Formalisations of recursive arithmetic.

In this thesis we shall present a new formalisation of the theory of primitive recursive functions, which is called Ternary Recursive Arithmetic. In a recent paper, Alonzo Church described a formalisation of recursive arithmetic in which single axioms of recursion and composition (i.e. definition by explicit substitution) took the place of an infinity of such axioms in earlier codifications. Church's system, however, postulates axioms of the propositional calculus and of mathematical induction, in Ternary Recursive Arithmetic these axioms have been eliminated in the manner of Goodstein. In chapter 1 a full statement of the primitive basic of the system will be given and in chapters 2, 3 and 4 we shall present a development of it and state precisely in what sense it may be considered a formalisation of the theory of primitive recursive functions. The main motivation of this work is that it enable us to give a proof of the consistency of primitive recursive arithmetic in a much simpler system than was hitherto possible; that is, in the system consisting of Ternary Recursive Arithmetic with one additional axiom. This proof and a discussion of the Godel incompleteness theorems are given in chapters 6 and 7. In presenting these results we have given the more routine work, which is necessary but does not form an essential part of the development, at the ends of the corresponding chapters, sections 3.7, 4.5 and 5.4 fall into this category. (Abstract shortened by UMI.).