Indestructible strong compactness but not supercompactness
نویسندگان
چکیده
Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ’s strong compactness, but not its supercompactness, is indestructible under arbitrary κ-directed closed forcing.
منابع مشابه
Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions
We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes [3, Theorem 1], but without the restriction that no cardinal is supercompact up to an inaccess...
متن کاملIndestructibility, Strong Compactness, and Level by Level Equivalence
We show the relative consistency of the existence of two strongly compact cardinals κ1 and κ2 which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for κ1. In the model constructed, κ1’s strong compactness is indestructible under arbitrary κ1-dir...
متن کاملIndestructibility and The Level-By-Level Agreement Between Strong Compactness and Supercompactness
Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or...
متن کاملIndestructibility, instances of strong compactness, and level by level inequivalence
Suppose λ > κ is measurable. We show that if κ is either indestructibly supercompact or indestructibly strong, then A = {δ < κ | δ is measurable, yet δ is neither δ+ strongly compact nor a limit of measurable cardinals} must be unbounded in κ. The large cardinal hypothesis on λ is necessary, as we further demonstrate by constructing via forcing two models in which A = ∅. The first of these cont...
متن کاملSuperdestructibility: A Dual to Laver's Indestructibility
After small forcing, any <κ-closed forcing will destroy the supercompactness and even the strong compactness of κ. In a delightful argument, Laver [L78] proved that any supercompact cardinal κ can be made indestructible by <κ-directed closed forcing. This indestructibility, however, is evidently not itself indestructible, for it is always ruined by small forcing: in [H96] the first author recen...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Ann. Pure Appl. Logic
دوره 163 شماره
صفحات -
تاریخ انتشار 2012