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On definition trees of ordinal recursive functionals: Reduction of the recursion orders by means of type level raising

Published online by Cambridge University Press:  12 March 2014

Jan Terlouw
Jan Terlouw*
Affiliation:
State University of Utrecht, Utrecht, The Netherlands
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It is known that every < ε0-recursive function is also a primitive recursive functional. Kreisel has proved this by means of Gödel's functional-interpretation, using that every < ε0-recursive function is provably recursive in Heyting's arithmetic [2, §3.4]. Parsons obtained a refinement of Kreisel's result by a further examination of Gödel's interpretation with regard to type levels [3, Theorem 5], [4, §4]. A quite different proof is provided by the research into extensions of the Grzegorczyk hierarchy as done by Schwichtenberg and Wainer: this yields another characterization of the < ε0-recursive functions from which easily appears that these are primitive recursive functionals (see [5] in combination with [6, Chapter II]).

However, these proofs are indirect and do not show how, in general, given a definition tree of an ordinal recursive functional, transfinite recursions can be replaced (in a straightforward way) by recursions over wellorderings of lower order types. The argument given by Tait in [9, pp. 189–191] seems to be an improvement in this respect, but the crucial step in it is (at least in my opinion) not very clear.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 2 , June 1982 , pp. 395 - 402
Copyright
Copyright © Association for Symbolic Logic 1982

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References

REFERENCES

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