Diagonal Prikry extensions

It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent of the tree property, the extent of stationary reflection, and the existence of non-free almost-free abelian groups. The explanation for this phenomenon lies in inner model theory, in particular core models and covering lemmas. If W is an inner model of V then (1) W strongly covers V if every uncountable set of ordinals is covered by a set of the same V -cardinality lying in W . (2) W weakly covers V if W computes the successor of every V singular cardinal correctly. Strong covering implies weak covering. In inner model theory there are many theorems of the general form “if there is no inner model of large cardinal hypothesis X then there is an L-like inner model KX which Y covers V ”. Here the L-like properties of KX always include GCH and Global Square. Examples include (1) X is “0 exists”, KX is L, Y is “strongly”. (2) X is “there is a measurable cardinal”, KX is the Dodd-Jensen core model, Y is “strongly”. (3) X is “there is a Woodin cardinal”, KX is the core model for a Woodin cardinal, Y is “weakly”. If V is strongly covered by an inner model with GCH then the SCH holds in V . If V is weakly covered by a model with Global Square then κ holds in V for every V -singular cardinal κ, and this also exerts a strong influence on the combinatorics of κ and κ; for example there is a special κ-Aronszajn tree. and there is a non-reflecting stationary set in κ.