Chapter 7 - Some theorems about Church's system (Three unpublished manuscripts)

Abstract

This chapter presents a discussion on Turing's three unpublished manuscripts that have some theorems about Church's system. First manuscript entitled, “Consequences of the Peano Axioms” has pages (numbered by AMT) 37-43. The second manuscript entitled, “Finite models of Church's and Zermelo's systems” has pages numbered 60-73, 84. Third manuscript entitled, “Some Theorems about Church's system” consists of 19 unnumbered pages running consecutively. In first manuscript, Turing proves various simple theorems and makes connections with “A Formal Theorem,” in particular, to show that Church's axioms 7 and 8 (with β replacing i) are satisfied. In second manuscript, Turing describes a system of tables over a finite system of individuals for Church's system, entirely similar to that given in “Practical Forms.” His version of Church's system includes axioms 7, 10 (extensionality), and 11 (choice), while axiom 8 is replaced by an axiom of finitude similar to (IX)N of “Practical Forms.’ He then proves that this system is sound and complete for the given interpretation. In third manuscript, Turing discusses some theorems about Church's system.