Covering lemma
|
|
In mathematics, under various anti-large cardinal assumptions, one can prove the existence of the canonical inner model, called the Core Model, that is, in a sense, maximal and approximates the structure of V. A covering lemma asserts that under the particular anti-large cardinal assumption, the Core Model exists and is maximal in a way.
For example, if there is no inner model for a measurable cardinal, then the Dodd-Jensen core model, KDJ satisfies the Covering Property, that is for every uncountable set x of ordinals, there is y such that y⊃x, y has the same cardinality as x, and y ∈KDJ. (If 0# does not exist, then KDJ=L.) This implies that if there is no inner model for a measurable cardinal, then KDJ correctly computes successors of singular cardinals.
If there is no inner model with a Woodin cardinal and either every set has a sharp or a subtle cardinal exists or every inner model with a proper class of strong cardinals does not have a sharp (the assumptions other than absence of an inner model with a Woodin cardinal are believed to be unnecessary), then the Mitchell-Steel Core Model K exists and satisfies the Weak Covering Proerty: If κ is a singular cardinal, then κ+=(κ+)K. Moreover, for every κ ≥ ω2, cofinality of (κ+)K is ≥ |κ|. If K exists and no ordinal is measurable in K, then K satisfies the Covering Property.