# new dissertation and talk



Dear all,

I have just completed and successfully defended my dissertation
Linear Set Theory,'' which is to be submitted to the department of
philosophy at Stanford University. I would like to post the abstract
of the thesis in this newsgroup. This is a LaTeX fle.

This dissertation work was supervised by Grigori Mints and
Solomon Feferman. The result on LZF will be presented in
the coming 1993-1994 ASL annual meeting in Gainesville, Florida,
as one of the contributed papers (Sunday afternoon).

Please note that my talk is NOT in the recently posted program of the
meeting since it was scheduled after the program had been in print.

Thank you very much,

Masaru Shirahata

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\documentstyle[11pt]{article}

\begin{document}

\title{Linear Set Theory}
\author{Masaru Shirahata\\
Center for the Study of Language and Information\\
Stanford CA 94305\\
masaru@csli.stanford.edu}
\date{February 28, 1994}
\maketitle

\begin{large}
\centerline{\bf Abstract }
\end{large}

\medbreak

\noindent
In this thesis, we develop four systems of set theory based
on linear logic.  All of those systems have the principle of
unrestricted comprehension but they are shown to be consistent. The
consitency proofs are given by establishing the cut-elimination
theorems.

Our first system of linear set theory SMALL is formulated in full
linear logic, {\em i.e.}, with exponentials. However we do not allow
exponentials to appear inside of set terms.

Secondly, we formulate a system of set theory in linear logic with
exponentials. This system is called AS$^{\infty}$.

Thirdly, we present the system of linear set theory LZF which is a
conservative extension of Zermelo-Fraenkel set theory without the
axiom of regularity or ZF$^{-}$. The idea is to build up a linear set
theory on top of ZF$^{-}$ in a style similar to SMALL. We establish a
partial cut-elimination result for LZF, and derive from it that LZF is
a conservative extension of ZF$^{-}$, and therefore consistent
relative to ZF$^{-}$.

Our last system of linear set theory AZF further extends LZF with
weakening and the substitutivity principle, which are not available in
LZF.

In addition, we present the phase-valued model of linear set theory
which is an extension of the Boolean-valued model of classical set
theory.  This only gives a model of linear set theory with the axiom
of separation and not with the unrestricted comprehension.

Finally, we explore possible applications of linear set theory,
particularly to the foundations of category theory.

\end{document}