Large Continuum, Oracles Sh895

Our main theorem is about iterated forcing for making the continuum larger than ℵ 2. We present a generalization of [Sh:669] which dealt with oracles for random, (also for other cases and generalities), by replacing ℵ 1 , ℵ 2 by λ, λ + (starting with λ = λ <λ > ℵ 1). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ + but we can get cov(meagre) = λ and we give some applications. As in " non-Cohen oracles " , [Sh:669], it is a " partial " countable support iteration but it is c.c.c. P of cardinality λ + , by a partial CS iteration but the result is a c.c.c. forcing. The general iteration theorems (treated in §1) seem generally suitable for constructing universes with MA <λ + 2 ℵ0 = λ + , and taking more care, we should be able to get universes without MA <λ , see 0.4 below. Our method is to imitate [Sh:669]; concerning the differences, some are inessential: using games not using diamonds in the framework itself, (inessential means that we could have in [Sh:669] imitate the choice here and vice versa). An essential difference is that we deal here with large continuum-λ + ; we concentrate on the case where we shall (in V P) have MA <λ but e.g. non(null) = λ and b = λ + (or b = λ). It seems to us that generally: {h.4} Thesis 0.1. The iteration theorem here is enough to get results parallel to known results with 2 ℵ0 = ℵ 2 replacing ℵ 1 , ℵ 2 by λ, λ +. To test this thesis we have asked Bartoszy´nski to suggest test problems for this method and he suggests: {h.7} Problem 0.2. Prove the consistency of each of the (A) ℵ 1 < λ < 2 ℵ0 and the λ-Borel conjecture, i.e. A ⊆ ω 2 is of strong measure zero iff |A| < λ (B) ℵ 1 < non(null) < 2