%trite.tex: An Infinite Sequence of True but Trite Sentences
%a Plain TeX file of an article D. Zeilberger
%begin macros
\def\L{{\cal L}}
\baselineskip=14pt
\parskip=10pt
\def\Tilde{\char126\relax}
\def\halmos{\hbox{\vrule height0.15cm width0.01cm\vbox{\hrule height
0.01cm width0.2cm \vskip0.15cm \hrule height 0.01cm width0.2cm}\vrule
height0.15cm width 0.01cm}}
\font\eightrm=cmr8 \font\sixrm=cmr6
\font\eighttt=cmtt8
\magnification=\magstephalf
\def\lheading#1{\bigbreak \noindent {\bf#1}
}
\parindent=0pt
\overfullrule=0in
%end macros
\bf
\centerline
{An Infinite Sequence of Trite but True Sentences}
\rm
\bigskip
\centerline{ {\it Doron Zeilberger}\footnote{$^1$}
{\eightrm \raggedright
Department of Mathematics, Temple University,
Philadelphia, PA 19122, USA.
%\break
{\eighttt zeilberg@math.temple.edu} \hfill \break
{\eighttt http://www.math.temple.edu/\Tilde zeilberg .}
Jan. 30, 1998. Exclusive to the Pesonal Journal of
Shalosh B. Ekhad and Doron Zeilberger.
{\eighttt http://www.math.temple.edu/\Tilde zeilberg/pj.html .}
}
}
{\it \qquad \qquad ``...but trite is not the opposite
of true, Hanna, also the sentence two times two
is four is trite, and nevertheless...'',
----
from `My Michael' by Amos Oz, p. 179}.
{\it \qquad \qquad
``Although it was well understood that linguistic
processes are in some sense ``creative'', the technical
devices for expressing a system of recursive processes
were simply not available until much more recently.
In fact, a real understanding of how a language can
(in Humboldt's words) ``make infinite use of finite means''
has developed only within the last thirty years,
in the course of studies in the foundation of mathematics''
----
Noam Chomsky, `Aspects of the Theory of Syntax',
1965, p. 8}.
The worst clich\'e is `that's a clich\'e'. Hence
{\bf Prop. 1.} S is trite implies that `S is trite' is trite.
\halmos \quad We also have
{\bf Prop. 2.} S is true implies that `S is true' is true.
\halmos \quad Hence
{\bf Corollary.} S is trite but true implies that
`S is trite but true' is trite but true.
Define
$S_0 :=$ two times two is four, and for $i>0$,
$S_i :=$ `$S_{i-1}$ is trite but true' .
Then $\{ S_i \}$ is the desired infinite sequence.
Of course the present construction is trite,
but it is, {\it bekhol zot} (nevertheless) true!
\bye