Combinatorics on Ideals and Forcing

A natural generalization of Cohen's set of forcing conditions (the t w ~ valued functions with domain a finite subset of w) is the set of twovalued functions with domain an element of an ideal J on ~. The I~roblem treated in this paper is to determine when such forcing yields a generic real of minimal degree of constructibility. A :;imple decomposition argument shows that the non-maximality of J implies the non-minimality of the generic real which is obtained. In § 3 and 4 we look at the case J is maximal and we show that the minimality of the generic real depends on a combinatorial property of J. In fact the minimality result uses the notioa of T-ideal and the nonminhnality result that of selective ultraf'dter (a notion studied in Booth [ 1 ] ). These notions are generalized to 1:he case of non-maximal ideals and shown to be oquivalent in § 1. Ash ort study of them is also made in § 2 and in the appendix. Tt:e notion of ~ideal, without any hypothesis of maximality, is 'used in § 5 where we generalize Silver's set of forcing conditions (described in Mathias [ 3 ] p. 4). In fact Silver's forcing is related to the above in the following way: first force to get a m~,ximal ideal, which is shown to be a T-ideal, and then force with this ideal in the above manner. ! would like to thank J.L. Krivine for simplifying many proofs in this work.