Some Uses of Dilators in Combinatorial Problems. II

Journal of Symbolic Logic 55 (1):32 - 40 (1990)

We study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with "+ 1" instead of "- 1"). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator F the point where the increasing F-sequence terminates. We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id) (ω) -sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in ID 1 . For a general presentation of the results stated in this paper, see [1]. We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25-31]
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DOI 10.2307/2274952
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[Product]¹2-Logic, Part 1: Dilators.J. Girard - 1981 - Annals of Pure and Applied Logic 21 (2):75.
Π12-Logic, Part 1: Dilators.Jean-Yves Girard - 1981 - Annals of Mathematical Logic 21 (2-3):75-219.

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