some $\tau$. Then (among others) the following ccc \cBas\ have the $\aleph_1$-FN: $\calC_\kappa$ ($\cong$ the completion of $\Fn(\kappa,2)$) for any $\kappa$; $\psof{\omega}$ (hence also $\psof{\omega}/{\rm fin}$); $Borel\/(\reals)/\mbox{Null-sets}$.\qed \end{Cor} In connection with \Thmof{cohen-model-rev}, we would like to mention the following open problems: %% %% \begin{Problem} Assume that $V[G]$ is a model obtained by adding Cohen reals to a model of $\ZFC +\CH$. Is it true that $\powersetof{\omega}$ has the $\aleph_1$-FN in $V[G]$\,? \end{Problem} By \Thmof{cohen-model-rev}, the answer to this question is positive if the number of added Cohen reals is less than $\aleph_\omega$. \begin{Problem} Does \Thmof{cohen-model-rev} hold also without the assumption of tameness\,? \end{Problem} Or, more generally: \begin{Problem} Are the following equivalent?\\ \assert{i} $\powersetof{\omega}$ has the $\aleph_1$-FN;\\ \assert{ii} every ccc \cBa\ has the $\aleph_1$-FN. \end{Problem} %% %% \section{Lusin gap}\label{lusin} For a regular $\kappa$, an almost disjoint family $\calA\subseteq[\omega]^{\aleph_0}$ is said to be a {\em $\kappa$-Lusin gap} if $\cardof{\calA}=\kappa$ and for any $x\in[\omega]^{\aleph_0}$ either $\cardof{\setof{a\in\calA}{\cardof{a\setminus x}<\aleph_0}}<\kappa$ or $\cardof{\setof{a\in\calA}{\cardof{a\cap x}<\aleph_0}}<\kappa$. \begin{Thm}\label{lusin-gap} Assume that $\psof{\omega}$ has the $\aleph_1$-FN. Then there is no $\aleph_2$-Lusin gap. \end{Thm} \prf Let $\mapping{f}{\psof{\omega}}{[\psof{\omega}]^{\aleph_0}}$ be an $\aleph_1$-FN mapping. We may assume that $f(a)=f(b)=f(\omega\setminus b)$ for all $a$, $b\in\psof{\omega}$ with $a=^*b$. Thus $x\subseteq^*y$ implies that there is $z\in f(x)\cap f(y)$ \st\ $x\subseteq^* z\subseteq^* y$ and $\cardof{x\cap y}<\aleph_0$ implies that there is $z\in f(x)\cap f(y)$ \st\ $x\subseteq^* z$ and $\cardof{z\cap y}<\aleph_0$. Suppose that $\calA=\setof{a_\alpha}{\alpha<\omega_2}$ is an almost disjoint family of subsets of $\omega$. We show that $\calA$ is not an $\aleph_2$-Lusin gap. Let $\chi$ be sufficiently large regular cardinal and consider the model $\calH=(\calH(\chi),\in,\unlhd)$ where $\unlhd$ is any well-ordering on $\calH$. Let $N$ be an elementary submodel of $\calH$ \st\ $\calA$, $f\in N$, $N\cap\omega_2\in\omega_2$ and $\cf(\delta)=\omega_1$ for $\delta=N\cap\omega_2$. For $\alpha\in N$ we have $\cardof{a_\alpha\cap a_\delta}<\aleph_0$ and hence $a_\alpha\subseteq^*(\omega\setminus a_\delta)$. Thus there is $b_\alpha\in f(a_\alpha)\cap f(a_\delta)$ \st\ $a_\alpha\subseteq^*b_\alpha\subseteq^*(\omega\setminus a_\delta)$. Since $f(a_\delta)$ is countable and $\cf(\delta)=\omega_1$ there is $b\in f(a_\delta)$ \st\ $I=\setof{\alpha<\delta}{b_\alpha=b}$ is cofinal in $\delta$. We show that $b$ witnesses that $\calA$ is not an $\aleph_2$-Lusin gap, i.e., $J=\setof{\alpha<\omega_2}{a_\alpha\subseteq^* b}$ and $K=\setof{\alpha<\omega_2}{\cardof{b\cap a_\alpha}<\aleph_0}$ both have cardinality $\aleph_2$. \begin{Claim} $\cardof{J}=\aleph_2$. \end{Claim} \prfofClaim First note that $b\in N$ since $b\in f(a_\alpha)$ for any $\alpha\in I\subseteq N$. Hence we have $J\in N$ and $I\subseteq J$. Since $I$ is cofinal in $N\cap\omega_2$, we have $N\models{J\mbox{ is cofinal in }\omega_2}$. By elementarity it follows that $\calH\models{J\mbox{ is cofinal in }\omega_2}$. Hence $J$ is really cofinal in $\omega_2$. \qedofClaim \begin{Claim} $\cardof{K}=\aleph_2$. \end{Claim} \prfofClaim Since $b\in N$ it follows that $K\in N$. For $\beta\in N\cap\omega_2=\delta$, we have $\calH\models{\delta\in K\wedge\beta<\delta}$. Hence $\calH\models{K\not\subseteq\beta}$ and $N\models{K\not\subseteq\beta}$ by elementarity. It follows that $N\models{K\xmbox{ is unbounded in }\omega_2}$. Hence, again by elementarity, $\calH\models{K\xmbox{ is unbounded in }\omega_2}$. Thus $K$ is really unbounded in $\omega_2$. \qedofClaim\\ \qedofThm \begin{Cor} ${\bf b}=\aleph_1$ or even the statement ``\,$\psof{\omega}$ does not contain any strictly increasing $\subseteq^*$-chain of length $\omega_2$'' does not imply that $\psof{\omega}$ has the $\aleph_1$-FN. \end{Cor} \prf Suppose that our ground model $V$ satisfies the CH. Koppelberg and Shelah \cite{koppelberg-shelah} proved that the forcing with $\Fn(\omega_2,2)$ can be represented as a two step iteration $A*\dot{B}$ where $\forces{A}{\psof{\omega}\xmbox{ contains an }\aleph_2\xmbox{-Lusin gap}}$. Thus, by \Thmof{lusin-gap}, we have $\forces{A}{\psof{\omega}\xmbox{ does not habe the }\aleph_1\xmbox{-WF}}$. On the other hand, we have $\forces{A*\dot{B}}{\psof{\omega}\xmbox{ does have the }\aleph_1\xmbox{-WF}}$ by \Thmof{cohen-model-rev}. Hence $\forces{A*\dot{B}}{% \xmbox{there is no strictly increasing }\subseteq^* \xmbox{-chain in }\psof{\omega}\xmbox{ of length }\omega_2}$. 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