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.
Partially supported by an NSF grants and (the first author) by the Letts-Villard Chair at Mills College.
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Blum, L., Smale, S. (1993). The Gödel Incompleteness Theorem and Decidability over a Ring. In: Hirsch, M.W., Marsden, J.E., Shub, M. (eds) From Topology to Computation: Proceedings of the Smalefest. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2740-3_32
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DOI: https://doi.org/10.1007/978-1-4612-2740-3_32
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