Successor systems. An investigation into the primitive recursive functions of generalised multisuccessor arithmetics, with applications to constructive algebra.

An investigation into the primitive recursive functions of generalised multisuccessor arithmetics, with applications to constructive algebra.' Submitted for the degree of Doctor of Philosophy by Paul Hudson Stanford* at Leicester University, England, in 1975. The above named thesis is concerned with the extension of the notion of primitive recursion to structures other than the natural numbers. Successor systems are generalisations of the arithmetics of Vu?kovi? [2], and as a class are closed under operations corresponding to direct products and quotient formation. Given a system ? we can also define a system a* which has successor functions Sax for each numeral a of ?. The formalisation used is derived from the free variable calculus of Goodstein [1]. Various forms of recursion are considered, none of which employ more than a small number of known functions. For example, given a function g from ? x ? to ? we can define f from ?* to ? as follows. f(0) = 0; f(Sax) = g(a,f(x)) Algebraic applications include the construction of groups and rings: actual examples range from the integers and polynomials to permutations, finite sets and ordinal numbers. Several relations which may hold between systems are investigated, as are the notions of anchored and decidable systems.*(supported by a Science Research Council grant) One chapter deals with the case of commuting successor functions, and another considers systems with only one successor. In an appendix we briefly investigate the further generalisation obtained by using non-unary successor functions. The author expresses his thanks to all concerned, especially his supervisor. Professor R. L. Goodstein. Contents of thesis: (1) Introduction, (2) The Integers, (3) Products, (4) Recursion, (5) The Star Operation, (6) Commutative systems, (7) Homomorphisms, (8) Groups, (9) Further recursion, (10) Decidable systems, (11) Single successor systems, (12) Polynomials; (A1) Small systems, (A2) Joint successor arithmetics, (A3) Polish Circles, (A4) A Formalisation of the Integers. References to abstract: [1] Goodstein, R.L., Recursive Number Theory, Amsterdam (1957) [2] Vu?kovi?, V., Partially ordered recursive arithmetics, Math.Scand. 7 (1959), 305-320.