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Relative constructivity

Published online by Cambridge University Press:  12 March 2014

Ulrich Kohlenbach
Ulrich Kohlenbach*
Affiliation:
Brics, Department of Computer Science, University of Århus, NYMunkegade, DK-8000 Århus C, Denmark. E-mail: [email protected]
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Extract

In a previous paper [13] we introduced a hierarchy (GnAω)n∈ℕ of subsystems of classical arithmetic in all finite types where the growth of definable functions of GnAω corresponds to the well-known Grzegorczyk hierarchy. Let AC-qf denote the schema of quantifier-free choice.

[11], [13], [8] and [7] study various analytical principles Γ in the context of the theories GnAω + AC-qf (mainly for n = 2) and use proof-theoretic tools like, e.g., monotone functional interpretation (which was introduced in [12]) to determine their impact on the growth of uniform bounds Φ such that

which are extractable from given proofs (based on these principles Γ) of sentences

Here A0(u, k, v, w) is quantifier-free and contains only u, k, v, w as free variables; t is a closed term and ≤p is defined pointwise. The term ‘uniform bound’ refers to the fact that Φ does not depend on v ≤ptuk (see [12] for the relevance of such uniform bounds in numerical analysis and for concrete applications to approximation theory).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 4 , December 1998 , pp. 1218 - 1238
Copyright
Copyright © Association for Symbolic Logic 1998

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References

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