A Practical Form of Type Theory II: (Unpublished manuscript of 81 unnumbered pages of typescript together with 22 manuscript pages inserted in various places.)

Abstract

This chapter discusses about Turing's unpublished paper, “A Practical Form of Type Theory II,” that is a collection of manuscript of 81 unnumbered pages of typescript together with 22 manuscript pages inserted in various places. In this paper, theorems A and B enunciated in “A Practical Form of Type Theory” are proved. Theorem A states that the nested-type system with an axiom of finitude (Rule IXN of Practical Forms) is sound and complete for the model whose elements in any type r are the tables of type r over a set of individuals of cardinality N. Theorem B is the equivalence of Church's system (system 1) with the “practical system” (system 2)—that is the nested-type system as formulated in “A Practical Form of type theory.” There are seven numbered sections in this paper: (1) preliminary definitions and conventions (pp. 1–17); (2) formal development of the [practical] system (pp. 18–23A); (3) justification and completeness in the finite case (Theorem A) (pp. 24–33); (4) Church's system derived from the practical system (pp. 34–49); (5) the practical system developed in the Church system (pp. 50–62); (6) continuation of theorem B (pp. 63–70); and (7) conclusion of Theorem B (pp. 71–81).