This book is first author's dissertation that is submitted in accordance with the requirements for the degree of Doctor of Philosophy to The University of Leeds, Department of Pure Mathematics in January 2000 under the direction of second author with the title "Tiered Arithmetic, its Functional Interpretation and Slow Growing Bounds." A two-sorted version of Peano Arithmetic is developed, with proof-rules corresponding to the normal/safe recursion schemes of Bellantoni and Cook. Classical methods of proof theory still apply, but now the provably recursive functions are brought down to more...