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If ฮบ < ฮป are such that ฮบ is a strong cardinal whose strongness is indestructible under ฮบ-strategically closed forcing and ฮป is weakly compact, then we show that A = {ฮด < ฮบ | ฮด is a non-weakly compact Mahlo cardinal reflecting stationary sets} must be unbounded in ฮบ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a Mahlo cardinal in which the least supercompact cardinal ฮบ is also the least strongly compact cardinal, ฮบ's strongness is indestructible under ฮบ-strategically closed forcing, ฮบ's supercompactness is indestructible under ฮบ-directed closed forcing not adding any new subsets of ฮบ, and ฮด is Mahlo and reflects stationary sets iff ฮด is weakly compact. In this model, no strong cardinal ฮด < ฮบ is indestructible under ฮด-strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal ฮบ whose strongness is indestructible under ฮบ-strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under ฮบ-directed closed forcing not adding any new subsets of ฮบ.
๐ SIMILAR VOLUMES
In this work we introduce a class of commutative rings whose defining condition is that its lattice of ideals, augmented with the ideal product, the semi-ring of ideals, is isomorphic to an MV-algebra. This class of rings coincides with the class of commutative rings which are direct sums of local A