Killing Ideals and Adding Reals
The relationship between killing ideals and adding reals by forcings is analysed. ?0. Introduction. In [Z] I have introduced a new class of forcing notions designed to kill a given a-ideal on col, that is, to add a countable collection of sets in the ideal whose union is the whole coi, all of this without collapsing co1. The ideals for which such a task can be performed were essentially classified, however interesting questions remained. Among them, it was not clear whether new reals do or do not have to be added. In this paper I address this question for normal ideals. Forcings for killing ideals are defined for which the mechanism of adding reals can be closely analysed. FACT 1. Suppose S C co1 is a stationary costationary set and the ideal NS r CO1 \ S is saturated and there is a measurable cardinal. Then any forcing killing the ideal NS [ S has to add a real. This was proved in [Z]. It turns out that in the situation from Fact 1 a special kind of a new real is obtained-one that increases the boldface ordinal c51. No explicit construction of such a real was known before. This paper provides contrasting results. FACT 2. Suppose S c co1 is a stationary costationary set and 0,,,, \S holds. Then there is aforcing killing the ideal NS [ S and adding no reals. FACT 3. It is possible to kill normal ideals without disturbing unbounded or dominating sequences of reals of length coi and similar objects. In order to specify the meaning of the word "similar" in the previous Fact, I define a notion of generic stability in Section 2. The techniques of the paper leave the case of nonnormal a-ideals wide open. The notation used in the paper follows the set theoretical standard of [J]. We only consider ideals containing all singletons. ?1. Not adding reals. I will start by defining a variant of the forcing notions used in [Z]. Let 3 be a normal ideal on coi. The forcing P(3) is defined as the set of all nonempty trees T of finite sequences of sets in the ideal such that Received June 10, 1998; revised October 26, 1998. 1991 Mathematics Subject Classification. 03E40. Author's research is partially supported by grant AV OR 201/97/0216. ? 2000, Association for Symbolic Logic 0022-4812/00/6502-0015/$1.90