Some uses of dilators in combinatorial problems. II

Abstract

AbstractWe 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 theoremevery inverse Goodstein sequence terminates(a combinatorial theorem about ordinal numbers) is not provable in ID1.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].