Advanced chapters cover the Ackerman function, Tarskiâs theorem on the non-representability of truth, Goedelâs incompleteness and Rosserâs incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob's deliverability conditions, Churchâs thesis, the second recursion theorem and applications, a provably recursive universal function for the primitive recursive functions, Oraclecomputations and various classes of computable functionals, the Arithmetical hierarchy, Turing reducibility and Turing degrees and the priority method, a thorough exposition of various versions of the first recursive theorem, Blumâs complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham's feasibly computable functions.