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The autonomous ordinal of the infinitary system Z and the limits of predicativity

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1407)

Abstract

By the soundness theorem 12.1. for the infinitary system ZΩ and lemma 9.2. we have

. That means that the ordinal Ω has the following closure property: For α,ρ ∈ Ω and

we also have |F| ∈ Ω. By Ω we usually denote the first uncountable regular ordinal. Later on we are going to use Ω as a formal symbol, whose standard interpretation is the first regular ordinal ℵ1. But we also will alternatively interpret Ω by other ordinals (cf. chapter III). By recursion theoretic methods (cf. exercise 13.13) it can be shown that Ω keeps the above closure property even in its recursive standard interpretation where Ω is interpreted as ω CK1 , the first recursively regular ordinal. It is now obvious to ask if ω CK1 already is the smallest ordinal above ω having this closure property. By purely recursion theoretic methods this question hardly is to answer. By proof theoretic methods, however, we will establish that there are in fact smaller such ordinals. The small estone will be ω. But of course our real interest is the question if there are ordinals between ω and ω CK1 having this closure property. If there exist such ordinals, then we already know that they have to be larger than εo. This follows from the proof theoretic analysis of Z1 where we noticed that for every ordinal α<εo there is a Π 11 -sentence of norm α which is provable in Z1 and therefore provable with a derivation of length smaller than ω·2 and finite cut rank. It is also easy to see that there also is a Π 11 -sentence of norm εo provable with a derivation of length smaller than εo and cut rank ω.

Keywords

  • Induction Hypothesis
  • Infinitary System
  • Closure Property
  • Structural Rule
  • Class Term

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1989 Springer-Verlag Berlin Heidelberg

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(1989). The autonomous ordinal of the infinitary system Z and the limits of predicativity. In: Proof Theory. Lecture Notes in Mathematics, vol 1407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-46825-7_3

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  • DOI: https://doi.org/10.1007/978-3-540-46825-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51842-6

  • Online ISBN: 978-3-540-46825-7

  • eBook Packages: Springer Book Archive