On Wednesday 10 November 2010 1:42:00 pm Petr Pudlak wrote:
I was reading the paper "Total Functional Programming" [1]. I encountered an interesting note on p. 759 that primitive recursion in a higher-order language allows defining much larger set of function than classical primitive recursion (which, for example, cannot define Ackermann's function). And that this is studied in in Gödel's System T. It also states that this larger set of primitive functions includes all functions whose totality can be proved in first order logic.
I was searching the Internet but I couldn't find a resource (a paper, a book) that would explain this in detail, give proofs etc. I'd be happy if someone could give me some directions.
Girard's book, Proofs and Types, has some stuff on System T. A translation is freely available: http://www.paultaylor.eu/stable/Proofs+Types.html Skimming, it looks like he gives an argument that T can represent all functions that are provably total in Peano arithmetic. The rest of the book is also excellent. -- Dan