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Reflecting on incompleteness1

Published online by Cambridge University Press:  12 March 2014

Solomon Feferman
Solomon Feferman*
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305
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Extract

To what extent can mathematical thought be analyzed in formal terms? Gödel's theorems show the inadequacy of single formal systems for this purpose, except in relatively restricted parts of mathematics. However at the same time they point to the possibility of systematically generating larger and larger systems whose acceptability is implicit in acceptance of the starting theory. The engines for that purpose are what have come to be called reflection principles. These may be iterated into the constructive transfinite, leading to what are called recursive progressions of theories. A number of informative technical results have been obtained about such progressions (cf. Feferman [1962], [1964], [1968] and Kreisel [1958], [1970]). However, for some years I had hoped to give a more realistic and perspicuous finite generation procedure. This was first done in a rather special way in Feferman [1979] for the characterization of predicativity, which may be regarded as that part of mathematical thought implicit in our acceptance of elementary number theory. What is presented here is a new and simple notion of the reflective closure of a schematic theory which can be applied quite generally.

Two examples of schematic theories in the sense used here are versions of Peano arithmetic and Zermelo set theory.

Type
Survey/expository paper
Information
The Journal of Symbolic Logic , Volume 56 , Issue 1 , March 1991 , pp. 1 - 49
Copyright
Copyright © Association for Symbolic Logic 1991

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Footnotes

1

Retiring Presidential Address. For the background to this paper and acknowledgements, see the end of the Introduction, below.

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