All uncountable cardinals in the Gitik model are almost Ramsey and carry Rowbottom filters

نویسندگان

  • Arthur W. Apter
  • Ioanna M. Dimitriou
  • Peter Koepke
چکیده

Using the analysis developed in our earlier paper [ADK], we show that every uncountable cardinal in Gitik’s model of [Git80] in which all uncountable cardinals are singular is almost Ramsey and is also a Rowbottom cardinal carrying a Rowbottom filter. We assume that the model of [Git80] is constructed from a proper class of strongly compact cardinals, each of which is a limit of measurable cardinals. Our work consequently reduces the best previously known upper bound in consistency strength for the theory ZF + “All uncountable cardinals are singular” + “Every uncountable cardinal is both almost Ramsey and a Rowbottom cardinal carrying a Rowbottom filter”.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Some Applications of Supercompact Extender Based Forcings to Hod

Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V . In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.

متن کامل

Making All Cardinals Almost Ramsey ∗ † ‡ Arthur

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + ¬ACω in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular c...

متن کامل

On the splitting number at Regular Cardinals

Let κ, λ be regular uncountable cardinals such that λ > κ+ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with s(κ) = λ starting from a ground model in which o(κ) = λ and prove that assuming ¬0¶, s(κ) = λ implies that o(κ) ≥ λ in the core model.

متن کامل

Silver type theorems for collapses

The classical theorem of Silver states that GCH cannot break for the first time over a singular cardinal of uncountable cofinality. On the other hand it is easy to obtain a situation where GCH breaks on a club below a singular cardinal κ of an uncountable cofinality but 2 = κ. We would like here to investigate the situation once blowing up power of singular cardinals is replaced by collapses of...

متن کامل

Easton's theorem and large cardinals from the optimal hypothesis

The equiconsistency of a measurable cardinal with Mitchell order o(κ) = κ++ with a measurable cardinal such that 2κ = κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ (see [8]). In [5], we formulated and proved Easton’s theorem [4] in a large cardinal setting, using slightly stronger hypothese...

متن کامل

ذخیره 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 نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Math. Log. Q.

دوره 62  شماره 

صفحات  -

تاریخ انتشار 2016