Abstract
§1. Introduction. It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then1. W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W.2. W weakly covers V if W computes the successor of every V-singular cardinal correctly.Strong covering implies weak covering.In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model Kx which Y covers V”. Here the L-like properties of Kx always include GCH and Global Square. Examples include1. X is “0# exists”, Kx is L, Y is “strongly”.2. X is “there is a measurable cardinal”, Kx is the Dodd-Jensen core model, Y is “strongly”.3. X is “there is a Woodin cardinal”, Kx is the core model for a Woodin cardinal, Y is “weakly”.
Highlights
It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals
In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X there is an L-like inner model KX which Y covers V ”
If V is weakly covered by a model with Global Square κ holds in V for every V -singular cardinal κ, and this exerts a strong influence on the combinatorics of κ and κ+; for example there is a special κ+-Aronszajn tree. and there is a non-reflecting stationary set in κ+
Summary
It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. In Theorem 2.14, we prove that in the model of [14] there is a cofinal set of regular cardinals in κ which has order type ω and carries a nongood scale of length κ+. This provides an alternative proof that the approachability property fails in that model. Theorem 2.14 gives the first example known to us of a model where some cofinal subsets of a singular cardinal carry good scales, while others do not. In Remark 5.4 we show that his construction can be viewed in a PCF theoretic light: to be a bit more explicit we show that extender based Prikry forcing adds a good scale in a canonical way, and we identify Sharon’s non-reflecting set with a natural PCF theoretic object
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
