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Did Turing prove the undecidability of the halting problem?

Joel David Hamkins, Theodor Nenu

TL;DR

The paper examines claims that Turing proved the undecidability of the halting problem in 1936 and argues that while Turing did not explicitly formulate or prove it as a halting decision problem in that paper, he developed foundational tools and decoupled problems (notably the circle-free and printing problems) that lead directly to undecidability results. It shows that the printing problem is computably equivalent to the halting problem and uses this to connect to Hilbert's Entscheidungsproblem, providing a nuanced historical account of attribution. The work ultimately defends a qualified attribution to Turing for the undecidability of the halting problem, while clarifying what exactly appears in his original text and how his ideas translate to the standard self-referential proof known today.

Abstract

We discuss the accuracy of the attribution commonly given to Turing's 1936 paper "On computable numbers..." for the computable undecidability of the halting problem, coming eventually to a nuanced conclusion.

Did Turing prove the undecidability of the halting problem?

TL;DR

The paper examines claims that Turing proved the undecidability of the halting problem in 1936 and argues that while Turing did not explicitly formulate or prove it as a halting decision problem in that paper, he developed foundational tools and decoupled problems (notably the circle-free and printing problems) that lead directly to undecidability results. It shows that the printing problem is computably equivalent to the halting problem and uses this to connect to Hilbert's Entscheidungsproblem, providing a nuanced historical account of attribution. The work ultimately defends a qualified attribution to Turing for the undecidability of the halting problem, while clarifying what exactly appears in his original text and how his ideas translate to the standard self-referential proof known today.

Abstract

We discuss the accuracy of the attribution commonly given to Turing's 1936 paper "On computable numbers..." for the computable undecidability of the halting problem, coming eventually to a nuanced conclusion.
Paper Structure (17 sections, 1 equation)

This paper contains 17 sections, 1 equation.

Table of Contents

  1. The halting problem
  2. Did Turing prove the undecidability of the halting problem?
  3. The prima facie case against the Turing attribution
  4. The circle-free problem
  5. Computable real numbers
  6. Circle-freeness
  7. The circle-free problem is computably undecidable
  8. Circle-free problem is harder than the halting problem
  9. Refining the argument
  10. The printing problem
  11. Alternative halting criterion
  12. Turing's clever proof of undecidability
  13. A simple self-referential proof of undecidability for printing
  14. The Entscheidungsproblem
  15. Mathematical attribution practice
  16. ...and 2 more sections