Some uses of dilators in combinatorial problems

Abstract

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)(.)-sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in ID,. 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]. ?1. Increasing F-sequences, F a dilator. 1.0. DEFINITION. For every dilator F and for every ordinal a, (AF'+x)xe~n (the increasing F-sequence with respect to at) is defined as follows: for all x ? a sup (F(Eyx)(AyF'a) + 1), if, for all c < y < x, AyF' < F(y), AF'a 7 ay<x x F(x), otherwise. 1.1. DEFINITION. For every dilator F, the function v(F, *) from On (the class of all the ordinal numbers) to On is defined as follows: for every ordinal a, v(F, a) = yux ? a such that AXF = F(x). 1.2. REMARK. (i) AF = 0. (ii) If a < x < v(F, c), then AXj+ 1 = F(Exx+ 1)(AXFa) + 1. (iii) If x is a limit ordinal, and a < x < v(F, a), then AxF' = sup (F(Ey x)(AF')) Received October 6, 1988. ( 1990, Association for Symbolic Logic 0022-4812/90/5501 -0003/$0 1.90