### Creature iteration for inaccessibles

by Shelah. [Sh:1100]

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 bold U_x, e.g. = lambda^+ to deal, e.g. with the big at universal graphs in lambda^+ < 2^lambda, while d_lambda = lambda^+ . The decision of ``weakly compact'' underline {or} demand? is done via the choice of bold 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 underline {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 {E} on it. (D) For some purposes we need stronger changes in the framework: allowing H 's in the bold i 's. This includes (f,g)-bounding.

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