# Turing Machines and Gödel Numbers

## Abstract

In §3 of Chapter III, we gave a procedure for determining whether or not an element *p* of *P*(*X*) is a theorem of Prop(*X*). In §4 of Chapter IV, we asserted that no such procedure exists for Pred(*V*, *ℛ*). Before attempting to prove this non-existence theorem, we must say more precisely what we mean by “procedure”. The procedures we shall discuss are called decision processes, and informally we think of a decision process as a list of instructions which can be applied in a routine fashion to give one of a finite number of specified answers. A decision process for Pred(*V*, *ℛ*) is then a finite list of instructions such that for any element *p* ∈ *P*(*V*, *ℛ*) there corresponds a unique finite sequence of instructions from the list. The sequence terminates with an instruction to announce a decision of some prescribed kind (e.g., *“p* is a theorem of Pred(*V*, *ℛ*).”). Thus at each step of the process, exactly one instruction of the list is applicable, producing a result to which exactly one instruction is applicable, until after a finite (but not necessarily bounded) number of steps, the process stops and a decision is announced.

## Keywords

Internal State Turing Machine Recursive Function Relation Symbol Axiom Scheme## Preview

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