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The Gödel Incompleteness Theorem and Decidability over a Ring

In: From Topology to Computation: Proceedings of the Smalefest

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  • Lenore Blum
  • Steve Smale

Abstract

Here we give an exposition of Gödel’s result in an algebraic setting and also a formulation (and essentially an answer) to Penrose’s problem. The notions of computability and decidability over a ring R underly our point of view. Gödel’s Theorem follows from the Main Theorem: There is a definable undecidable set ovis Z. By way of contrast, Tarski’s Theorem asserts that every definable set over the reals or any real closed field R is decidable over R. We show a converse to this result: Any sufficiently infinite ordered field with this latter property is necessarily real closed.

Suggested Citation

  • Lenore Blum & Steve Smale, 1993. "The Gödel Incompleteness Theorem and Decidability over a Ring," Springer Books, in: Morris W. Hirsch & Jerrold E. Marsden & Michael Shub (ed.), From Topology to Computation: Proceedings of the Smalefest, chapter 32, pages 321-339, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-2740-3_32
    DOI: 10.1007/978-1-4612-2740-3_32
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