Turing Machines and Gödel Numbers

  • Donald W. Barnes
  • John M. Mack
Part of the Graduate Texts in Mathematics book series (GTM, volume 22)

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, ). 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, ) is then a finite list of instructions such that for any element pP(V, ) 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, ).”). 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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1975

Authors and Affiliations

  • Donald W. Barnes
    • 1
  • John M. Mack
    • 1
  1. 1.Department of Pure MathematicsThe University of SydneySydneyAustralia

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