Fragments of arithmetic

Annals of Pure and Applied Logic 28 (1):33-71 (1985)
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Abstract

We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Minc [14]; each has been shown to be of considerable interest for both mathematical practice and metamathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems. The results are generalized to relate a hierarchy of subsystems, all contained in the theory of arithmetic properties, to a corresponding hierarchy of fragments of arithmetic. The proof theoretic tools employed there are used to re-establish in a uniform, elementary way relationships between various fragments of arithmetic due to Parsons, Paris and Kirby, and Friedman

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Wilfried Sieg
Carnegie Mellon University

Citations of this work

Partial realizations of Hilbert's program.Stephen G. Simpson - 1988 - Journal of Symbolic Logic 53 (2):349-363.
Hilbert’s Program.Richard Zach - 2003 - In Edward N. Zalta (ed.), The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information, Stanford University.
Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics.Solomon Feferman - 1992 - PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:442 - 455.

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References found in this work

Systems of predicative analysis.Solomon Feferman - 1964 - Journal of Symbolic Logic 29 (1):1-30.
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On n-quantifier induction.Charles Parsons - 1972 - Journal of Symbolic Logic 37 (3):466-482.

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