科研之友
微信
新浪微博
Facebook
分享链接
摘要
We show first that it is consistent that kappa is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(kappa (+)). Then with further forcing we show that it is consistent that GCH fails at a%26quot;mu (omega) , a%26quot;mu (omega) strong limit, while there is a lightface definable wellorder of H(a%26quot;mu (omega+1)) (%26quot;definable failure%26quot; of the singular cardinal hypothesis at a%26quot;mu (omega) ). The large cardinal hypothesis used is the existence of a kappa (++)-strong cardinal, where kappa is kappa (++)-strong if there is an embedding j: V -%26gt; M with critical point kappa such that H(kappa (++)) aS dagger M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the %26quot;tuning-fork%26quot; method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new %26quot;definable-collapse%26quot; coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a kappa (++)-strong cardinal is used throughout.
- 出版日期2012-11
