CYCLIC INDISCERNIBLES AND SKOLEM FUNCTIONS by D. A. ANAPOLITANOS in Athens (Greece)l)
In this paper we introduce the notion of cyclic indiscernibles which appeared firstly in [l]. The new notion is fundamentally different from the notions of order indiscernibles and absolute indiscernibles. More specifically we show that there exist first order theories which have models with cyclic indiscernibles but no models with absolute ones. On the other hand it is obvious that the theory of linear ordering has models with order indiscernibles but no models with cyclic ones. But the difference between the order indiscernibles and the cyclic ones, is deeper than that. I n the classical paper [4] A. EHRENFEUCRT and A. MOSTOWSKI used the notion of order indiscernibles in order to produce, for first order theories with an infinite model, models which admit lots of automorphisms. So we introduced the new notion with the hope that the proof of [a] could be adapted for producing automorphisms of a finite order. Unfortunately this could not be done. I n this paper we show that there exist theories which do not have models with automorphisms of finite order but have models with cyclic indiscernibles. We also show exactly how much of the proof of [a] can be carried out and where the program breaks down. More specifically we show that the program breaks down a t the point where we have to construct cyclic indiscernibles using RAMSEY'S theorem. So cyclic indiscernibles appear to be a really intermediate notion between the notions of order indiscernibles and absolute ones. Finally we have to say that we follow the proof in [3], which is more mondern than the original one which appeared in [4]. D e f i n i t i o n s 1. (a) We say that the set X + 0 is n-cycle ordered by the relation B$
(iii) for every {al, as, . . ., a,} g X , such that ai + aj if i + j , there exists a permutation (a;, a;, . . ., a;) of {al, a2, . . ., a,] which belongs to B", and moreover (a;, . . .,a:) E I35 and {a:, . . ., a:} = {al, . . ., a,> iff (a;, . . ., a:) is a cyclic per-, mutation of (a;, . . . , a:), (iv) if (a,, . . . , a,) E B$ then ai -+ aj for i + j .
(b) We say that the set X -+ 0 is sn-cycle-ordered by the set of relations {B$}2dkcn
ordered by the relation Bifor all 2 s k 5 n, (ii) for 3 5 k 5 n and a,, . . ., ak E X if B$(al, . . ., ak) then for any a i l , . . .,aim (3 5 m k) from a,, . . ., ak such that ai, + ai, for j p 1 and such that i,, . . ., i, keep the order induced by 1,. . ., k we have Bg(ai,, . . ., aim).
(c) We say that the set X is <w-cycle-ordered (or simply cycle-ordered) by the set of relations {BF}2sk<o iff x is In-cycle-ordered by the set of relations {B$}2sksn for any n E O . l ) This paper is an improved version of a certain part of the author's Ph. D. Thesis. 23 Ztschr. f. math. Log&