Homogeneous sequences of cardinals for ordinal definable partition relations

• Main Author: Kafkoulis, George
• Format: Others
• Language: en
• Published: 1990
• Online Access: https://thesis.library.caltech.edu/2572/1/Kafkoulis_g_1990.pdf; Kafkoulis, George (1990) Homogeneous sequences of cardinals for ordinal definable partition relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/hchw-ca34. https://resolver.caltech.edu/CaltechETD:etd-06132007-073731 <https://resolver.caltech.edu/CaltechETD:etd-06132007-073731>

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this dissertation we study the consistency strength of the theory ZFC & ([...] strong limit)([...] < [...])([...]) (*), and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it. If U is a normal measure on [...], then [...] denotes the Supercompact Prikry forcing induced by U. [...] is the partition relation [...] except that we consider only OD colorings of [...]. Theorems 1,2 are the main results of our thesis. Theorem 1. If there exists a model of ZFC in which [...] is a supercompact cardinal and [...] is an innaccessible above [...], then we can construct a model V of the same properties with the additional property that if U is a normal [...]-measure and G is [...] - generic over V, then V[G] does not satisfy the [...] partition property. [...] If G is a [...]-generic over V filter, then we define H to be the set H: [...], and we consider the inner model V(H), which is the smallest inner model of ZF that contains H as an element. We prove that V(H) satisfies the above partition property (*). Moreover, V(H) satisfies < [...] - DC and using this fact we define a forcing [...], which is almost-homogeneous, < [...] - closed forcing that forces the AC over V(H) and does not add any new sets of rank < [...]. Theorem 2. If [...] is [...] -generic over V(H) and V[...], then [...] + [...] strong limit + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & [...] inaccessible & is [...]) [...] Con(ZFC + ([...] strong limit)[...]. [...]