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7 - Predicative Foundations of Arithmetic

from Part II - Predicative Mathematics and Beyond

Published online by Cambridge University Press:  26 January 2021

Geoffrey Hellman
By
Solomon Feferman  and
Geoffrey Hellman
Geoffrey Hellman
Affiliation:
University of Minnesota
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Summary

Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.

Type
Chapter
Information
Mathematics and Its Logics
Philosophical Essays
 , pp. 103 - 116
Publisher: Cambridge University Press
Print publication year: 2021

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References

Dedekind, R. [1888] “The nature and meaning of numbers,” reprinted in Beman, W. W., ed., Essays on the Theory of Numbers (New York: Dover, 1963), pp. 31–115, translated from the German original, Was sind und was sollen die Zahlen? (Brunswick: Vieweg, 1888).Google Scholar
Feferman, S. [1964] “Systems of predicative analysis,” Journal of Symbolic Logic 29: 1–30.Google Scholar
Feferman, S. [1968] “Autonomous transfinite progressions and the extent of predicative mathematics,” in Logic, Methodology, and Philosophy of Science III (Amsterdam: North Holland), pp. 121–135.CrossRefGoogle Scholar
Feferman, S. [1988] “Weyl vindicated: Das Kontinuum 70 years later,” in Temi e Prospettive della Logica e della Filosofia della Scienza Contemporanee (Bologna: CLUEB), pp. 59–93.Google Scholar
Hellman, G. [1989] Mathematics without Numbers: Towards a Modal-Structural Interpretation (Oxford: Oxford University Press).Google Scholar
Isaacson, D. [1987] “Arithmetical truth and hidden higher-order concepts,” in Paris Logic Group, eds., Logic Colloquium ‘85 (Amsterdam: North-Holland,), pp. 147–169.CrossRefGoogle Scholar
Monk, J. D. [1976] Mathematical Logic (New York: Springer,).CrossRefGoogle Scholar
Nelson, E. [1986] Predicative Arithmetic (Princeton, NJ: Princeton University Press,).Google Scholar
Parsons, C. [1983] “The impredicativity of induction,” in How Many Questions? Essays in Honor of S. Morgenbesser (Indianapolis, IN: Hackett), pp. 132–154.Google Scholar
Shapiro, S. [1991] Foundations without Foundationalism: The Case for Second-Order Logic (Oxford: Oxford University Press).Google Scholar
Shoenfield, J. [1954] “A relative consistency proof,” Journal of Symbolic Logic 19: 21–28.CrossRefGoogle Scholar
Sieg, W. [1985] “Fragments of arithmetic,” Annals of Pure and Applied Logic 28: 33–72.Google Scholar
Simpson, S. G. [1987] “Subsystems of Z2 and reverse mathematics,” in Takeuti, G., ed., Proof Theory, 2nd edn. (Amsterdam: North-Holland), pp. 432–446.Google Scholar

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