Induction, pure and simple

Abstract

Induction is the process by which we reason from the particular to the general. In this paper we use ideas from the theory of abstract machines and recursion theory to study this process. We focus on pure induction in which the conclusions "go beyond the information given" in the premises from which they are derived and on simple induction, which is rather a stark kind of induction that deals with computable predicates on the integers in rather straightforward ways. Our basic question is "What are the relationships between the kinds of abstract machinery we bring to bear on the job of doing induction and our ability to do that job well?" Our conclusions are as follows: (1) If we use only the abstract machinery of the digital computer in a computing center (which we assume to be capable of only evaluating totally computable functionals or functionals in Σ0 of the Arithmetic Hierarchy) then a single inductive procedure can only develop finitely many sound theories. (2) If we use only the abstract machinery of the mathematician (which we assume to be the machinery required to evaluate a functional in Σ2 of the Arithmetic Hierarchy) then we can develop inductive procedures that generate infinitely many sound theories but no procedure that is analytic in the sense that the theories it puts forth as sound will be sound in "all possible worlds". (3) If we use the machinery of the trial and error machine (or the machinery required to evaluate functionals in Σ2 of the Arithmetic Hierarchy) then we can develop consistent procedures for induction but none that are complete in the sense that they generate all useful theories in a fairly natural sense of "useful". (4) Finally, if we use the machinery of the hyper-trial-anderror machine (or the machinery required to evaluate functionals in Σ3 of the Arithmetic Hierarchy) then we can develop procedures that are both consistent and complete. © 1977 Academic Press, Inc.