## Induction, Pure and Simple (1977)

Venue: | INFORMATION AND CONTROL 35, 276--336 (1977) |

Citations: | 13 - 7 self |

### BibTeX

@MISC{Kugel77induction,pure,

author = {Peter Kugel},

title = { Induction, Pure and Simple},

year = {1977}

}

### OpenURL

### 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 210 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 271 of the Arithmetic Hierarchy) then we can develop inductive