Turing Machines and Gödel Numbers
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.
KeywordsInternal State Turing Machine Recursive Function Relation Symbol Axiom Scheme
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