Some Structural Issues in Iterated Forcing
by
Te-han Ko
Victoria University of Wellington

Forcing is a powerful technique of adjoining to a given model of set theory a certain "generic" object tightly controlled by a set of approximations. A similar enlargement procedure can be applied to the generic extension itself to obtain a further extension, and so on. Naturally, we want to be able to iterate the extension procedure for any transfinite number of times. Nevertheless, unlike many other set-theoretic constructions, here it is not always clear what should be done at limit stages. Today this issue is usually handled by defining an iterated forcing via an ascending sequence of complete Boolean algebras, where each earlier term in the sequence is a complete subalgebra of all latter ones. Towards evaluating the adequacy of this now orthodox conception, we will raise certain structural questions and try to settle some of them.

Date received: June 23, 1998