Applications of iterated perfect set forcing

Abstract

The major part of the results in this paper appeared in the author’s thesis. The author gratefully acknowledges the invaluable support and advice of her thesis supervisor, A. Kanamori. She also wishes to thank U. Abraham and H. Friedman for asking a number of thought-provoking questions, some of which are answered here. This paper will address questions about two different but closely related partial orders. The first is the partial order of degrees of constructibility (L-degrees) ordered by relative constructibility (cL), or more generally, degrees of constructibility relative to some inner model (M-degrees ordered by sM). The second is the partial order of countable, standard, transitive model of ZF of a given height, ordered by inclusion. The theory of the L-degrees is highly non-absolute. The notion of relative constructibility originates with Gijdel’s constructible universe L, in which there is exactly one (trivial) L-degree. Cohen’s introduction of forcing allowed many non-trivial consistency results to be obtained, beginning with Sacks’ forcing construction of a minimal L-degree [15]. This prototypical result can be extended in two directions. Z. Adamowicz shows that any upper semi-lattice satisfying certain properties in a model of V = L can be isomorphic to the L-degrees in a generic extension [2], [3]. Further results on the possibilities for L-degrees in generic extensions are due to P. Balcar and A. Hajek [4] and J. Truss [17]. In [9], Hajek draws conclusions about the richness of the structure of L-degrees from assumptions “~4 is countable” or “Ott exists”. The partial order of countable, standard, transitive ZF models of given height p may, of course, be empty, but in case it is not, the method of forcing shows this structure has certain richness attributes. Although this partial order has for the most part not been addressed in the literature, many results can be interpreted in this context. Giidel’s work implies the existence of a minimal model (L,, which models ZFC + V = L) and Sacks’ minimal degree construction produces a minimal model above L,. The forcing arguments of Adamowicz show the