Sh:1100

• Shelah, S. Creature iteration for inaccessibles. Preprint.
• Abstract:
Our starting point is [Sh:1004]. As there we concentrate on forcing for inaccessibles and our definition is by induction when we like to get a nice forcing. (A) Mainly we deal with iterations for \lambda inaccessible of creature forcing (so getting appropriate forcing axioms). We concentrate on the case the forcing is strategically (< \lambda)-complete \lambda^+-c.c. (even \lambda-centered) and mainly (i.e. in (A)) on cases leading to \lambda-bounding forcing. In this case we can start with 2^\lambda > \lambda^+ and the forcing preserves various statements. We allow \mathbf U_{\mathfrak x}, e.g. = \lambda^+ to deal, e.g. with the big at universal graphs in \lambda^+ < 2^\lambda, while {\mathfrak d}_\lambda = \lambda^+. The decision of “weakly compact" or demand? is done via the choice of \mathbf j. (B) A different case is in the same framework but naturally assuming 2^\lambda = \lambda^+. The forcing satisfies only the \lambda^{++}-c.c. and is \kappa-proper so do not collapse \lambda^+. We may make 2^\lambda arbitrarily large or weaken the demands on forcing and get 2^\lambda = \lambda^{++}. ’ We only later do something concerning this. (C) Changing the frame somewhat, we allow adding unbounded \lambda-reals (i.e. \eta \in {}^\lambda \lambda) without adding \lambda-Cohens. For this we need to assume \lambda is measurable and use a fix normal ultrafilter \mathbb{E} on it.

(D) For some purposes we need stronger changes in the framework: allowing H’s in the \mathbf i’s. This includes (f,g)-bounding.

@article{Sh:1100,
}