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\begin{center}{\huge Minimal Upper Bounds in the Theory of 
$\Delta^1_3$-degrees}\end{center}
\begin{center}{\large Philip Welch}\end{center}
\begin{center}{\large Kobe University, Japan}\end{center}

\bigskip
\large

We consider   the interaction between core model theory, 
{\boldmath $\Delta^1_2$}-determinacy and the theory of
 the $\Delta^1_3$-degrees, 
(where a degree is an equivalence class in the reducibility ordering
``$r \leq_{\Delta^1_3} s" \Leftrightarrow r \mbox{ (as a set of integers)
 is a } \Delta^1_3$-relation in the predicate $ s$),
establishing a result on minimal upper bounds
for countable collections of such degrees.\\

\noindent
We prove: \\
\mbox{}\\
\noindent
$ ZFC +  \mbox{{\boldmath{$\Pi^1_1$}}}\mbox{-determinacy} \vdash$

Either  {\boldmath{$\Delta^1_2$}}-determinacy holds, or 
there is a real  $d_0$ so that if $D = \{\mbox{{\boldmath $d_n$}}\}_{n\in \omega}$
is a countable collection of $\Delta^1_3$-degrees above that of $d_0$,
then $D$ has a minimal upper bound.

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