A Mathematical Incompleteness in Peano Arithmetic

Publisher Summary This chapter investigates a reasonably natural theorem of finitary combinatorics, a simple extension of the Finite Ramsey Theorem. The chapter demonstrates that this theorem, while true, is not provable in Peano arithmetic. The chapter works extensively with the partition calculus. In the proof of the main theorem, the chapter relies on various proof-theoretic results, in particular, on Godel's Second Incompleteness Theorem. It is possible, however, to prove the main theorem using only model-theoretic methods. This is the approach where a general model-theoretic methodology (called indicator functions) for producing such results is developed.