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ON THE ITERATED ω-RULE

✍ Scribed by Grzegorz Michalski

Publisher
John Wiley and Sons
Year
1992
Tongue
English
Weight
235 KB
Volume
38
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

Let Γ~n~(φ) be a formula of L~PA~ (PA = Peano Arithmetic) meaning “there is a proof of φ from PA‐axioms, in which ω‐rule is iterated no more than n times”. We examine relations over pairs of natural numbers of the kind.

(n, k) ≦~H~ (n', k') iff PA + RFN~n'~ (H~k'~) ⊩ RFN~n~ (H~k~).

Where H denotes one of the hierarchies ∑ or Π and RFN~n~(C) is the scheme of the reflection principle for Γ~n~ restricted to formulas from the class C(Γ~n~(φ) implies “φ is true”, for every φ ∈ C). Our main result is that.

(n, k) ≦H (n', k') if n ≦ n' and k ≦ max (k', 2n' + 1).

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