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Transfinite Progressions: A Second Look at Completeness

Published online by Cambridge University Press:  15 January 2014

Torkel Franzén
Torkel Franzén*
Affiliation:
Department of Computer Science and Electrical Engineering, Luleå University of Technology, SE-97187, Luleå, SwedenE-mail: [email protected]
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§1. Iterated Gödelian extensions of theories. The idea of iterating ad infinitum the operation of extending a theory T by adding as a new axiom a Gödel sentence for T, or equivalently a formalization of “T is consistent”, thus obtaining an infinite sequence of theories, arose naturally when Godel's incompleteness theorem first appeared, and occurs today to many non-specialists when they ponder the theorem. In the logical literature this idea has been thoroughly explored through two main approaches. One is that initiated by Turing in his “ordinal logics” (see Gandy and Yates [2001]) and taken very much further in Feferman's work on transfinite progressions, which also introduced the more general study of extensions by reflection principles, of which consistency statements are a special case. This approach starts from an assignment of theories to ordinal notations, and extracts sequences of theories through a suitable choice of a path in the set of ordinal notations. The second approach, illustrated in particular by the work of Schmerl and Beklemishev, starts instead from a suitably well-behaved primitive recursive well-ordering, which is used to define a sequence of theories. This second approach has led to precise results about the relative proof-theoretical strength of sequences of theories obtained by iterating different reflection principles. The Turing-Feferman approach, on the other hand, lends itself well to an investigation in qualitative and philosophical terms of the relevance of such iterated reflection extensions to mathematical knowledge, in particular because of two developments associated with this approach.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 10 , Issue 3 , September 2004 , pp. 367 - 389
Copyright
Copyright © Association for Symbolic Logic 2004

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References

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