### 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 naturallyassuming 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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