The Gödel Incompleteness Theorem and Decidability over a Ring

• Lenore Blum
• Steve Smale
Chapter

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.

Keywords

Finite Extension Incompleteness Theorem Real Closed Field Computable Isomorphism Weak Conjecture
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© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

• Lenore Blum
• Steve Smale

There are no affiliations available

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