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

摘要

Using the analysis developed in our earlier paper [5], we show that every uncountable cardinal in Gitik's model of [8] 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 [8] 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". (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim