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\begin{center}{\huge Continuous Functionals 
of Dependent and Transfinite Type}\end{center}
\begin{center}{\large Ulrich Berger}\end{center}
\begin{center}{\large University of Munich}\end{center}

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\large

Dag Normann generalized the Kleene--Kreisel continuous functionals
[1],[2] to a constructive set theoretic interpretation of dependent
types and their basic constructors [3].  For almost all applications
of this interpretation the density theorem is crucial [4],[5],[6].  We
review Normann's model in a more abstract domain-theoretic setting [7],[8],
and prove density theorems for iterated super universe operators.
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\noindent
References:
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\noindent
[1] S.~C.~Kleene.
Countable functionals.
In A.~Heyting, editor, 
{\it Constructivity in Mathematics}, 
pages 81--100,
North--Holland, Amsterdam, 1959.

\noindent
[2] G.~Kreisel.
Interpretation of analysis by means of
constructive functionals of finite types.
In A.~Heyting, editor, 
{\it Constructivity in Mathematics}, 
pp.~101--128,
North--Holland, Amsterdam, 
1959.

\noindent
[3] L.~Kristiansen, D.~Normann.
Semantics for some constructors of type theory.
In Behara, Fritsch, Lintz, editors, 
{\it Symposia Gaussiana}, 
pp.~201--224,
de Gruyter,
1995.

\noindent
[4] D.~Normann.
Closing the gap between the continuous functionals and recursion in ${}^3E$.
To appear in the Archive of Mathematical Logic, 
1995.

\noindent
[5] D.~Normann.
Continuity, proof systems and the theory of transfinite computations.
{\it Preprint Series, Inst.~Math.~Univ.~Oslo No.~19},
1996. 

\noindent
[6] D.~Normann.
Representation theorems for transfinite computability and definability.
{\it Preprint Series, Inst.~Math.~Univ.~Oslo No.~20},
1996. 

\noindent
[7] E.~Palmgren and V.~Stoltenberg--Hansen.
Domain interpretations of Martin--L{\"o}f's
partial type theory.
{\it Annals of Pure and Applied Logic} vol.~48,
pp.~135--196, 
1990.

\noindent
[8] U.~Berger.
Total sets and objects in domain theory.
{\it Annals of Pure and Applied Logic} vol.~60,
pp.~91--117, 1993.

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