I: A General Method in Proofs of Undecidability

Abstract

A theory T is called decidable or undecidable depending on whether the solution of the decision problem is positive or negative. The decision problem is one of the central problems of contemporary metamathematics. Because only few theories turn out to be decidable, most endeavors are directed toward a negative solution. The aim of this chapter is to set up theoretical foundations for the general method, which is referred to as theories with standard formalization. They can be briefly characterized as theories that are formalized within the first-order predicate logic. The symbols that occur in expressions of a given theory T are divided into variables and constant. With every predicate and every operation symbol, a positive integer is correlated, which is called the rank of the symbol. The identity symbol, though regarded as a logical constant, is included in the set of binary predicates. In practice, in addition to variables and constants, the so-called technical symbols, like parentheses and commas, are also used in constructing expressions; theoretically, however, these technical symbols can be dispensed with.