  %\documentclass[11pt, amsfonts,amssymb, amsmath]{amsart}
\documentclass{article}

\usepackage{times}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amscd}
\usepackage{url}

\def\mcolon{\! : \!}

\newcommand{\qed}{\hfill\mbox{$\Box$}}
\newfont{\msbm}{msbm10 scaled\magstephalf}
\sloppy
\def\ind #1#2#3{#1 \mathbin{\mathop{\downarrow}_{#2}} #3}
\def\LS{\rm LS}
\newcommand\myrestriction{\mathord\restriction}
\def\gaS{\rm ga-S}
\def\aut{\rm aut}
\def\TV{\rm TV}
\def\otp{{\rm otp}}
\def\cf{\mathop{\rm cf}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{convention}[theorem]{Convention}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{lemma2}[theorem]{Lemma*}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{definition2}[theorem]{Definition*}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{observation}[theorem]{Observation}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}[theorem]{Example}
\newtheorem{example2}[theorem]{Example*}
\newtheorem{construction}[theorem]{Construction}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{question}[theorem]{Question}
%variants for chapters with sections
\newtheorem{stheorem}{Theorem}[section]
\newtheorem{scorollary}[stheorem]{Corollary}
\newtheorem{sproposition}[stheorem]{Proposition}
\newtheorem{slemma}[stheorem]{Lemma}
\newtheorem{sdefinition}[stheorem]{Definition}
\newtheorem{sclaim}[stheorem]{Claim}
\newtheorem{sfact}[stheorem]{Fact}
\newtheorem{sremark}[stheorem]{Remark}
\newtheorem{squestion}[stheorem]{Question}
\newtheorem{sassumption}[stheorem]{Assumption}
\newtheorem{sobservation}[stheorem]{Observation}
\newtheorem{snotation}[stheorem]{Notation}
\newtheorem{sconjecture}[stheorem]{Conjecture}
\newtheorem{sexample}[stheorem]{Example}
\newtheorem{sexercise}[stheorem]{Exercise}
\newtheorem{sapp}[stheorem]{Applications}
%\def\beth{]}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\def\indd #1#2#3{#1 \mathbin{\mathop{\down}^d_{#2}} #3}
\def\bar#1{\overline{#1}}
\def\implies{\rightarrow}
\def\td{d_f}
\def\betabar{{\boldmath\beta}}
\def\alphabar{{\boldmath\alpha}}
\def\gammabar{{\boldmath\gamma}}
\def\b1K{\mbox{\boldmath $K$}_{-1}}
\def\bK{\mbox{\boldmath $K$}}
\def\boK{\mbox{\boldmath $K$}_{0}}
\def\Mscr{{\MM }}
\def\Rscr{{\cal R }}
\def\Cscr{{\cal C }}
\def\Escr{{\cal E }}
\def\Ascr{{\mathcal A }}
\def\Tscr{{\mathcal T }}
\def\Aind{A_{\rm ind}}
\def\bcf{\mbox{\boldmath $c$}_f}
\def\bdf{\mbox{\boldmath $d$}_f}
\def\bc{\mbox{\boldmath $c$}}
\def\bd{\mbox{\boldmath $d$}}
\def\bP{\mbox{\boldmath $P$}}
\def\bbK{\mbox{\boldmath ${\overline { K}}$}_{-1}}
\def\bboK{\mbox{\boldmath ${\overline { K}}$}_{0}}
\def\subi{\leq_{i}}
\def\Th{\mathop{\rm Th}}
\def\th{\mathop{\rm Th}}
\def\Ex{\mathop{\rm Ex}}
\def\bD{\mathop{\bf D}}
\def\stp{\mathop{\rm stp}}
\def\span{\mathop{\rm span}}
\def\hqt{\rm ht}
\def\at{\rm at}
\def\icl{\rm icl}
\def\diag{\rm Diag}
\def\rest{\mid}
\def\Nbar{\overline{N}}
\def\Mbar{\overline{M}}
\def\Kbar{\overline{K}}
 \def\dom{\mathop{\rm dom}}
\def\join{\vee}
\def\meet{\wedge}
\def\kappabar{\overline{\kappa}}
\def\grp{{\rm grp}}
\def\span{{\rm span}}
\def\spec{{\rm spec}}
\def\Spec{{\rm Spec}}
\def\fld{{\rm fld}}
\def\dcl{{\rm dcl}}
\def\rcl{{\rm rcl}}
\def\acl{{\rm acl}}
\def\cl{{\rm cl}}
\def\pr{{\rm pr}}
\def\st{{\rm st}}
\def\gp{{\rm gp}}
\def\ccl{{\rm ccl}}
\def\acf{{\rm acf}}
\def\str{{\rm str}}
%\def\tcl{{\rm tcl}}
\def\tcl{{\rm acf}}
\newbox\noforkbox \newdimen\forklinewidth
\forklinewidth=0.3pt \setbox0\hbox{$\textstyle\smile$}
\setbox1\hbox to \wd0{\hfil\vrule width \forklinewidth depth-2pt
 height 10pt \hfil}
\wd1=0 cm \setbox\noforkbox\hbox{\lower 2pt\box1\lower
2pt\box0\relax}
\def\unionstick{\mathop{\copy\noforkbox}\limits}
\def\nonfork_#1{\unionstick_{\textstyle #1}}
\def\nonfork_#1{\unionstick_{\textstyle #1}}
\setbox0\hbox{$\textstyle\smile$} \setbox1\hbox to \wd0{\hfil{\sl
/\/}\hfil} \setbox2\hbox to \wd0{\hfil\vrule height 10pt depth
-2pt width
               \forklinewidth\hfil}
\wd1=0 cm \wd2=0 cm
\newbox\doesforkbox
\setbox\doesforkbox\hbox{\lower 2pt\box1 \lower
2pt\box2\lower2pt\box0\relax}
\def\nunionstick{\mathop{\copy\doesforkbox}\limits}

\def\add{}
\def\LS{\rm LS}
\def\gatp{\rm ga-tp}
%\def\gaS{\rm ga-S}
\def\aut{\rm aut}
\def\Lscr{\mathcal L}
\newcommand{\GG}{\mbox{\msbm G}}
\newcommand{\TT}{\mbox{\msbm T}}
\newcommand{\ZZ}{\mbox{\msbm Z}}
\newcommand{\QQ}{\mbox{\msbm Q}}
\newcommand{\NN}{\mbox{\msbm N}}
\newcommand{\MM}{\mbox{\msbm M}}
\newcommand{\CC}{\mbox{\msbm C}}
\newcommand{\Fscr}{\mbox{\msbm F}}
\def\subm{\prec_{\bK}}
\def\sub'm{\prec_{\bK'}}
\def\grpf #1 #2{{\rm grp}_{#2}(#1)}
\def\spanf #1 #2{{\rm span}_{#2}(#1)}
\def\fldf #1 #2{{\rm fld}_{#2}(#1)}
\def\dclf #1 #2{{\rm dcl}_{#2}(#1)}
\def\rclf #1 #2{{\rm rcl}_{#2}(#1)}
\def\aclf #1 #2{{\rm acl}_{#2}(#1)}
\def\acff #1 #2{{\rm acf}_{#2}(#1)}
\def\strf #1 #2{{\rm str}_{#2}(#1)}
%\def\tclf #1 #2{{\rm tcl}_{#2}(#1)}
\def\tclf #1 #2{{\rm acf}_{#2}(#1)}
\def\abar{\mbox{\boldmath $a$}}
\def\bbar{{\bf b}}
\def\cbar{{\bf c}}
\def\dbar{{\bf d}}
\def\ebar{{\bf e}}
\def\fbar{{\bf f}}
\def\gbar{{\bf g}}
\def\hbar{{\bf h}}
\def\ibar{{\bf i}}
\def\jbar{{\bf j}}
\def\kbar{{\bf k}}
\def\lbar{{\bf l}}
\def\mbar{{\bf m}}
\def\nbar{{\bf n}}
\def\obar{{\bf o}}
\def\pbar{{\bf p}}
\def\qbar{{\bf q}}
\def\rbar{{\bf r}}
\def\sbar{{\bf s}}
\def\tbar{{\bf t}}
\def\ubar{{\bf u}}
\def\vbar{{\bf v}}
\def\wbar{{\bf w}}
\def\xbar{{\bf x}}
\def\ybar{{\bf y}}
\def\zbar{{\bf z}}
 \def\restrict{|}
\def\tp{{\rm tp }}
\def\Qbar{\overline  Q}
\def\bG{\mbox{\boldmath $G$}}
\def\bk{\mbox{\boldmath $k$}}
\def\bH{\mbox{\boldmath $H$}}
\def\bI{\mbox{\boldmath $I$}}
\def\bR{\mbox{\boldmath $R$}}
\def\bT{\mbox{\boldmath $T$}}
\def\bF{\mbox{\boldmath $F$}}
\def\bZ{\mbox{\boldmath $Z$}}
\def\bt{\mbox{\boldmath $t$}}
\def\Nperp{\mbox{$^{\perp}N$}}
\def\subN{{\prec_N}}
\def\Cperp{\mbox{$^{\perp}{\mathcal C}$}}
\def\subC{{\prec_{\mathcal C}}}
\def\Ext{\mbox{{\rm Ext}}}
\def\tc{\mbox{{\rm tc}}}
\def\Hom{\mbox{{\rm Hom}}}
\def\bpi{\mbox{\boldmath $\pi$}}
% \sidebar{..} puts .. in comment form
\newcommand{\sidebar}[1]{\vskip10pt\noindent
 \hskip.70truein\vrule width2.0pt\hskip.5em
 \vbox{\hsize= 4truein\noindent\footnotesize\relax #1 }\vskip10pt\noindent}
\date{\today}
\begin{document}


\title{A Hanf number for saturation and omission}

\author{ John T. Baldwin\thanks{We give special thanks to the Mittag-Leffler Institute where this research was
conducted. This is paper 958    in Shelah's bibliography.  Baldwin was partially supported by
NSF-0500841.  Shelah thanks the Binational Science Foundation for partial
support of this research.}
\\University of Illinois at Chicago\\ \and Saharon
Shelah\\
Hebrew University of Jerusalem\\Rutgers University}

\maketitle

\begin{abstract} Suppose $\bt =(T,T_1,p)$ is a
triple of two countable theories $T\subseteq T_1$ in vocabularies
$\tau \subset \tau_1$  and   a $\tau_1$-type $p$ over the empty set.
We show the Hanf number for the property: There is a model  $M_1$  of
$T_1$ which omits $p$, but $M_1 \restriction \tau$ is saturated is
essentially equal to the L\"owenheim number of second order logic.
In Section~\ref{exact} we make exact computations of these Hanf
numbers and note some distinctions between `first order' and `second
order quantification'.  In particular, we show that if $\kappa$ is
uncountable,
 $h^3(L_{\omega,\omega}(Q), \kappa) = h^3(L_{\omega_1,\omega}, \kappa)$, where $h^3$ is the `normal' notion
 of Hanf function (Definition~\ref{hndef}.)
\end{abstract}


Newelski asked  in \cite{Newelskihanf} whether it is possible to calculate the Hanf number of
the following property $P_N$.  In a sense made precise in Theorem~\ref{big}, we show the answer is no.
In accordance with the original question, we focus on
countable vocabularies for the first three sections. We deal with extensions to larger vocabularies in
Section~\ref{exact}.

\begin{definition}\label{notation}\label{keynot}  We say $M_1 \models \bt$ where   $\bt =(T,T_1,p)$ is a
triple of two %countable
theories in vocabularies $\tau \subset
\tau_1$, respectively, $T \subseteq T_1$ and $p$ is a $\tau_1$-type over the empty set if $M_1$ is
a model of $T_1$ which omits $p$, but $M_1 \restriction \tau$ is
saturated. Let $\bK_{\bt}$ denote the class of models $M_1$ which satisfy $\bt$.

For $\bK = \bK_{\bt}$, with $\bt$ in a countable vocabulary,
%with
%cardinality ${\kappa}$,
let $P^c_N(\bK_{\bt},\lambda)$ hold
if $|\tau_1| \leq \aleph_0$ and for some $M_1$ with $|M_1|=\lambda$,
$M_1 \models \bt$.
%If $\bt$ is in a countable vocabulary, we
%Finally,
 $P^{f}_N(\bK_{\bt},\lambda)$ is the same property restricted to
triples where $T_1$ and $T$ are finitely axiomatizable in finite
vocabularies and $p$ is definable in second order logic.



\end{definition}



%Fix a {\em set} of classes $\bK$ of
%a given kind
%(e.g. the classes of models defined by sentences of
%$L_{\mu,\nu}$ for some fixed $\mu, \nu$ of a given similarity
%type).
Recall Hanf's observation \cite{Hanf} that for any such  property
$P(\bK,\lambda)$, where $\bK$  ranges over a {\em set} of
classes of models, there is a cardinal $\kappa =H(P)$ such that $\kappa$ is the least cardinal satisfying:
if $P(\bK,\lambda)$ holds for some $\lambda\geq \kappa$ then
$P(\bK,\lambda)$ holds for arbitrarily large $\lambda$. $H(P)$ is
called the Hanf number of $P$. E.g. $P(\bK,\lambda)$ might be the
property that $\bK$ has a model of power $\lambda$. Similarly the
L\"owenheim number $\ell(P)$ of a set $P$  of %non-empty
classes is the least
cardinal $\mu$ such that any class $\bK \in P$ that has a model
has one of cardinality $\leq \mu$.

\begin{theorem}\label{big} Assume the collection of $\lambda$ with $\lambda^{<\lambda} = \lambda$ is a proper class.
$H(P^f_N) = \ell(L^{II})$ where $L^{II}$ denotes
the collection of sentences of
second order logic.
\end{theorem}

Since $H(P^c_N) \geq H(P^f_N)$, this shows that the Hanf
number in the abstract is at least $\ell(L^{II})$, as asserted.
In Section~\ref{II} we introduce a variant $\ell^2(L^{II})$ on the L\"owenheim number of second order logic
which is `essentially   equal' to $\ell(L^{II})$ (i.e.$\!$ equal modulo a mild set theoretic hypothesis:
Assumption~\ref{assumpt}). It is fairly easy to show (Claim~\ref{iiltnew}) $\ell^2(L^{II}) \geq H(P^f_N)$ giving the
`essentially equal' of the abstract.  We will replace this `essential equality' with an exact computation and deal with
uncountable languages in  Section~\ref{exact}.




Jouko Vaananen provided the following summary of the effect of
this result by indicating the size of $\ell(L^{II})$.
$\ell(L^{II})$ is bigger than the first (second, third, etc) fixed
point of any normal function on cardinals that itself can be
described in second order logic. For example it is bigger than the
first $\kappa$ such that $\kappa=\beth_\kappa$, bigger than the
first $\kappa$ such that there are $\kappa$ cardinals $\lambda$
below $\kappa$ such that $\lambda=\beth_\lambda$, etc. It is easy
to see that if there are measurable (inaccessible, Mahlo, weakly
compact, Ramsey, huge) cardinals, then the Lowenheim number of second
order logic exceeds the first of them (respectively, the first
inaccessible, Mahlo, weakly compact, Ramsey, huge) (and second, third,
etc). So even under $V=L$, the L\"owenheim number is bigger than
any `large' cardinal that is second order definable and consistent
with $V=L$. Such results are discussed in Vaananen's paper ``Hanf
numbers of unbounded logics''\cite{Vanhanf}. A result of Magidor
\cite{Magidor} shows the Lowenheim number of
 second order logic is always below the first supercompact. Vaananen's paper ``Abstract logic and
 set theory II:
Large cardinals'' gives lower bounds for the Lowenheim number of
equicardinality quantifiers and thus {\em a fortiori} for second
order logic \cite{Vanlow}. In simple terms, if $E(\kappa)$ is the
statement that $2^\kappa\geq \kappa^{++}$ then the first $\kappa$
cardinals (if any) such that $E(\kappa)$ holds is less than the
Lowenheim number of second order logic. This shows that by forcing
we can push the Lowenheim number up at will.


{\bf We make the following assumption throughout.}
\begin{assumption} \label{assumpt} Assume the collection of $\lambda$ with $\lambda^{<\lambda} = \lambda$ is a proper
class.
\end{assumption}

This assumption follows from GCH, but if GCH fails badly the only such cardinals are
strongly inaccessible.
%In our discussion of the context for Newelski's problem we
%will argue for the appropriateness of this assumption for answering it.
The key point for our use of the condition is that
$\lambda^{<\lambda} = \lambda >|\tau_T| +\aleph_0$ is a sufficient condition for the
existence of a saturated model in $\lambda$ of a complete theory $T$; if $T$ is unstable $\lambda^{<\lambda} = \lambda$
is also necessary.   We will explore this issue for stable theories, in the absence of Assumption~\ref{assumpt},
elsewhere.  In Section~\ref{II}
we review some properties of second order logic and show the
equality of two `L\"owenheim numbers'
%; this equality demonstrates
%the assumption is harmless
in our context.  In
Section~\ref{concl}, we state two technical results, prove one,
and deduce Theorem~\ref{big} from them.  In Section~\ref{tech}, we
prove the more difficult technical result.
In Section~\ref{exact}, we code syntax more carefully and  obtain a uniform equivalence for
vocabularies of all cardinalities. %And in Section~\ref{context}, we discuss the origin of

Newelski's question arose in the study of the model theory of
groups and the existence of groups with bounded orbits.
%However the
%same property $P(N)$ has also appeared in the study of Peano
%arithmetic \cite{EngKaye}.
%with dependent theories (without NIP)
%and its connections to fundamental issues about the existence of monster models.
The authors acknowledge very fruitful discussions with Jouko V\"a\"an\"anen and Tapani Hyttinen concerning the material
of this paper.  We also thank the referee for an unusually detailed and helpful series of reports.


\section{Some Second Order Logic}\label{II}


By (pure) second order logic, $L^{II}$, we mean the logic with
individual variables and variables for relations of all arities but
no non-logical constants. The atomic formulas are equalities between
variables and expressions $X(\xbar)$ where $X$ is an $n$-ary relation
variable and $\xbar$ is an $n$-tuple of individual variables.  Note
that a structure $A$ for this logic is simply a set so is determined
entirely by its cardinality.  But we use the full semantics; the
$n$-ary relation variables range over all $n$-ary relations on $A$.


%binary relation symbols for $x \in X$ and for equality between %elements and between sets.  $L^{II}$ denotes the collection of %finite formulas in that language.

We explain the connection of our restriction to $\lambda$ with $\lambda=\lambda^{<\lambda}$ to the computation of some
variants on the Hanf and L\"owenheim numbers.. In general for any class $\bK$ of models write
$\spec(\bK)$ for the collection of $\lambda$ such that there is a
model in $\bK$ with cardinality $\lambda$.
%We describe some
%technical variants for the second order case that are relevant
%here.



\begin{definition} Let $\psi$ be a sentence of  second order logic.
\begin{enumerate}
\item $\spec^1(\psi) = \{\lambda: \lambda \models \psi\}$.
\item $\spec^2(\psi) = \{\lambda:\lambda=\lambda^{<\lambda} \wedge \lambda \models \psi\}$.
\end{enumerate}
\end{definition}

Note that there is a sentence $\chi$ in second order logic which
has a model of size $\lambda$ if and only if
$\lambda^{<\lambda}=\lambda$.  Namely, let $\chi$ assert there is
an extensional relation $R$ on sets such that each element
denotes, via $R$, a set of smaller cardinality than the universe
and each such set is coded by $R$. We will generally write
$\lambda^{<\lambda}=\lambda$ to denote this sentence.


%Recall we assume there are proper class of $\lambda$ with


\begin{definition} Define $H^2$ and $\ell^2$ to  be the Hanf and Lowenheim numbers with respect to $\spec^2$.
\end{definition}


We'll write $\ell^1$ for $\ell$ and $H^1$ for $H$ where it is convenient for comparison.
Note the following easy transformations in second order logic.

\begin{fact}\label{f1}
 Fix $\phi \in L^{II}$.
\begin{enumerate}

\item $\spec^1(\phi)$:
\begin{enumerate}

\item
 There is a $\phi_1 \in L^{II}$ with $\min (\spec(\phi) )<
\min (\spec(\phi_1) )$.
\item If $\spec(\phi)$ is bounded and nonempty there is a $\phi_2 \in L^{II}$ with $\spec(\phi_2)$ bounded and
    nonempty and
$\sup (\spec(\phi) )< \sup (\spec(\phi_2) )$.

\end{enumerate}
\item $\spec^2(\phi)$:
\begin{enumerate}


\item  There is a $\phi_3 \in L^{II}$ with $\min (\spec(\phi) )< \min (\spec(\phi_3) )$ and if $\lambda \models
    \phi_3$, $\lambda^{<\lambda}=\lambda$.

\item  If $\spec^2(\phi)$ is bounded and nonempty there is a $\phi_4 \in L^{II}$ with $\spec^2(\phi_4)$ bounded
    and nonempty and
$\sup (\spec^2(\phi_4) )< \sup (\spec^2(\phi_4) )$.

% there is a $\phi^\dag \in L^{II}$ with $\min (\spec(\phi) )< \min (\spec(\phi^\dag) )$ and if $\lambda \models \phi^\dag$, $\lambda^{<\lambda}=\lambda$.
\end{enumerate}

\end{enumerate}
\end{fact}

These transformations imply:

\begin{fact}\label{f2}
\begin{enumerate}
\item
$H^1(L^{II}), H^2(L^{II}), \ell^1(L^{II}),  \ell^2(L^{II})$ are strong limit cardinals.
\item There is no sentence attaining any of these values exactly. (E.g., there is no  $\phi \in L^{II}$
with $\sup(\spec(\phi)) = H^1(L^{II})$.)
\item For either
spectrum, $\ell^i(L^{II}) = \sup\{\min\{\spec^i(\phi)\}\mcolon \phi
\in L^{II} \hbox{ has a model}\}$ and similarly $H^i(L^{II}) = \sup\{\sup\{\spec^i(\phi)\}\mcolon \phi
\in L^{II} \hbox{ is bounded}\}$.
\end{enumerate}
\end{fact}

Note that any logic satisfying Fact~\ref{f1} will also satisfy Fact~\ref{f2}. We use this observation without comment in
studying infinitary second order logics in Section~\ref{exact}.

Using Assumption~\ref{assumpt} we can show:

%\sidebar{I changed this as you wrote on July 4. But I don't see the relevance of the assumption to the new  conclusion.
%For any $\theta$ there is a $\theta'$ saying each model %is in the successor cardinal of model of $\theta$ and a
%$\theta''$ saying a model
%%is the same size as the power set of a model of $\theta$.
%This seems to imply the strong limit conclusion without the Assumption .3}

\begin{lemma}\label{eqlow}  $H(L^{II}) = H^2(L^{II})$, $\ell(L^{II}) = \ell^2(L^{II})$
%.and these cardinals are strong limit cardinals.
\end{lemma}

Proof. %We prove the L\'owenheim number assertion.
One direction is
easy. For every sentence $\psi$ of second order logic, there is a
sentence $\psi^*$ such that:
$$\spec^2(\psi) = \spec^1(\psi^*).$$  $\psi^*$ just expresses the conjunction of $\psi$
with $\lambda^{<\lambda} = \lambda$. Recall
Fact~\ref{f2}.3
Since every 2-spectrum is a 1-spectrum $\ell^2(L^{II})\leq \ell^1(L^{II})$ and $H^2(L^{II})\leq H^1(L^{II})$.

But the opposite inequality also holds. Let $ \phi$ be a sentence
with a non-empty 2-spectrum.
Let $f(\lambda)$ denote the least $\mu > \lambda$ with $\mu^{<\mu} =\mu$.  It is easy to construct for each second order
sentence $\phi$ a sentence $\phi^*$ such that $$\spec(\phi^*) =\spec^2(\phi^*)=\{f(\lambda): \lambda \in
\spec(\phi)\}.$$
Clearly the map $\phi \mapsto \phi^*$ shows
 $\ell^2(L^{II})\geq \ell^1(L^{II})$ and $H^2(L^{II})\geq H^1(L^{II})$.
% For example if $\phi \in L^{II}$ has a bounded spectrum, by Fact~\ref{f2}, the bound is less than
  %$H^1(L^{II})$ and the spectrum of $\phi^*$ is also bounded by $H^1(L^{II})$ so

$\qed_{\ref{eqlow}}$



%Then let $\hat \phi$ express $(\exists
%U) (\phi^U \wedge \lambda^{<\lambda} = \lambda)$. Now $\hat %\phi$ has
%a non-empty 2-spectrum so it has a model $M$ with cardinality
%$|M|= \kappa \leq \ell^2(L^{II})$. But then $\phi$ has a %model
%with cardinality $|U^M| \leq \kappa \leq \ell^2(L^{II})$.  So
%$\ell^1(L^{II})\leq \ell^2(L^{II})$.




% similar argument works for Hanf numbers.
%$\qed_{\ref{eqlow}}$


\section{The main result}\label{concl}


 Recall our
notation from Definition~\ref{notation}. % more specific.


\begin{notation}\label{deftrip}%\begin{enumerate}
 We will write $\bt$ (possibly with subscripts) for a triple $(T,T_1,p)$.
The expression `$\bt$ has a model in $\lambda$' means there is a
model of $T_1$ with cardinality $\lambda$ that omits $p$ and whose
reduct to $L(T)=\tau$ is saturated.

\end{notation}


\begin{convention}\label{fix} When $\tau_1$ is finite we consider it to be a subset of $\omega$. Thus the set
of first order $\tau_1$-sentences is recursive and we can code them as natural numbers.
\end{convention}




We concentrate first on  $P^{f}_N(\bK_{\bt},\lambda)$ from
Definition~\ref{notation}.  We need some additional coding to
handle non-finitely axiomatizable theories and consider this generalization in Section~\ref{exact}.
We begin by clarifying a notion from Definition~\ref{keynot}.

\begin{definition}
A type $p$ in a vocabulary satisfying Convention~\ref{fix} is {\em definable in second order logic}  if we can code the
type as a subset $A_p$ of $\omega$ so
that  in the vocabulary with constant symbol $0$ and  relation symbol $S$
there is a second order sentence  $\psi$
 and second order formula $\varphi (x)$ satisfying the following condition.
For the first cardinal $ \lambda $ which satisfies $ \psi $
if $M$ is
$ ( \lambda , 0, S)$, with $0$  interpreted
as $0$ and $S$ as successor on the natural numbers, then
 $A_p= \{ n\mcolon M \models \varphi (n) \} $.
\end{definition}

Now for convenience we restrict our triples to those satisfying the convention. Formally:


\begin{notation} $\Tscr^f$ denotes the set of triples $\bt$ such that
 $T$ is finitely axiomatizable and $p$ is second order definable.
\end{notation}



\begin{theorem}\label{IInew1}  For every second order sentence $\phi$, there is a triple $\bt_\phi\in \Tscr^f$ such that
if $\lambda^{<\lambda} =
\lambda$, then the following are equivalent:

\begin{enumerate}
\item $\bt_\phi$ has a model in $\lambda$.

\item $\phi$ has a model in every cardinal strictly less than $\lambda$.

    \end{enumerate}
% and

 %if $\phi$ is true in all $\mu \leq \lambda$ and %$\lambda^{<\lambda} = \lambda$  then $\lambda %\models\bt_\phi$.
 \end{theorem}
 \medskip

 We prove Theorem~\ref{IInew1} in Section~\ref{tech}.


 %if $\phi$ is true in all $\mu \leq \lambda$ and %$\lambda^{<\lambda} = \lambda$  then $\lambda %\models\bt_\phi$.
 %\end{theorem}


% A variant of this result can
%be proved for countable vocabularies.



 \begin{lemma}\label{newtoii}  For every  $\bt\in \Tscr^f$ %with finitely axiomatizable $T_1$ and
 %second order definable $p$,
 there is a second order
  $\phi_{\bt}$,  such that $\phi_{\bt}$  has a model in $\lambda$ if and only if $\bt$ has a model in $\lambda$.

    \end{lemma}

%\sidebar{This seems slightly less general than needed because %of the restriction finitely axiomatized $T$. This is minor and %will be fixed.}
Proof.
%Since $T_1$ is finitely axiomatizable,
 Recalling the restrictions involved in $\Tscr^f$, it is easy to write a second order sentence $\theta$ such that $M
 \models \theta$ if and only if $M\models T_1$, $M$ omits $p$ and $M\myrestriction \tau$ is saturated.
$\qed_{\ref{newtoii}}$

%\sidebar{In considering Section 4, I realized that the last sentence of the previous proof is false as stated.
%For, on the left side of the equivalence $M$ is a structure for pure equality and on the right it is a $\tau_1$-structure.  The correction would be to replace the rhs by:
%
%$$M \models (\exists \tau_1)(\exists N) [|N|=|M| \wedge $$
% \hbox{\rm \  `$N$  is a $\tau_1$-structure and   $N$ omits $p$ and $N\restriction \tau$ is saturated.
% '$]$\ }
%      }

%\begin{remark} In fact, the proof of Theorem~\ref{IInew1} shows that the type $p$ can be chosen of the form:
%
%$$\{P(x) \} \cup \{x \neq c_n:n<\omega\}$$
%where the $c_n$ are a recursive sequence of terms in $\tau_1$.
%\end{remark}

\medskip
We could strengthen Lemma~\ref{newtoii} by restricting  the second order quantification to sets of size strictly less
than the size of the model, but that is not important here.
We now deduce Theorem~\ref{big} from Theorem~\ref{IInew1}   and Lemma~\ref{newtoii}.
We use the following notation.

\begin{notation}
 $\Spec(\bt)$ is the
collection of cardinals $\lambda$ such that there is an $M_1$
satisfying $\bt$ with $|M_1| = \lambda$.
\end{notation}

%\sidebar{The Nov 18 version of the paper had the inequality reversed in the following claim .
%The reversed inequality is the succeeding lemma where I have corrected the proof.
%More crucially, both the proof here of the claim and the one in the November version are wrong.  They assume that $\neg\phi_{\bt}$ has no models below the bound on $spec(\bt)$.}

%\sidebar{I believe the objections you raised to the July 4 version were `seeing ghosts'.  I have added a parenthetical sentence, exorcising the ghost.}
 We have not established that the Hanf and Lowenheim numbers for the $P_N$ satisfy Fact~\ref{f2}.  This complicates the
 argument for the following two results.

\begin{claim}\label{newltii}
$H(P^f_N) \leq \ell^2(L^{II})$ where $L^{II}$ denotes second order
logic.
\end{claim}

Proof.
  Lemma~\ref{newtoii} shows that for any $\bt\in \Tscr^f$, there is a $\phi_{\bt}\in L^{II}$ with
 $\spec(\bt) =\spec(\phi_{\bt})$.
Suppose for contradiction that $H(P^f_N) > \ell^2(L^{II})$.
Then there is a triple $ {\bf t} \in {\mathcal T}^f $ such that
$\sup( \spec( {\bf t}))  \geq   \ell^2(L^{II}) $ .

Let $ {\bf C} = \{\mu : \mu = \mu ^{ < \mu }\}  $.
Choose  $ \psi _ {\bf t} \in L^{II} $ so that
$ \mu \models \psi_ {\bf t} $ iff
for  every infinite cardinal
$ \kappa \in {\bf C} \cap  \mu $
there is $ \theta \in [\kappa , \mu )\cap \spec( {\bf t} ) $.


Let $ \lambda _ {\bf t} $ be the minimal element    of $ {\bf C} \cap \spec( {\bf t} )^c$. Then
$\lambda _ {\bf t} \geq \ell^2(L^{II})$.
%such that
%$ \theta \in spec( {\bf t}) \Rightarrow
%\theta < \lambda _ {\bf t}$
For any $\bt$, the definition of $P^f_N$ guarantees that if $ \lambda^{<\lambda} =\lambda,  \mu > \lambda$
 and some $M \models \bt$ has cardinality $\mu$ then some $N \models \bt$ has cardinality $\lambda$
  (take a saturated elementary submodel). Thus,
$\mu \models \psi _ {\bf t}$ if and only if  every $ \kappa \in {\bf C} \cap \mu  $
belongs to $ \spec( {\bf t} ) $.
%Thus,
%(By the Low.Sko.Tar theorem this is equivalent to
%$ \theta \in {\bf C} \cap  \spec( {\bf t} )  \Rightarrow
%\theta < \lambda _ {\bf t}$).

Now $ \spec ( \psi_ {\bf t}  ) $
is exactly $\{  \mu \mcolon \mu \le \lambda _ {\bf t} \} $, whence      $ \spec( \neg\psi_{\bf t} ) $  is
$ \{ \mu\mcolon \mu > \lambda _ {\bf t} \} $.
So the L\"owenheim number of $ \neg \psi _ {\bf t} $
is $  ( \lambda _ {\bf t} )^+  >   \ell^2(L^{II})$ and this contradiction completes the proof.

$\qed_{\ref{newltii}}$


%By the definition of $ \ell^2(L^{II}) $,
  %this sentence  $\neg \psi _ {\bf t}  $  exemplifies
%$ \ell^2(L^{II}) <  \lambda_ * $


\begin{lemma}\label{iiltnew}
$H(P^f_N) \geq \ell^2(L^{II})$ where $L^{II}$ denotes second order
logic.
\end{lemma}

Proof. Suppose for contradiction that there is a second order
sentence $\psi$ such that $\lambda_0 = \min(\spec^2 (\psi)) \geq
H(P^f_N)$. By the definition of $\spec^2$, $\lambda_0^{<\lambda_0} =
\lambda_0$.  Let $\hat \psi$ express $(\exists
U) (\psi^U \wedge |U|^{<|U|} = |U|)$.
We apply Theorem~\ref{IInew1} to ${\neg(\hat\psi)}$.
%(asdefined the proof of Lemma~\ref{eqlow}).
Note that $\hat \psi$ is true on all cardinals
$\geq \lambda_0$
 and
false  on all
$\mu < \lambda_0$. By Theorem~\ref{IInew1}, $\lambda_0 \models
\bt_{\neg(\hat\psi)}$ and $\lambda_0 \geq H(P^f_N)$. So
$\bt_{\neg(\hat\psi)}$ and therefore $\neg(\hat\psi)$ has arbitrarily large
models. But $\neg(\hat\psi)$ has no models larger than $\lambda_0$. This
contradiction yields the theorem. $\qed_{\ref{iiltnew}}$


%On the one hand $\psi^*$
%has a model in cardinality $\lambda_0$ where
%$\lambda_0^{<\lambda_0}=\lambda_0$ and $\neg\psi$ is true on all
%$\mu < \lambda_1$. By Theorem~\ref{IInew1}, $\lambda_0 \models
%\bt_{\neg \psi}$ and $\lambda_0 \geq H(P^f_N)$. So
%$\bt_{\neg\psi}$ and therefore $\neg\psi^*$ has arbitrarily large
%models. But $\neg\psi$ has no models larger than $\lambda_0$. This
%    contradiction yields the theorem. $\qed_{\ref{iiltnew}}$


\medskip
%\begin{remark} {\rm




In the next section we prove the crucial Theorem~\ref{IInew1}.  In the last section we
remove the restrictions to finitely axiomatizable theories and countable languages.


\section{Essential Lemmas}\label{tech}

Now we prove Theorem~\ref{IInew1}.  For convenience, we list here
the two vocabularies. We describe the axioms of $T$ and $T_1$
below.



%Proof of 2) implies 1) of Theorem~\ref{equiv}.

%Assume $T_1$ is finitely axiomatizable. Then it is easy to %write second order sentence $\theta$ such that if $M \models %\theta$, $M\models T_1$, $M$ omits $p$ and $M\restriction %\tau$ is saturated.
%$\qed_{\ref{equiv} 2 \rightarrow 1}$

\begin{notation}
\begin{enumerate}
\item $\tau$ contains unary predicates $Q_1, Q_2$, a binary relation $R$ and
partial binary functions $F$ and $F_2$.  It contains two constant
symbols $c_0,c_\omega$ and a unary function symbol $g$.% $\{c_i:i\leq
%\omega\}$.
    \item $\tau_1$ adds a unary predicate $Q_0$ and a binary relation $<_1$.
    %, a partial binary function $F$ and a binary relation %$<_0$.  It contains constant symbols $\{d_i:i< \omega\}$.
\end{enumerate}
\end{notation}


%\sidebar{This version reverses the notes where most of the %information is in the expanded language. That doesn't seem %right because we have to apply the saturation to these %relations.}


\begin{remark}[Proof Sketch] {\rm For each second order $\phi$, we construct a triple $\bt_{\phi}$.
But most of the construction is independent of the particular
$\phi$ and so we first construct a theory $T_1$ which does not
depend on $\phi$. The vocabulary  $\tau$ will contain unary
predicates $Q_1,Q_2$. The axioms will assert that $Q_1,Q_2$
partition the universe. $Q_0$ is in $\tau_1$. Omission of the type
$p$ will guarantee that $Q_0\subset Q_1$ is countable. Omission of
the type in a model $M$ of $T_1$ whose $\tau$-reduct is
$\aleph_1$-saturated and some coding involving the partial order
$<_0$ in $\tau$
 will guarantee that  $Q_1(M)$  is well-ordered by a relation symbol $<_1$ in $\tau_1$.
 A relation symbol $R$ in $\tau$ will code subsets of $Q_1$ by elements of $Q_2$. Thus
 first order quantification on $Q_2$ will encode second order quantification on $Q_1$.
 In particular, we can code a given second order sentence $\phi$ and thus extend $T_1$ to $T_\phi$.
 But the encoding guarantees
 `correctness' only on  subsets whose every subset is coded in $Q_2$.  The construction will guarantee that if  $\mu <
 \lambda$
 and
 $M$ is $\lambda$-saturated, then $\mu$ is a $<_1$-initial segment $Q_1$. Since $\mu< \lambda$ each subset of $\mu$ is
 coded by a type of size $\mu$ so the  encoded semantics is correct and $\mu$ is a model of $\phi$.
}\end{remark}

\noindent
{\bf Beginning the Proof of Theorem~\ref{IInew1}.} We gradually introduce the
vocabulary and theory explaining the use of various predicates as
they are introduced; we repeat a bit of the proof sketch. Below we
say certain conditions hold to mean they hold in any model of $T$. We first describe $\tau$ and $T$.
In particular, $\tau$ contains unary  predicates $Q_1,Q_2$ that
partition the universe.
 %elements of $Q_1$.





%\sidebar{Originally the functions below had first argument in %$Q_0$.  But this doesn't allow the use of saturation in %$\tau$. Now the restriction to $Q_0$ is given by the type %$p$.}


There is a binary relation $<_0$, which is a partial order of $Q_1$.
There is a partial function $F$ mapping $Q_1 \times Q_1$ into $Q_1$.
We write $F_a$ for the partial function from $Q_1$ into $Q_1$ indexed by $a$.
Any model of $T$ satisfies: $a \leq_0 b$ implies $F_a \subset F_b$.

We have two further properties of $F$. $F_{c_0}$ is the empty function.
For every $a \in Q_1$ and every $e \in Q_1$, if $e \not \in \dom F_a$, then there are
$b,d\in Q_1$ with $a <_0  b$  and $F_b = F_a \cup \{\langle e,d\rangle\}$.  %.
%$<_3$ is a linear order of $Q_1$.

Further there is a pairing function $F_2$ on $Q_1$ and an
extensional relation $R$ between $Q_1$ and $Q_2$ so that each
element of $Q_2$ codes a subset of $Q_1$ via $R$.  We write $U_b$
for $\{a\mcolon R(b,a)\}$ (for $a\in Q_1$ and $b\in Q_2$).

$T$ asserts that $Q_1$ is preserved by $g$, that $g$ is a
permutation, and $Q_1(c_0)$.

The set of $\{U_a\mcolon a \in Q_2\}$ is closed under Boolean
operations and if $U_b$ is such a set so is $F_a(U_b)$ for any $a
\in Q_1$. For each $a \in Q_1$, there is $b \in Q_2$ such that
$U_b =\{c\mcolon c<_1 a\}$.




Secondly, we turn to the description of $\tau_1$ and $T_1$. In $\tau_1$,
 there is a binary
$\tau_1$-relation $<_1$, which is a linear order of $Q_1$ and such
that   $Q_1(x)$ implies  $x<_1 g(x)$ and no element of $Q_1$ lies
between $x$ and $g(x)$. Moreover, there are elements $c_0,c_1$ of
$q_1$ such that $c_0$ is the $<_1$-least element of $Q_1$ and
 %$x<_1 g(x)$ and
  $ x<_1c_\omega$ implies $g(x) <_1 c_\omega$. $T_1$ further asserts $(Q_1,
<_1)$ is `internally well-ordered' in the following sense.  For every
$a \in Q_2$, if $U_a$ is non-empty, it has a $<_1$-least element.
Finally, there is a unary relation $Q_0$ such that $Q_0 \subset Q_1$
and $T_1$ asserts
%$c_0<c_\omega$ are in $Q_0$,
%$c_0$ is the $ <_1$-least element of $Q_0$, and $\neg Q_0(c_\omega)$.
%$Q_0(x)$ implies $ x<_1 c_\omega$,
%$Q_0$ is preserved by $g$,
%if $Q_0(x)$ then $x<_1
%g(x)$
 $c_0\leq_1 x<_1c_\omega$ if and only if $Q_0(x)$. ($Q_0$ is
just an
abbreviation and $<_1$ is the crucial symbol added to create $\tau_1$.) % satisfying $g^i(c_0) <_1 g^j(c_0)$
Thus, each $g^i(c_0) \in Q_0$ and $\{ g^i(c_0) \mcolon i < \omega \}$
%\cup \{c_\omega\}$
name countably many elements of $Q_0$ which are $<_1$- ordered in
order type $\omega$.
%There are constants $\langle c_i\mcolon i \leq \omega \rangle$
%which name countably many elements of $Q_1$ which are $<_1$-
%ordered in order type $\omega+1$.


%The extra information in $\tau_1$ is a countable list of %constants $d_i$.
The type $p$ asserts $Q_0(x)$ and $x$ is not a $g^i(c_0)$ for any $i<
\omega$. Thus if $p$ is omitted in a model $M$, $Q_0(M) = \{ g^i(c_0)
\mcolon i < \omega \}$.
% \cup\{c_\omega\}$.
% and
%$(\exists y)(\exists z)F(x,y,z) \rightarrow Q_0(x)$.


%\sidebar{Note the added information in $p$. This seems to be %essential otherwise the saturation hypotheses of the next %Lemma~\ref{wo} would not be in $\tau$.}



%Q_3 = Q_1 \cup Q_2$ with initial segment $Q_1$.
%\sidebar{Notes say $F_b$ adds only one tuple. But this would %imply each $F_a$ has finite domain.}



\begin{claim}\label{wo}
% In any $\aleph_1$-saturated model of $T$,
 %$(Q_1, <_1)$ is a well-ordering.
 %that omits$p$,

If a model $M$ of $T_1$ is such that its reduct to $\tau$ is an $\aleph_1$-saturated model of
 $T$ but $M$ omits $p$, $(Q_1, <_1)$ is a well-ordering in $M$.
\end{claim}

Proof. Suppose there is a countable $<_1$-descending chain $B = \{b_i\mcolon i<\omega\}$
in $(Q_1, <_1)$.  Using the properties of $F$,
we can define a $<_0$-increasing chain of $a_n$ in $Q_1$ %in $Q_0$
such that $F_{a_n} = \{\langle c_0,b_0\rangle, \ldots , \langle
g^n(c_0),b_n\rangle\}$, where the $g^i(c_0)$ are images of $c_0$ by
iterating $g$. Since the model is $\aleph_1$-saturated there is an
$a_\omega \in Q_1$ such that each $F_{a_n} \subset F_{a_\omega}$. But
then $B = F_{a_\omega}(\{g^i(c_0)\mcolon i< \omega\})$.  Note that
while the choice of $b_i$ involved the $\tau_1$-symbol $<_1$, the
existence of $a_\omega$ is by the consistency of a $\tau$-type so the
use of saturation is legitimate.


Since $M$ omits $p$, $\{g^i(c_0)\mcolon i< \omega\}= Q_0(M) =
\{a\mcolon c_0 \leq a<_1 c_\omega\}$ and therefore is coded by an
element of $Q_2$. By the closure properties of the coded sets, $B =
U_d$ for some $d \in Q_2$. This contradicts the internal
well-ordering of $Q_1$. $\qed_{\ref{wo}}$


\medskip

Now translate $\phi$ to the first order formula $\phi^*(v)$ by
translating each bound second  order variable X to a first order
formula in $x$ and $v$. Replace each occurrence of $X(z)$ by
$R(z,v) \wedge R(z,x)$. This translation has the following
consequence.  (This is immediate for monadic second order but we
included a pairing function $F_2$ on $Q_1$ so it extends to
arbitrary sentences.)

\begin{fact}\label{key} If $M \models T$, $a \in Q_2(M)$ and each subset of
$U_a$ is coded by an element of $Q_2(M)$, then $M \models
\phi^*(a)$ if and only $U_a(M) \models \phi$.
\end{fact}




Add the following axiom to $T_1$ to obtain the theory $T_\phi$:

$$(\forall u)(\forall w)[((\forall z)R(z,w) \leftrightarrow z<_1 u )%\wedge `u \hbox{\rm \ is a cardinal'\ })
 \rightarrow \phi^*(w)
].$$
%$\{y \in \{u: R(u,v)\}: y \in \{u: R(u,x)\}$.

This completes ${\bf t}_\phi$ as $\langle T, T_\phi, p\rangle$.
%We may need $M$ has cardinality $2^\lambda$ to gua

\begin{claim}\label{end} If  $\mu < \lambda=\lambda^{<\lambda}$ and $M$ is
a model of $T_{\phi}$ with cardinality $\lambda$ that omits $p$ but
whose reduct to $\tau$ is saturated then $\mu \models \phi$.

Conversely, if $\phi$ is true on all  $\mu< \lambda=\lambda^{<\lambda}$, there is a
 model $M_1$  of $T_\phi$ with cardinality $\lambda$ that omits $p$ but
 whose reduct to $\tau$ is saturated.

%for every $\chi< \mu$, $\chi \models \phi$.
\end{claim}

Proof. Since $\mu< \lambda$, $\mu$ is an initial segment of $Q_1$
so $\mu =\{a\in Q_1\mcolon R(y,d)\}$ for some $d\in Q_2$.  But then each
subset $Y$ of $\mu$ gives rise to a type:
$$q_Y(x) =\{R(y,d)\}\cup
\{R(y,x): y\in Y\} \cup \{\neg R(y,x): y\not \in Y\}.$$ For each
$Y$ the $\tau$-type $q_Y(x)$ has cardinality less than $\lambda$
and so is realized by saturation. We finish by Fact~\ref{key}.

%First choose a saturated model $M$ of $T$ with $|M|^{<|M|} %=|M|=\lambda$.
For the converse, well-order $Q_1$ by $<_1$ in order type
$\lambda$. Add in $Q_2$ a code for each subset of cardinality
$<\lambda$. Let the $F_a$ list the partial functions of
cardinality less than $\lambda$ from $Q_1$ to $Q_1$ and let $<_0$
denote the natural partial ordering on $Q_1$ induced by inclusion
of the named functions. Since $\phi$ is true below $\lambda$, each
infinite initial segment in $\lambda$ defines a model of $\phi$
and the definition of $T_\phi$ shows that we have a saturated
model of $T$ when we take the reduct to $\tau$.  Finally, let
$Q_0$ include exactly the first $\omega$ elements of $Q_1$.
% $\phi$ is true on $\lambda$, we can choose Skolem functions %for $\phi$ and add to $Q_2$ names for any set in the range of %the Skolem functions.


%Note that by saturation in $\lambda$, $Q_2$ codes every subset %of every initial segment of $\lambda$ and so



$\qed_{\ref{end}}$

\noindent{\bf Concluding the Proof of Theorem~\ref{IInew1}.}


Letting $\bt_{\phi}$ be the triple $(T,T_\phi,p)$ we have a triple satisfying
Theorem~\ref{IInew1}.
$\qed_{\ref{IInew1}}$


\section{The exact strength}\label{exact}

In this section we remove the restrictions to finitely axiomatizable theories and countable languages.  In
Theorem~\ref{sumthm} we prove actual equality of  the Hanf number studied here (for any triple of theories
and types of any cardinality) with a L\"owenheim number of second order logic; the cost is that we must move
into infinitary second order logic and  (in the proof) allow relation constants (i.e. predicate  symbols
other than equality) in the vocabulary of the second order sentence.


Instead of Theorem~\ref{IInew1} we  could slightly more easily prove
$$H(P^f_N) \leq \ell^2(L^{II}) \leq H(P^c_N),$$
which gives our answer to Newelski's question but is not quite as
sharp.  That is, if we had just required ${\bt}_\phi$ in
Theorem~\ref{IInew1} to be in a countable language rather than
finitely axiomatizable, this would have no effect on the proof of
Lemma~\ref{iiltnew} and it would have simplified the proof of
Theorem~\ref{IInew1} since we could have worked with countably
many constants and omitted the function $g$.
Similarly the arguments of Sections \ref{concl} and \ref{tech} extend from finitely axiomatizable to
`arithmetic' by coding a model of arithmetic in the second order
sentence.  And it is easy to see that the theory constructed in
Theorem~\ref{IInew1} is recursive.
This observation is
generalized in Theorem~\ref{sumthm} to remove the restrictions on
axiomatizability.
The key idea is to see that we can use the same ideas as in Section~\ref{tech} to code the syntax of infinitary second
order logic by a triple $\bt$.

 %For simplicity of exposition we deal first with the case of finite axiomatizabilty.
%\sidebar{Note however, that coding of syntax and satisfaction is used in the proof of
%Lemma~\ref{newtoii}.}

We extend our notion of second order logic in two ways. First we allow infinite
conjunctions and strings of quantifiers.
Secondly we  now allow some relation constants instead of dealing with `pure' second
order logic.



\begin{definition} \begin{enumerate}
\item $L_\theta$ denotes
the $\theta$ stage in the construction of the inner model $L$.
 \item Let $L_{\theta^+,\kappa}(II)$ denote second order logic allowing strings of
 second order quantifiers of
     cardinality $<\kappa \leq \theta^+$ and
conjunctions and disjunctions of cardinality $\leq\theta$.


%\item   $\Tscr^\theta$ denotes the set of triples $\bt$ as in Notation~\ref{deftrip} (except allowing $|\tau_1| \leq \theta$) such
% that $\tau_1$ satisfies $\tau_1\subset L_\theta$ and we can now omit the requirement of
%  second order definability of $p$.
%    %$T$ is finitely axiomatizable and $p$ is second order definable.

\end{enumerate}
\end{definition}



\begin{remark}  Again using Assumption~\ref{assumpt}, note that as in Fact~\ref{f1} the L\"owenheim number of
$L_{\theta^+,\kappa}(II)$
is a strong limit cardinal of cofinality $> \theta$ and is an accumulation point of
$\{\mu: \mu = \mu^{<\mu}\}$.
\end{remark}


\begin{notation} We denote by $L(II,\tau_*)$  second order logic in the vocabulary $\tau_*$ consisting of
constant symbols $c_\kappa, c_\theta, c_\tau, c_\phi$, a unary predicate $Q$, % and $P_1$
 and a binary relation $R_1$.
\end{notation}

%\begin{definition}\label{deftype}
%A type $p$ in a vocabulary of cardinality $\theta$ is  {\em definable in $L(II,\tau_*)$}  if we can code the type as a subset $B_p$ of $\theta$ so
%that
%there is a  sentence  $\psi\in L(II,\tau_*)$
% and a formula $\varphi (x)\in \in L(II,\tau_*)$ satisfying the following condition.
%For $L(II,\tau_*)$-structure $M$ with cardinality  $ \lambda\geq \theta $ which satisfies $ \psi $,
%$B_p= \{ \alpha <\theta: M \models \varphi (\alpha) \} $.
%\end{definition}

\begin{notation}
Let $\bK = \bK_{\bt}$,  where the set of triples $\bt$ is as in Notation~\ref{deftrip} but requiring
 that  $\tau_1\subset L_\theta$ and omitting the requirement that $p$ is second order definable.
Then,  $P^{\kappa}_N(\bK_{\bt},\lambda)$ holds
if $|\tau_1| \leq \kappa$ and for some $M_1$ with $|M_1|=\lambda$,
$M_1 \models \bt$.
\end{notation}

In the following $H(\mu)$ denotes the set of all sets  whose transitive closure has cardinality  less than $\mu$.  We
use implicitly that $\mu^{<\mu}=\mu$ implies that $\mu$ is regular and so $(H(\mu),\epsilon)$ satisfies all axioms of
ZFC except power set.

We now construct a sentence  $\psi \in L({II, \tau_*})$ and   for every $\tau \subset L_{\theta}$ and every
$\phi$ in $L_{\theta^+,\kappa}(II)(\tau)$ a set $A_{\tau,\phi}$ of ordinals so that $\psi$ and $A_{\tau,\phi}$ codes
$\phi$.

%We now show how to code the L\"owenheim number of sentences $\phi$ of $L_{\theta^+,\kappa}(II)$ by using   a sentence
%in $\psi \in L({II, \tau_*})$ and a set $A_{\tau,\phi}$ of ordinals.  We begin with a more general lemma concerning the coding of transitive sets.

\begin{definition}\label{defcode} Let $X$ be a  transitive subset of $H(\theta^+)$ with cardinality $\theta$.
 %such that
  %$\{\emptyset\}\subset X$.
  We say that
$X$ {\em is coded by } $A\subseteq \theta$ if there is an injection $f$ from $X$ into $\theta$ such that:
$$A = \{\pr(f(a),f(b))\mcolon a\in b \in X\}.$$
Here $\pr$ is the standard pairing function on ordinals.
%Here $\pr$ is the standard pairing function on ordinals which satisfies the condition $\pr(\alpha_1,\beta_1) \leq \pr(\alpha_2,\beta_2)$  ?????
\end{definition}

Now we show that the coding of $X$ by $A$ does {\em not} depend on the choice of $f$.

\begin{lemma}\label{codewdef}  If $A\subseteq \theta$ codes $X_1$ by $f_1$ and $X_2$ by $f_2$ then $X_1 = X_2$.
\end{lemma}

Proof.  We first note that for any $X,A$ if $A$ codes $X$ by $f$ then $f$ is a bijection between $X$ and
$B = \{\alpha\mcolon \pr(\alpha,\beta)\in A \vee \pr(\beta,\alpha)\in A\}$.
%The nontriviality conditions on $X$
%guarantee that $f$ maps into $B$; $f$ is $1$-$1$ by definition of $f$ and onto $B$ by the definition of $B$.
Applying this remark to $f_1,f_2$, we see $g=f_2^{-1}f_1$ is a bijection from $X_1$ onto $X_2$. But then $g$ is an
isomorphism with respect to $\epsilon$ as it is easy to check (from the definition of coding) that for any $y_1,z_1 \in
X_1$ with $y_2 = g(y_1)$ and $z_2 = g(z_1)$, $y_1 \in z_1$ if and only if $y_2\in z_2$.
 That is, $y_1 \in z_1$ if and only $\pr(f_1(y_1),f_1(z_1))\in A$ if and only $\pr(f_2(y_2),f_2(z_2))\in A$
  if and only $y_2 \in z_2$.
  But then since $\emptyset \in X_1 \cap X_2$, $\epsilon$-induction yields that $X_1 = X_2$.
$\qed_{\ref{codewdef}}$




% Given an $L_{\theta^+,\kappa}(II)$ sentence $\phi$
%we construct a sentence $\psi \in L^{II}$ and a set $A$ such that
%if $ M \models\psi$ and $A\subset M$ satisfy the codes the sentence $\phi$.  We write $L_\theta$ for
%the $\theta$'d set in the construction of the inner model $L$.

\medskip


%\sidebar{My first reaction to the referees comment on Section 4 was to think that burying the crucial definition
%in the statement of the Lemma had made it impossible for him to parse it.  So I moved it outside in the following definition.  However, after a long discussion today, I find that I really do not understand the claim in Lemma 4.6.
%My lack of understanding mainly revolves around why the $A_{\tau,\phi}$ should be a parameter. How can the second order sentence be wrong about a statement but it thinking about why I discovered what I think is a counterexample to the Lemma as claimed. (But one that is false for trivial reasons.)  Namely.

%Let $\theta$ be very much greater than $\lambda$. Suppose $\phi$ is a sentence in $\phi\in L_{\theta^+,\kappa}(II)$ which (perhaps because of great redundancy) has a standard code whose cardinality is greater than $\lambda$.  Then
%Lemma~\ref{kappatoII} fails for $\phi$ because $|A_{\tau,\phi}| > \lambda$ and $A_{\tau,\phi} \not \subseteq M$ when $|M| = \lambda$.

%It would be nice if you could explain the idea this in a short way so I could rewrite the proof. However, if you prefer to write out the argument, please do so from scratch. Part of the difficulty is that I cannot read the crucial lines of the current notes from you on this topic.}

%\sidebar{ 1. In proving 3 implies 4, we are moving from $\phi\in L_{\theta^+,\kappa}(II)$ to  $\psi_\phi \in L(II,\tau)$. There is no
%non-logical vocabulary in $\phi$. The $\tau$ in the second logic is defined in notation 4.3.

%2.  Note the change from `satisfies' to `asserts' in clause 2:

%I hope I have correctly interpreted the rest of your remarks.}

%\begin{lemma}\label{codeform}  If $\tau \subseteq L_\theta$ then each $\phi \in L_{\theta^+,\kappa}(II)(\tau)$
%has at most $\theta$ subformulas, each in $\tc(\phi)$, and $\tc(\phi) \cup \{\phi\}$ is coded (in the sense of
%Definition~\ref{defcode}) by a set $A_{\tau,\phi} \subset \theta$.
%\end{lemma}
%
%Proof. The standard construction of  $\phi \in L_{\theta^+,\kappa}(II)(\tau)$ yields that each formula is in
%$H(\theta^+)$, each subformula of $\phi \in \tc(\phi)$, and $\phi$ has at most $\theta$ subformulas.  Define
%$A_{\tau,\phi}$ by Definition~\ref{defcode}.
%$\qed_{\ref{defcode}}$

\medskip

We first define certain set of ordinals $A_{\tau,\phi} \subseteq \theta$ in $V$ that codes $\tau$ and  $\phi\in
L_{\theta^+,\kappa}(II)(\tau)$.   Here $\tc(X)$ denotes the transitive closure of $X$.




\begin{definition}\label{Adef}
 For  every vocabulary $\tau \subseteq L_\theta$ and every sentence $\phi\in L_{\theta^+,\kappa}(II)(\tau)$
we define $A_{\tau,\phi} \subseteq \theta$ to be a set which codes
$\tc(\{\phi,\tau\}) \cup \{\phi,\tau\}$ in the sense of
Definition~\ref{defcode}.
\end{definition}

Such a code exists since the standard construction of  $\phi \in
L_{\theta^+,\kappa}(II)(\tau)$ yields that each formula is in
$H(\theta^+)$, each subformula of $\phi \in \tc(\phi)$, and $\phi$
has at most $\theta$ subformulas.

Now we define the sentence $\psi \in L(II,\tau_*)$, which does not
depend on $\phi$ or $\tau$, so that the interpretation of a predicate
$Q$ as $A_{\tau,\phi}$ in a model $M$ of $\psi$ will identify $\phi$
as the sentence under consideration. The function $G$ in
Definition~\ref{psidef}.3 simply formalizes the normal definition of
truth.

\begin{definition}\label{psidef}
 %For  every vocabulary $\tau \in L_\theta$
 %and every sentence $\phi\in L_{\theta^+,\kappa}(II)(\tau)$
We define a sentence $\psi \in L(II,\tau_*)$  $M \models \psi$ if and
only if $M$ satisfies:
%and for every $\kappa \leq \theta^$ and every sentence $\phi\in L_{\theta^+,\kappa}(II)$
%we define   a set  $A_{\tau,\phi}$ %,\psi_\phi)$of ordinals  and
%to assert the following:




    %\item

    %Define $\psi_\phi$
%    $M \models \psi_\phi$ iff $M$ satisfies
%the properties we now describe.
\begin{enumerate}
%\item $(P_1^M,R_1^M)$ is a well ordering with $P_1^M \subseteq P^M$

\item %There is a binary relation $\epsilon$ such that
$(M,R^M_1) \approx (H(\mu),\epsilon)$ for some $\mu$ with
$\mu^{<\mu} = \mu$ and % by a map $g$.
\item  the following $\tau^*$-axioms.
\begin{enumerate}
\item $c_\kappa$ and $c_\theta$ are cardinals in the sense of $M$.
% satisfying
%$c_\theta^{c_\kappa} =c_\theta$.

%$g(c_\kappa)$ and $g(c_\theta)$ are cardinals in the sense of $M$ satisfying
%$c_\theta^{c_\kappa} =c_\theta$.
%denoted $\theta_M$ and $\kappa_M$ such that $\theta_M^{<\kappa_M} =\theta_M$.
\item $c_\tau$ is a vocabulary of cardinality $c_\theta$ contained in $L_\theta$.
\item The set of    formulas of $L_{c^+_{\theta},c_\kappa}(II)(c_\tau)$ are given their usual inductive
    definition in $M$ by a formula $\mu(x,c_\theta,c_\kappa,c_\tau)\in L(II)(\tau_*)$.
    \item $c_\phi \in L_{c^+_{\theta},c_\kappa}(II)(c_\tau)$,
        i.e. $M \models
        \mu(c_\phi,c_\theta,c_\kappa,c_\tau)$.
        \item $Q(x) \rightarrow x R_1 c_\theta$.
        \item $Q$ is a set of codes (in the sense of $M$) of
            $\tc(\{c_{\phi},c_\tau\}) \cup
            \{c_\phi,c_\tau\}$, where $\tc$ is with respect
            to $R_1$.
\end{enumerate}
%\item $P^N = P^M$ and $Q^N = Q^M$.
%\item  $Q^M$ is the set of
%ordinals (contained in the ordinal $P^M$) coding subformulas of
%$\phi$ under the standard inductive
%definition of $L_{\theta^+,\kappa}(II)$.
\item There is a definable function $G$  which defines truth  of
    sentences of $L_{\theta^+,\kappa}(II,c_\tau)$ on
    $c_\tau$-structures $b^*\in M$:
\begin{enumerate}
\item for every sentence $\chi$, $M \models G(b^*,\chi^*) =
    1$ if
    and only if
   % $$(M \myrestriction \{a\mcolon a R_1^M b\})
    $$ b\models \chi.$$
%for false and $1$ for true.
(Here $\chi$ is an actual sentence, $b$ is a $c_\tau$-structure
 in $H(\mu)$, and $\chi^*$ respectively, $b^*$ is the member of
 $M$ mapped to $\chi$, respectively $b$ under the isomorphism in
 Definition~\ref{psidef}.1. )
\item $G^M(b^*,c_\phi) = 0$ for every $c_\tau$-structure $b^*
    \in M$.
\end{enumerate}



%subformulas of our given formula $\phi$ on subsets $b$ of $M$ and by this coding, $M$ satisfies `$b \models \neg\phi$' if $M$ \models models |b|<|N|$.


\end{enumerate}



%\end{enumerate}
\end{definition}

The goal of the following lemma is to compute the L\"owenheim number of $L_{\theta^+,\kappa}(II)$. Since it is certainly
greater than $\theta$, we may assume $\lambda^{<\lambda}> \theta$.


\begin{lemma}\label{kappatoII} Fix $\kappa \leq \theta^+$, a vocabulary $\tau \subseteq L_{\theta}$ and $\phi\in
L_{\theta^+,\kappa}(II)(\tau)$.   Choose
  $A_{\tau,\phi}$ and
 $\psi \in L(II,\tau_*)$ satisfying Definitions~\ref{Adef} and \ref{psidef}, respectively.
For any cardinal $\lambda = \lambda^{<\lambda}> \theta$, the following are equivalent.
\begin{enumerate}
\item  $\phi$ has no model of cardinality $<\lambda$.
\item
 There is a
 %model $(M,P^M,P_1^M,R_1^M,c^M_\theta,c^M_\kappa,c^M_\phi )$
 model $(M,R_1^M,Q^M,c^M_\theta,c^M_\kappa,c^M_\phi,c^M_\tau )$
 with cardinality $\lambda$ of the sentence $\psi$ defined in
 Definition~\ref{psidef}  such that letting $P^M$ denote $$\{b
 \mcolon M \models  \hbox{\rm\ `$b$  is an ordinal' \ } \meet b
 R_1^M c_\theta \}$$ and $P^M_1$ denote $$\{b \mcolon M \models
 \hbox{\rm\ `$b$  is an ordinal' \ }\meet b  R_1^M c_\kappa \};$$
 \begin{enumerate}
 \item %, R_1^M)$$ has order type $\theta$,
 $(P^M,R_1^M)$ has order type $\theta$;
  \item $(P^M_1, R_1^M)$ has order type $\kappa$;
 \item
  $$A_{\tau,\phi}=\{\alpha<\theta: {\rm for\  some \  } a\in Q^M, %{\rm we  have \ }
    \alpha = \otp(\{b R_1 c_\theta\mcolon b R_1^M a\},R_1^M)\}.$$
   % \item $c_\tau$ denotes $\tau$.
\end{enumerate}
\end{enumerate}
\end{lemma}


Proof.
%\sidebar{The next sentence should apparently be deleted since it is part of the hypothesis.}
%By Lemma~\ref{eqlow}, we may assume $\lambda^{<\lambda} = \lambda$.
%will be actually true on $b\subset N$
%with $|b|<|N|$
    Suppose 2). Without loss of generality, we identify $(M,R^M_1)$ with
     $H(\lambda,\epsilon)$. $\tau \in H(\lambda)$ since $\tau \subset L_\theta$. Then  $|M|= \lambda$, $(P^M,R_1^M)$ has order type
    $\theta$, and $A_{\tau,\phi}$ is the image of $Q^M$ under an
isomorphism from $(P^M,R_1^M)$ to $\theta$.  By the choice of $A_{\tau,\phi}$ ,
 the model $M$ correctly recognizes
 the
vocabulary $\tau$ and the formula $\phi$. The function $G^M$
correctly represents truth in $M$ by Definition~\ref{psidef}.3a.
% and
%Definition~\ref{psidef}.4.
 So $\phi$  fails on all subsets of $M$
with cardinality $<\lambda$ by Definition~\ref{psidef}.3b. Thus 2)
implies 1). Clearly if 1) holds we can construct a model $M$
satisfying 2).

%Further $M \models \psi$ implies `for all $a$
%with cardinality $< |M|$, $M \models$ `$a \models \neg \phi$'.

$\qed_{\ref{kappatoII}}$


%\sidebar{probably drop:





%In the language for $\phi$, add a sort $R_1$ for the formulas of $\phi$ and an inductive definition for the logic  $L_{\theta^+,\kappa}(II)$.  In the vocabulary for $\psi$ there is a predicate $P$ for $\theta^+$ and $Q \subset P$ is to be interpreted as $\kappa$.  Then the usual inductive definition identifies the elements of $R_1$ as the formulas of $L_{\theta^+,\kappa}(II)$.   Another sort $R_2$ is taken for the interpretation of sentences of $L_{\theta^+,\kappa}(II)$. Now in any model $N$ of $\psi$ of where $P^N_1$ has cardinality $\theta^+$ and $P^N$ has cardinality $\kappa$, for any sentence $\chi \in  L_{\theta^+,\kappa}(II)$,

%$$R_2^N \models \psi^N {\rm iff\ } |Q^N| \models \psi.$$
%}

We continue to use the conventions regarding $P^M, P^M_1$ from the proof of Lemma~\ref{kappatoII}.

\begin{definition} For $\psi$  as in Definition~\ref{kappatoII},
$\spec(\psi,\theta,\kappa,A_{\tau,\phi})$ is the set of the cardinalities of models $M$ of $\psi$ with
$(P^M,P_1^M,Q,R_1^M) \approx (\theta, A_{\tau,\phi},<)$.
\end{definition}

%We generalize our notion of the L\"owenheim number of second order logic. In the first three sections second
%order logic was restricted to a vocabulary with only the equality symbol fixed. In this section we have
%considered arbitrary vocabularies and we introduce a L\"owenheim number to reflect this.






\begin{theorem}\label{sumthm}  For any cardinals $\theta,\kappa$ the following four cardinals are equal.
\begin{enumerate}
\item $\lambda_1$ is the Hanf number of $P^\theta_N$. \item $\lambda_2$ is the L\"owenheim number of
    $L_{\theta^+,\omega}(II)=\ell^2(L_{\theta^+,\kappa}(II))$.


%\item $\lambda_{2.5}$ is the   L\"owenheim number of $L_{\theta^+,\theta}(II)$ as defined in
%    Definition~\ref{withvocab}


\item $\lambda_3$ is the L\"owenheim number of
    $L_{\theta^+,\theta^+}(II)=\ell^2(L_{\theta^+,\theta^+}(II))$.




\item $\lambda_4 = \sup\{ %\sup
 \spec(\psi,\theta,\kappa,A_{\tau,\phi})  \mcolon $
 $\psi \in L(II,\tau_*)$, $\phi \in L_{\theta^+,\theta^+}(II)$,  \hbox{\rm and } $A_{\tau,\phi} \subset \theta$
 \hbox{\rm such that} $\spec(\psi,\theta,\kappa,A_{\tau,\phi})$ \hbox{\rm is bounded}$\}$.
\end{enumerate}
\end{theorem}



Proof. We chose the logic $L_{\theta^+,\omega}$ precisely so $\lambda_1 \leq \lambda_2$ (by a proof like that
of Lemma~\ref{newtoii} but now we have conjunctions of cardinality $\theta$). So in fact the demand that `a
model omits the type $p$' becomes `the model satisfies the sentence in $L_{\theta^+,\omega}$, $\neg(\exists
x) \bigwedge p$' so the worries in the first sections about the second order definability of $p$ disappear.
Clearly $\lambda_2 \leq \lambda_{3}$.
% because each sentence $\chi \in L_{\theta^+,\theta}(II)(\tau)$ with
%$|\tau| \leq \lambda$ induces a sentence with the same spectrum by existentially quantifying out the symbols
%of the vocabulary.


With $\psi$ from Lemma~{\ref{kappatoII}}  and applying that lemma with  $\tau$ as $\{=\}$ and with $\kappa=
\theta^+$) yields:$$\{\min(\spec^2(\phi)):\phi \in L_{\theta^+,\theta^+}\} \subseteq \{\sup (\spec^2(\theta,
\psi,A_{\tau,\phi})): \phi \in L_{\theta^+,\theta^+} \hbox{ is bounded}\}.$$ (We can replace $\phi$ by a
$\phi^*$ whose only model is the model of $\phi$ with minimum cardinality to guarantee the containment.)
Thus, $\lambda_3 \leq \lambda_4$.

The proof that $\lambda_4 \leq \lambda_1$ is
obtained by modifying
the proof of Theorem~\ref{IInew1}.
Add to the vocabulary in the $T_{\phi}$ from the proof in section 3 of Theorem~\ref{IInew1}, symbols $P,P_1,Q,R_1$ and
symbols $c_\alpha$ for each $\alpha \in A_{\tau,\phi}$ and
use the same coding ideas to guarantee that $P_1,Q $ are contained in $P$ and all three  are well-ordered by $R_1$.  Thus, for $\phi \in L_{\theta^+,\theta^+}(II)$
%for each $A_{\tau,\phi}$,
 we can construct $\bt_{\phi}$, encoding the second order sentence $\psi \in L(II,\tau_*)$ defined in
 Definition~\ref{psidef} and where the type $p$ also codes that ${Q}^M \approx A_{\tau,\phi}$ so that  the two spectra
 are related as in Theorem~\ref{IInew1}.  The type $p$ is just $\{Q(x)\}\cup \{x \neq c_\alpha\mcolon \alpha \in
 A_{\tau,\phi}\}$. This yields $\lambda_4 \leq \lambda_1$ by slightly modifying the argument for Lemma~\ref{iiltnew}.

%Since
%$\sup \{\sup( \spec^2(\theta, \psi_\phi,A)): \phi \in %L_{\theta^+,\theta^+} \hbox{ is bounded}\} \leq \lambda_4$ we %finish.
$\qed_{\ref{sumthm}}$


\medskip

Our discussion of the Hanf and L\"owenheim numbers of second order logic focused on two vocabularies: $\{=\}$
and $\tau_*$.  In contrast,  in many studies     of the Hanf and L\"owenheim numbers of logics the number is
taken as the supremum for a given logic over all vocabularies of a bounded cardinality. That is, a
L\"owenheim or Hanf function (with argument the cardinality of the vocabulary) is defined:

\begin{definition}\label{hndef}  For any logic $\Lscr$ and any cardinal $\theta$.
\begin{enumerate}
\item Let the  L\"owenheim function $\ell^3( \Lscr,\theta)$ be the least cardinal $\mu$ such that for any
    vocabulary $\tau$ of cardinality $\leq \theta$ and any $\phi \in \Lscr(\tau)$, if $\phi$ has a model
    it has one of cardinality less than or equal $\mu$.
\item Let the  Hanf function $h^3( \Lscr,\theta)$ be the least cardinal $\mu$ such that for any
    vocabulary $\tau$ of cardinality $\leq \theta$ and any $\phi \in \Lscr(\tau)$, if $\phi$ has a model
    of cardinality $\mu$, it has arbitrarily large models.
    \end{enumerate}
    \end{definition}



The cardinalities of the vocabularies play a significant role.
 A trivial example is that both the L\"owenheim function and Hanf function in the sense of Definition~\ref{hndef}
 of first order logic map  the
cardinality of the vocabulary to itself.  A more interesting example is that for the logic $L(Q)$ (with $Q$
interpreted as `there exists uncountably many)', the L\"owenheim number in $\tau =\{=\}$ is $\aleph_1$, while
for an arbitrary vocabulary $\tau$ it is $\aleph_1 + |\tau|$.  The Hanf number of $L(Q)$ for countable
vocabularies
        is  $ \beth _ \omega $ (See e.g. 3.3.12 of \cite{Schmerl}).  Thus,
        $$\beth_{\omega} =h^3(L_{\omega,\omega}(Q), \aleph_0) < h^3(L_{\omega_1,\omega}, \aleph_0)=\beth_{\omega_1}.$$

Computing the Hanf function of $L(Q)$ for vocabularies of cardinality $ \mu \ge  {\aleph_1} $ is considerably
more complicated. The key point is the following observation which does not depend on
Assumption~\ref{assumpt}. Since we have not been able to find it in print  we describe the key innovation
here.

\begin{theorem}\label{eqH} If $\kappa$ is uncountable,
 $h^3(L_{\omega,\omega}(Q), \kappa) = h^3(L_{\omega_1,\omega}, \kappa)$.
\end{theorem}

Proof.  Lopez-Escobar and Chang (e.g. \cite{Chang1} showed how to code sentences of $L_{\omega_1,\omega}$ as
first order theories omitting types.  Since each sentence will have only countably many subformulas, each
sentence regardless of the cardinality of the vocabulary can be coded by a first order sentence omitting
countably many types. And it is known (e.g. by the proof of Theorem 5.1.4 of \cite{Shelahbook}) that this
omitting types problem can be reduced to omitting one type $p$ of the form $P(x) \cup \{x\neq
c_n:n<\omega\}$. More precisely, we can find for any pair $(T,\Gamma)$ in a vocabulary $\tau$ (where $\Gamma$
is a countable collection of types) a pair $(T_1,\{p\})$ in an expanded vocabulary $\tau_1$ such that each
$\tau$-structure omitting $\Gamma$ can be expanded to a $\tau_1$-structure omitting $p$ and each model of
$T_1$ omitting $p$ also omits each type in $\Gamma$. This translation can be done for $\tau$ of any
cardinality. The crux of the current extension is that with an uncountable language one can further reduce
the omission of the type $p$ to the assertion $\neg (Qx) P(x)$. For this, add to the language unary function
symbols $F_{\alpha}$ for $\alpha< \aleph_1$ and a binary relation symbol $<$. Fix a family $\langle
\eta_\alpha:\alpha < \omega_1\rangle$ of distinct functions from $\omega$ to $2$. Now add axioms asserting
\begin{enumerate}
\item $<$ linear orders $P$;
\item $c_{n+1}$ is the $<$-successor of  $c_n$ and $c_0$ is $<$-minimal;
\item Each $F_\alpha$ maps $P$ onto $P$;
\item For $\alpha < \beta < \aleph_1$, if $(\eta_\alpha \myrestriction n \neq \eta_\beta \myrestriction n
    )$ then $(\forall x)(c_n< x \rightarrow F_\alpha(x) \neq F_\beta(x))$.

% $\{(\forall x)(c_n< x \rightarrow F_\alpha(x) \neq F_\beta(x) \mcolon (\eta_\alpha \myrestriction n
%    \neq \eta_\beta \myrestriction n ) \wedge (\alpha < \beta < \aleph_1)\}$
\end{enumerate}
Now in any model of these axioms if the type $p$ is realized by some $c$, the $F_{\alpha}(c)$ for $\alpha <
\aleph_1$ are uncountably many elements of $P$.  This establishes the reduction. $\qed_{\ref{eqH}}$

\medskip


\begin{remark}{\rm  This result can extended substantially; we could have replaced the study of sentences of
$L_{\omega_1,\omega}$ by theories of cardinality $\aleph_1$ without loss.   Basic facts concerning the  Hanf
number of $ {\Bbb L} _{
           \mu ^+, {\aleph_0}} $ for vocabularies of cardinality $ \mu \ge  {\aleph_1} $ appear in VII.5 of
           \cite{Shelahbook} and Chapter 4 of \cite{Baldwincatmon}. General sufficient conditions  to show
           $h^3(L_{\omega_1,\omega}, \kappa)=
           h^3(L_{\kappa^+,\omega}, \kappa)$ are studied in \cite{Sh831}.}
           \end{remark}
%\begin{enumerate}
%
%       \item at most the first measurable cardinal; \item at most the Hanf number of $ {\Bbb   L}   _{
%           \mu ^+, {\aleph_0}} $; \item if e.g. $ \mu= 2^ \kappa  < $ "first measurable" , then it is
%           equal to the cardinal in 2)
%           \end{enumerate}
%
% From VII.5 of \cite{Shelahbook}, we have $ HN({\Bbb   L}   _{ \mu ^+, {\aleph_0}}) \leq \beth_{(2^{\mu})^+}$.
% If $\mu$ is singular of
%           cofinality $\omega$, it is $\beth_{\lambda^+}$.

\medskip

However, the dependence on the size of the vocabulary disappears for the L\"owenheim
number of infinitary
second order logic.Without loss of generality we restrict to relational languages.
There is no loss in our
restriction to vocabularies of cardinality at most $\theta$ since no sentence of
$L_{\theta^+,\kappa}(II)$
can contain more than $\theta$ relation symbols.




%seem to me that for $ L(Q) $ : a. the Lowenheim number for countable vocabulary
%        is $ {\aleph_1} $  as is the case for the empty one
%b. the Lowenheim number for arbitrary vocabulary $ \tau $
%        is $ {\aleph_1} +| \tau | $
%
%d. the Hanf number for vocabularies of cardinality $ \mu \ge  {\aleph_1} $ is
%        d1. at most the first measurable cardinals
%        d2. at most the Hanf number of $ {\Bbb   L}   _{ \lambda ^+, {\aleph_0}}  $
%        d3. if eg $ \mu= 2^ \kappa  < $ "first measurable" , then it is
%        equal to  the cardinal in d2
%
%e.can say+quote more in e1. d e2. and  we can say more if we fix the vocabulary (eg if only unary predicates,
%if on unary function and unary predicates; rest; but probably


%\begin{definition}\label{withvocab} Let the  L\"owenheim number $\ell^3( L_{\theta^+,\kappa}(II))$
%be the least cardinal $\mu$ such that for any vocabulary $\tau$ of cardinality $\leq \theta$ and any $\phi
%\in L_{\theta^+,\kappa}(II)(\tau)$, if $\phi$ has a model it has one of cardinality less than or equal $\mu$.
%\end{definition}

%Without loss of generality we restrict to relational languages. There is no loss in our restriction to
%vocabularies of cardinality at most $\theta$ since no sentence of $L_{\theta^+,\kappa}(II)$ can contain more
%than $\theta$ relation symbols.

Note that just by existentially quantifying out the relation symbols, for any $\kappa\leq \theta^+$:
$$\ell^3( L_{\theta^+,\kappa}(II),\theta)\leq \ell^2(L_{\theta^+,\theta^+}(II)).$$

Combining this observation with $\lambda_2=\lambda_3$ in Theorem~\ref{sumthm}, we have:

\begin{corollary}\label{nocab} $\ell^3( L_{\theta^+,\kappa}(II),\theta ) =
\ell^2(L_{\theta^+,\kappa}(II))$.
\end{corollary}

We arrived at this result using our analysis of the Hanf number of $P^{\theta}_N$.  But a variant of
Corollary~\ref{nocab} can be obtained, using the same ideas of coding syntax, without the detour through
$P^{\theta}_N$.  It shows that with only a finite vocabulary one can code any reasonable similarity type.

\begin{notation} $\tau_{**}$ is the  vocabulary containing one unary predicate $P$, one binary relation
symbol $<$ and a ternary predicate $R$.
%sequence $\langle R_i\mcolon i \in \theta\rangle$ of binary relation symbols.
\end{notation}

\begin{theorem}\label{smcab} $\ell^3( L_{\theta^+,\kappa}(II),\theta) =
\ell^2(L_{\theta^+,\kappa}(II)(\tau_{**}))$.
\end{theorem}


Proof. Instead of considering an arbitrary vocabulary of size $\theta$ we can consider a `universal'
vocabulary $\sigma$ of $\theta$ binary relations. (It is easy to code the first in the second.) So the claim
is that for any sentence $\phi \in L_{\theta^+,\kappa}(II)(\sigma)$ there is a sentence $\phi^*$ of
$L_{\theta^+,\kappa}(II)(\tau_{**})$ such that $\spec(\phi) =\spec(\phi^*)$. (The converse is obvious.)

Let $\phi^*$ assert
\begin{enumerate}
\item $(P,<)$ is a well-order.
\item the formula obtained by replacing each occurence of $R_i(\sigma_1,\sigma_2)$ in $\phi$ by the
    formula (where $z$ does not occur in $\phi$):
    $$(\exists z) [P(z) \meet \otp(\{y\mcolon P(y) \meet y < z\}) =i \meet R(z,\sigma_1,\sigma_2)].$$
(Here $\sigma_1, \sigma_2$ are arbitrary terms from $\phi \in L_{\theta^+,\kappa}(II)(\sigma)$, i.e.
individual variables.)
\end{enumerate}

Now there is a model  $N$ with cardinality $\lambda \geq \theta$ of $\phi^*$ where $P^N$ has order type
$\theta$, such that $N\models \phi^*$ if and only if there is a $\sigma$-structure $M$ with cardinality
$\lambda$ satisfying $\phi$. $\qed_{\ref{smcab}}$.
%\bibliography{ssgroups}
%\bibliographystyle{alpha}
%%\bibliographystyle{plain}
%\end{document}
\begin{thebibliography}{Mag71}

\bibitem[Bal09]{Baldwincatmon} John~T. Baldwin.
\newblock {\em Categoricity}.
\newblock Number~51 in University Lecture Notes. American Mathematical Society,
  2009.
\newblock www.math.uic.edu/\~\ jbaldwin.

\bibitem[Cha68]{Chang1} C.~C. Chang.
\newblock Some remarks on the model theory of infinitary languages.
\newblock In J.~Barwise, editor, {\em The syntax and semantics of infinitary
  languages}, pages 36--64. Springer-Verlag, 1968.
\newblock LNM 72.

\bibitem[GS05]{Sh831} R.~Goebel and S.~Shelah.
\newblock How rigid are reduced products?
\newblock {\em J Pure and Applied Algebra}, 202:230--258, 2005.

\bibitem[Han60]{Hanf} William Hanf.
\newblock Models of languages with infinitely long expressions.
\newblock In {\em Abstracts of Contributed papers from the {F}irst {L}ogic,
  {M}ethodology and {P}hilosopy of Science Congress, Vol.1}, page~24. Stanford
  University, 1960.

\bibitem[Mag71]{Magidor} M.~Magidor.
\newblock On the role of supercompact and extendible cardinals in logic.
\newblock {\em Israel Journal of Mathematics}, 10:147--157, 1971.

\bibitem[New]{Newelskihanf} Ludomir Newelski.
\newblock Bounded orbits and measures on a group.
\newblock preprint.

\bibitem[Sch85]{Schmerl} J.~Schmerl.
\newblock Transfer theorems and their applications to logics.
\newblock In J.~Barwise and S.~Feferman, editors, {\em Model-Theoretic Logics},
  pages 177--209. Springer-Verlag, 1985.

\bibitem[She78]{Shelahbook} S.~Shelah.
\newblock {\em Classification {T}heory and the {N}umber of {N}onisomorphic
  {M}odels}.
\newblock North-Holland, 1978.

\bibitem[Vaa79]{Vanhanf} Jouko Vaananen.
\newblock On the {H}anf numbers of unbounded logics.
\newblock In B.Mayoh F.Jensen and K.Moller, editors, {\em Proceedings from 5th
  Scandinavian Logic Symposium}, pages 309--328. Aalborg University Press,
  1979.

\bibitem[Vaa82]{Vanlow} Jouko Vaananen.
\newblock Abstract logic and set theory {II}: Large cardinals.
\newblock {\em Journal of Symbolic Logic}, 47:335--346, 1982.

\end{thebibliography}




\end{document}


\section{Context}  Newelski deals with the following situation.
$T$ is a first order theory with countable vocabulary $\tau$.
It is assumed that the models of $T$ are in particular groups. Then each model of
$T$ acts on itself and therefore on its Stone space by left translation.
Let $\Mscr$ be a monster model of $T$.  We say $\Mscr$ has a {\em bounded orbit}
if there is a $p \in S(\Mscr)$ such that $|\{ap:a\in \Mscr\}|<<|\Mscr|$.  The
question is whether if we choose a new monster, it continues to have a bounded orbit.
In particular this is true (at least if we allow uncountable languages)
if $|\Mscr| > H(P_N)$. To see this, let $M\prec \Mscr$ be a model such that there
exist a family $\{q_{\alpha}\mcolon \alpha< \mu\}$ such that each ${q_{\alpha}$ has
a unique extension to some $ap$; each $ap$ restricts to some ${q_{\alpha}$ and (wolog)
each $a \in M$.  Now we are asking, can we extend $\Mscr$ to $\Mscr'$ which is saturated
and omits the type of a $c$ such that $c q_{0} \neq c_{\alpha}$ for $\alpha< \mu$.

\bibliography{ssgroups}
%\bibliographystyle{alpha}
\bibliographystyle{plain}

\end{document}
\section{Context}

We put our restriction to $\{\lambda=\lambda^{<\lambda}$ in a more general setting.

\begin{definition} Let $\psi$ be a sentence of  second order logic.
\begin{enumerate}
\item $\spec^1(\psi) = \{\lambda: \lambda \models \psi\}$.
\item $\spec^2(\psi) = \{\lambda:\lambda=\lambda^{<\lambda} \wedge \lambda \models \psi\}$
\end{enumerate}
\end{definition}

Recall we assume there are proper class of $\lambda$ with $\{\lambda=\lambda^{<\lambda}$.

\begin{lemma}\label{nodiff} For every sentence $\psi$ of  second order logic, there is a sentence $\psi^*$ such that:
$$\spec^2(\psi) = \spec^1_*(\psi).$$
\end{lemma}

Proof.  $\psi_*$ just expresses the conjunction of $\psi$ with $\lambda^{<\lambda} = \lambda$.
$\qed_{\ref{nodiff}}$


\begin{definition} Define $H^2$ and $\ell^2$ to  be the Hanf and Lowenheim numbers with respect to $\spec^2$.
\end{definition}

We can show

\begin{corollary}\label{eqlow}  $H(L^{II}) = H^2(L^{II})$ and $\ell(L^{II}) = \ell^2(L^{II})$
\end{corollary}

Proof. By Lemma~\ref{nodiff} and recalling that for either spectrum $\ell(L^{II}) = \sup\{\min\{\spec^i(\phi)\}\}$,
since every 2-spectrum is a 1-spectrum the $\ell^2(L^{II})\leq \ell^1(L^{II})$.

But the opposite inequality also holds. Let $\phi$ be a sentence with a non-empty 2-spectrum.  Then let $\phi^*$ express
$(\exists U) \phi^U \wedge \lambda^{<\lambda} = \lambda$.  Now $\phi^*$ has a non-empty 2-spectrum so it has a model $M$
with cardinality $|M|= \kappa \leq \ell^2(L^{II})$. But then $\phi$ has a model with cardinality $|U^M| \leq \kappa \leq
\ell^2(L^{II}$.  So $\ell^1(L^{II})\leq \ell^2(L^{II})$.

%A similar argument works for Hanf numbers.
$\qed_{\ref{eqlow}}$




%\thanks{The author is partially supported
%by NSF grant  DMS-0500841.  }
%\address{John T. Baldwin\\ Department of
%Mathematics, Statistics and Computer Science\\University of
%Illinois at Chicago }



Proof. We shall construct for each sentence $\phi$ is second order logic, whose smallest model $M_\phi$ has cardinality
$\kappa_\phi$, a pair $(T,L')$,

For simplicity, fix $\kappa = \aleph_0$.




















%\section{First attempt}



 Proof.
         \sidebar{What logic? What is role of $\kappa$ and $\mu$? What is the vocabulary for 1)}
         Proof: 2) implies 1. Write the sentence of second order logic asserting that situation of clause 2) holds in
         $\mu$.

         \sidebar{ What is role of $\kappa,\mu$?}

         1) implies 2).   There are an $A \subseteq \kappa$ and second order $\psi$ with $<, Q_1, Q_2, c_i:i< \kappa$
         contained in $\tau(\phi)$ such that $\psi$ has a model of cardinality $\lambda>\kappa$ iff there is a model $N$
         with
         cardinality $\lambda$ of $\phi$ such that $Q^N_2 -\langle c_i:i<\kappa\rangle$, $c_i<^N c_j$ if and only $i<j<
         \kappa$
          $Q^N_1 = \{c_i:i\in A\}$.

         \sidebar{What is $\phi$?  What is the role of preceding paragraph in the proof?}

         Now fix vocabularies $\tau_0,\tau_1$ and theory $T_1$ and type $p$ as follows:
     $\tau_0 = \{Q_0,<_0\}$,  $\tau_1 =  \tau_0\cup \{Q_1,Q_2,Q_3,<_1,F, F_1, R, c_i:i<\kappa\}$
         where $F$ is a binary function.

         The axioms of the first order theory $T_1$ assert:


         \sidebar{$Q^M$ notation in abeyance.}

         \begin{enumerate}
        \item $<_0$ is a partial order on $Q_0$.

         \sidebar{similar to partial finite functions from $\omega$ to $\omega$ ???}

       \item The first coordinate of $F$ is from $Q_0$, the second from $Q_3$ and for each $a \in Q_0$,
         $\lambda y F(a,y)$ denotes a partial function from $Q_3$ to $Q_3$.

         \item If $a< b$ then $F_a \subseteq F_b$.

         \item for some $a \in Q_0$, $F_a$ is the empty function.
         \item If $a \in Q_0$ and $b \in Q_3 - \dom F_a$, then there is an $a_1 > a$ with $b \in \dom F_{a_1}$.

         \item $Q_1 \subseteq Q_2 \subseteq Q_3$

         \item The $c_i:i<\kappa$ list $Q_2$ without repetition.



         \sidebar{This depends on specific model}

         \item $<_1$ linearly orders $Q_3$ and $Q_2$ picks out an initial segment.

           \item The $c_i:i\in A$ list $Q_1$.

       \item $F_1$ is a pairing function on $Q_3$ such that
       $F_1(a_1,b_1) <_1 F_1(a_2,b_2)$ iff $\min(a_1,b_1) <\min(a_2,b_2)$ or $a_1<_1 a_2$ or $a_1 =a_2$ and $b_1<_1
       b_2$.

       \item For $a \in Q_3$, $U_a = \{b:R(a,b)\}$.

       \sidebar{Is $R$ a new relation or varies over all formulas?}


       The family of subsets $U_a$ of $Q_3$
       is closed under Boolean operations, includes all $<_1$ initial segments, closed under images by $F_a$ for
       each $a \in Q_0$.

       \item \sidebar{condition  q is illegible}
\end{enumerate}
         \end{document}

         \begin{definition}  Let $L' \subseteq L$ and $T$ be a complete countable theory in $L$.  For any cardinal
         $\kappa$, $T$ is said to be
$\kappa$-saturated over $L$ if: for a model $M$ of $T$ satisfies that $M \restriction L$ is $\kappa^+$-saturated then
$T$ is $\kappa^+$-saturated.
\end{definition}

The usual soft Hanf argument shows that there is a $\kappa^*$ such that for every pair $(T,L')$ as above,
if $T$ is $\kappa^*$-saturated over $L$, then $T$ is
$\kappa'$-saturated over $L$ for arbitrarily large $\kappa' \geq \kappa$.

See Pabion in re eventual truth.

\begin{theorem} $\kappa$ is at least the L\"owenheim of second order logic.
\end{theorem}
\begin{theorem} For all cardinals
$\kappa < \lambda$ with $\lambda^{<\lambda} = \lambda$, the following are equivalent:

\begin{enumerate}
 \item There is a sentence $\psi$ in $L_{\kappa^+,\omega}$ (second order???) such that if $\chi< \mu = \mu^{<\mu}<
     \lambda$, there is a model $M$ of $\psi$ with $|M| = \mu$.


     \item  there are vocabularies $\tau_0 \subseteq \tau_1$ of cardinality $\kappa$ and a $\tau_1$ theory $T_1$ and
         type $p$ such that there is a model $M$ of $T_1$ of cardinality $\lambda$ which omits $p$ but $M
         \restriction \tau_0$ is saturated.

         \end{enumerate}

  \end{theorem}

\begin{remark} We reformulate the main direction. For any second order sentence $\phi$, we construct a triple
$(T,T_1,p)$ such that if there is a model of $T_1$ of cardinality $\lambda$ that omits $p$ but whose restriction to
$\tau$ is saturated,
then for any $\mu^{<\mu} =\mu < \lambda$, $\phi$ has a model in $\lambda$.

This shows the Hanf number of $\Phi$ is at least  the Hanf number of second order logic.


\end{remark}

old and probably wrong:

\begin{claim}\label{end} If $\mu = \mu^{<\mu}$ with $\mu < \lambda=\lambda^{<\lambda}$ and $M$ is model of $T_1$ that
omits $p$ but whose reduct to $\tau$ is saturated then
$\mu \models \phi$.

%for every $\chi< \mu$, $\chi \models \phi$.
\end{claim}

Proof. Since $\mu< \lambda$, $\mu$ is an initial segment of $Q_1$.  But then each subset $Y$ of $\mu$ gives rise to a
type $\{R(y,x): y\in Y\} \cup \{\neg R(y,x): y\not \in Y\}$ and this type has cardinality less than $\lambda$. We finish
by Fact~\ref{key}.
$\qed_{\ref{end}}$


The converse is more complicated.  We need the following reduction to deal with the restriction to cardinals of the form
$\lambda^{<\lambda}=\lambda$.

\sidebar{
begin{notation} For any second order sentence $\psi$, we denote by $\psi^*$ the sentence which true of $\lambda$ if and
only if


$$(\forall \mu< \lambda)[\mu^{<\mu}=\mu \rightarrow (\forall \theta < \mu) \theta \models \neg \psi].$$
end{notation}

Note $\psi^*$ is inconsistent if $\lambda_0^{<\lambda_0}=\lambda_0$. We should either declare $\psi_* =\psi$ in that
case or introduce a strict inequality.


Revised:}

\begin{notation}
Let $\lambda_0 = \min(\spec \psi)$ and
$\lambda_1 = \min\{\lambda:\lambda \geq \lambda_0 \wedge \lambda^{<\lambda}=\lambda\}$.

For any second order sentence $\psi$, we
let $\psi^* = \neg \psi$ if $\lambda_0 =\lambda_1$.  If $\lambda_0 <\lambda_1$
then $\psi^*$ is the sentence which is true of $\lambda$ if and only if


$$(\forall \mu< \lambda)[\mu^{<\mu}=\mu \rightarrow (\forall \theta < \mu) \theta \models \neg \psi].$$
\end{notation}



\begin{lemma}\label{fixspec}
If $\lambda^{<\lambda}=\lambda\}$ then
$$(\forall \chi<\lambda) \chi \models \psi^* \hbox{\rm \ if and only if \  } \lambda \leq \lambda_1.$$
\end{lemma}

Proof.  If $\lambda_0 = \lambda_1$, the claim is clear.  If $\lambda_0 < \lambda_1$, since $\lambda_0 \models \psi$, any
$\lambda \geq \lambda_1$ fails $\psi^*$.  So if $\lambda > \lambda_1$, it is not the case that $(\forall \chi<\lambda)
\chi \models \psi^*$.

But if $\lambda^{<\lambda}=\lambda$ and $\lambda \leq \lambda_1$, any cardinal $\mu < \lambda$ satisfying
$\mu^{<\mu}=\mu$ is less than $\lambda_0$ and the result follows.

$\qed{\ref{fixspec}}$


 Let
 $$\mu_1=\sup\{\sup(\spec^2(\phi_T))\mcolon \bt \in \Tscr^f {\ \rm and \ }
 \spec^2( \phi_{\bt}) \hbox{\ \rm is bounded} \}.$$

 There are two cases.  1) For some bounded  $\bt_1\in \Tscr^f$, $\mu_1\in \spec^2({\bt}_1)$.  Then
 $H(P^f_N)= \mu_1^+$ and $\mu_1 \geq \ell^2(L^{II})$.


 % Choose $\phi^1_{\bt}$ so that $\mu \models \phi^1_{\bt}$ if and only if
%$$\mu  \models \neg\phi_{\bt} \wedge (\forall \kappa <\mu)(\exists \lambda (\kappa \leq \lambda <\mu)[ \lambda \models \phi_{\bt}].$$




%Then $\min(\spec(\phi^1_{\bt})) = \mu_0^+ > \ell^2(L^{II})$ and we finish.

 2) 1) fails, then $H(P^f_N)= \mu_1$ and
 % Either for some $\bt$, $\sup(\phi_{\bt}) = H(P^f_N)$ or
  for every $\mu < H(P^f_N)$ there is a $\bt\in \Tscr^f $
with bounded $2$-spectrum such that
 $\mu<\sup(\spec^2(\phi_{\bt})) < H(P^f_N) $. Then, there is $\bt_2 \in \Tscr^f$
with a bounded $2$-spectrum  and  $\mu_2 =\sup(\spec^2(\phi_{\bt_2})) >\ell^2(L^{II})$.

For any $\bt$, the definition of $P^f_N$ guarantees that if $ \lambda^{<\lambda} =\lambda, \mu^{<\mu} =\mu, \mu >
\lambda$ and some $M \models \bt$ has cardinality $\mu$ then some $N \models \bt$ has cardinality $\lambda$ (take a
saturated elementary submodel). Thus for $i =1,2$,
 for every $\lambda < \mu_i$ if $ \lambda^{<\lambda} =\lambda$ then $\lambda \models   \phi_{\bt_i}$.


Define $\psi_i$ so that $\mu \models \psi_i$ if and only $(\exists \lambda)[ \lambda < \mu \meet \lambda^{<\lambda}
=\lambda \meet \lambda \models \neg  \phi_{\bt_i}]$.
Depending on the case we have:

 $\min \spec^2(\psi_1)  >\mu_1 \geq\ell^2(L^{II})$ or
 $\min \spec^2(\psi_2) \geq \mu_2 >\ell^2(L^{II})$. In either case the contradiction proves the theorem.

%Choose $\phi^2_T$ so that $\mu \models \phi^2_T$ if and only if
%$$(\forall \kappa <\mu)(\exists \lambda (\kappa \leq \lambda <\mu)[ \lambda \models \phi_T].$$
%Then $\min \spec\, (\phi^2_T) > \ell^2(L^{II}$ and this contradiction yields the theorem.
%This
%contradicts the definition of the L\"owenheim number.

%(Either there exists a $T$ with $\sup\spec(\bt) = H(P^f_N)$ or $H(P^f_N)$ is a limit of $\sup\spec(\bt)$ for various $T$ and one is strictly greater than $\ell^2(L^{II})$.) Trivially, $\spec^2(\phi_{\bt})$ is an initial segment of the cardinals satisfying $\mu^{<\mu}=\mu$ as we can choose a saturated elementary submodel of the given member of $\bt$ of the appropriate cardinality which necessarily omits $p$.  But then, $\neg\phi_{\bt}$ has a
%model and  $\min\spec(\neg\phi_{\bt})\geq \sup\spec(\bt)\geq \ell^2(L^{II})$.
 %But then
%$$\min{\bt} (\min\spec(\bt)) \geq \min{\phi} (\min\spec(\phi)) = \ell(L^{II}).$$
