When the continuum has cofinalityω1

Abstract

In this paper we consider models of set theory in which the continuum has cofinality ωv We show that it is consistent with -,CH that any complete boolean algebra B of cardinality less than or equal to c (continuum) there exists an ωλ-generated ideal / in P(ω) (power set of ω) such that B is isomorphic to P(ω)mod/. We also show that the existence of generalized Luzin sets every ωλ -saturated ideal in the Borel sets does not imply Martin's axiom. Introduction. In §1 we prove our main result that it is consistent with -ηCH that every complete boolean algebra of cardinality < c is isomorphic to P(ω)mod / some / ω^generated. We think of this as generalizing Kunen's theorem that it is consistent with -πCH that there is an ω1 generated nonprincipal ultrafilter on ω. For / an ideal in the Borel subsets of the reals we say that a set of reals X is a κ-/-Luzin set iff X has cardinality K and every A in /, A n X has cardinality less than /c. If c is regular, then it follows easily from Martin-Solovay (9) that MA is equivalent to the statement for every ωΓsaturated σ-ideal / in the Borels there is a c-/-Luzin In §2 we show that the regularity of c is necessary. This answers a question of Fremlin (5). We also show that it is consistent with -,CH that every such / there exists an ω Γ/-Luzin set. These results can be thought of as a weak form of the following conjecture.