The forcing technique to produce outer models has been exploited to a
great extent to show an enormous range of possibilities which may be
true of the universe. Relative to a fixed universe, some of the
same results can be true in inner and outer models, but different
techniques are used. S. Friedman introduced a programme which aims to
discover what questions may be answered in an inner model.
There are many consistency results achieved using
forcing which are not yet known to be obtainable via inner models,
since internal consistency results are harder to achieve. To obtain
any consistency result, one must show that a generic filter exists. To
achieve internal consistency, one needs an inner model with all the
ordinals of the ambient universe, therefore one cannot restrict to a
countable submodel, as is usual in set-forcing. The question then is
how to build generics, if one even exists.
One class of results which are studied in this context are called
``global properties''. These are statements which are true throughout
the universe (e.g. at every regular cardinal). Global properties are
achieved using class forcing. This adds an extra challenge to building
generics as all antichains of the forcing which exist in the universe
must be considered.
In this talk, we will examine the techniques used to build such
generics and the properties of the forcings which are sufficient
to utilise these methods.