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Choice principles, the bar rule and autonomously iterated comprehension schemes in analysis

Published online by Cambridge University Press:  12 March 2014

S. Feferman  and
G. Jäger
S. Feferman
Affiliation:
Stanford University, Stanford, California 94305
G. Jäger
Affiliation:
Ludwig-Maximilians-Universität München, D-8000 München 2, Federal Republic of Germany
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Extract

In [10] Friedman showed that (-AC) is a conservative extension of (-CA)<ε0 for -sentences where i = min(n + 2, 4), i.e., i = 2, 3, 4 for n = 0, 1, 2 + m. Feferman [5], [7] and Tait [11], [12] reobtained this result for n = 0, 1 and even with (-DC) instead of (-AC). Feferman and Sieg established in [9] the conservativeness of (-DC) over (-CA)<ε0 for -sentences (i as above) for all n. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem functionals.

In this paper we study the same choice principles in the presence of the Bar Rule (BR), which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result (Theorem 1 below) characterizes (-DC) + (BR) as a conservative extension of a system of the autonomously iterated -comprehension axiom for -sentences (i depending on n as above). For n = 0 this has been proved by Feferman in the form that (-DC) + (BR) is a conservative extension of (-CA)<Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger system Zω + μ + (QF-AC) + (BR) and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 1 , March 1983 , pp. 63 - 70
Copyright
Copyright © Association for Symbolic Logic 1983

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References

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