Subrecursive Programming Systems pp 221-222 | Cite as

# Further Problems

## Abstract

A difficulty with all the work on relative succinctness up to this writing (including this book) is that the programs shown to witness relative succinctness fail to compute anything particularly interesting. An analogous state of affairs held in the area of incompleteness of formal systems for about 45 years. In 1931, Gödel published his famous paper on incompleteness of formal systems [Göd86], which showed, among other amazing things, that there are true sentences in the language of first-order arithmetic which are independent of Peano Arithmetic (assuming PA is *ω*-consistent). A problem with these results is that the sentences shown independent of PA fail to have much import for mathematics outside of logic. In the late 70s, Paris and Harrington [PH77] showed that a version of Ramsey’s Theorem is independent of Peano Arithmetic, thus giving a finite combinatorial version of Gödel’s incompleteness theorem.^{1} It would be interesting to see some analog of the Paris-Harrington Theorem for relative succinctness of programming systems.^{2}