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.

Specifically, if there is no inner model for a measurable cardinal, then for every uncountable set x of ordinals, there is y such that y⊃x, y has the same cardinality as x, and y belongs to the Dodd-Jensen core model, K^DJ. (If 0^# does not exist, then K^DJ=L.)

This implies that if there is no inner model for a measurable cardinal, then K^DJ correctly computes successors of singular strong limit cardinals.

If there is no inner model with a Woodin cardinal and either every set has a sharp or a subtle cardinal exists (the large cardinal assumption is believed to be unnecessary), then the Mitchell-Steel Core Model K exists and satisfies the Weak Covering Lemma: If κ is a singular strong limit cardinal, then κ⁺=(κ⁺)^K.