\documentclass[a4paper]{article}
\title{More on $\kappa$-like models}
\author{Richard Kaye\\School of Mathematics and Statistics\\
        The University of Birmingham\\Edgbaston\\Birmingham\\B15 2TT\\U.K.\\
        \texttt{R.W.Kaye@bham.ac.uk}}
\date{26th April, 1996}

\newtheorem{theorem}{Theorem}

\begin{document}

\maketitle

In this document, I will list any extra observations since writing my two papers
on $\kappa$-like models~\cite{kappa1,kappa2}.  The notation follows that in
those papers.

\begin{theorem} $\mbox{\rm IPHP} \vdash \mbox{\rm IPHP}_{\rm cf}$ 
\end{theorem}
\noindent\textit{Proof.}  Let $M\models\mbox{\rm IPHP}$.  Then either
$M\models\mbox{\rm PA}$ in which case $M\models\mbox{\rm IPHP}_{\rm cf}$,
or $M\models \neg B\Sigma_n$ for some $n$.  Suppose $b\in M$ and
\[ M\models \forall {x{<}a}\, \exists y\, \theta(x,y) \wedge
            \forall z\, \exists {x{<}a}\, \forall {y{<}z}\, \neg\theta(x,y).
\]
Let $K\succ M$ be a proper extenstion, where the extension preseves
$\{ x: x<\max(a,b) \}$; $K$ exists by $\mbox{\rm IPHP}$.  It suffices
to show that $M\subseteq_{\rm cf} K$; but if not there is $c\in K$
greater than all $y\in M$, so by elementarity
\[ K\models \forall {x{<}a}\, \exists {y{<}c}\, \theta(x,y) \]
hence
\[ M\models \exists z\, \forall {x{<}a}\, \exists {y{<}z}\, \theta(x,y) \]
a contradiction.

\begin{thebibliography}{1}

\bibitem{kappa1} R. W. Kaye, `Constructing $\kappa$-like models of arithmetic', 
to appear in \textit{The Journal of the London Mathematical Society}, 1996. 

\bibitem{kappa2} R. W. Kaye, `The theory of $\kappa$-like models of arithmetic', 
\textit{Notre Dame Journal of Formal Logic} \textbf{36}, Fall 1995, 547--559. 

\end{thebibliography}

\end{document}
