Mathematics is about proofs, that is the derivation of correct statements; and calculations, that is the production of results according to well-defined sets of rules. The two notions are intimately related. Proofs can involve calculations, and the algorithm underlying a calculation should be proved correct. The aim of the author is to explore this relationship. The book itself forms an introduction to simple type theory. Starting from the familiar propositional calculus the author develops the central idea of an applied lambda-calculus. (...) This is illustrated by an account of Gödel's T, a system which codifies number-theoretic function hierarchies. Each of the book's 52 sections ends with a set of exercises, some 200 in total. These are designed to help the reader get to grips with the subject, and develop a further understanding. An appendix contains complete solutions of these exercises. (shrink)

This work is derived from the SERC "Logic for IT" Summer School Conference on Proof Theory held at Leeds University. The contributions come from acknowledged experts and comprise expository and research articles which form an invaluable introduction to proof theory aimed at both mathematicians and computer scientists.

I modify the standard coverage construction of the reals to obtain the irrationals. However, this causes a jump in ordinal complexity from ω+1 to Ω.

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Gödel’s T.

In Simmons (Arch Math Logic 43:65–83, 2004), I described a method of producing ordinal notations ‘from below’ (for countable ordinals up to the Howard ordinal) and compared that method with the current popular ‘from above’ method which uses a collapsing function from uncountable ordinals. This ‘from below’ method employs a slight generalization of the normal function—the fruitful functions—and what seems to be a new class of functions—the helpful functions—which exist at all levels of the function space hierarchy over ordinals. Unfortunately, (...) I was rather sparing in my description of these classes of functions. In this paper I am much more generous. I describe the properties of the helpful functions on all finite levels and, in the final section, indicate how they can be used to simplify the generation of ordinal notations. The main aim of this paper is to fill in the details missing from [7]. The secondary aim is to indicate what can be done with helpful functions. Fuller details of this development will appear elsewhere. (shrink)

I survey the syntactic technique of tiering which can be used to restrict the power of a recursion scheme. I show how various results can be obtained entirely proof theoretically without the use of a model of computation.