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

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References

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