On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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History and Philosophy of Logic 12 (2):225-233 (1991)
Presburger's essay on the completeness and decidability of arithmetic with integer addition but without multiplication is a milestone in the history of mathematical logic and formal metatheory. The proof is constructive, using Tarski-style quantifier elimination and a four-part recursive comprehension principle for axiomatic consequence characterization. Presburger's proof for the completeness of first order arithmetic with identity and addition but without multiplication, in light of the restrictive formal metatheorems of Gödel, Church, and Rosser, takes the foundations of arithmetic in mathematical logic to the limits of completeness and decidability
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Kevin Woods (2015). Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials. Journal of Symbolic Logic 80 (2):433-449.
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