Some Uses of Dilators in Combinatorial Problems, II

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].