An Infinite Sequence of Trite but True Sentences

Doron Zeilberger^{1}

“...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.

“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 cliché is ‘that’s a cliché’. Hence

Prop. 1. S is trite implies that ‘S is trite’ is trite. We also have

Prop. 2. S is true implies that ‘S is true’ is true. Hence

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, bekhol zot (nevertheless) true!

1 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. zeilberg@math.temple.edu

http://www.math.temple.edu/~zeilberg . Jan. 30, 1998. Exclusive to the Pesonal Journal of Shalosh B. Ekhad and Doron Zeilberger. http://www.math.temple.edu/~zeilberg/pj.html .