Covering lemma
In the foundations of mathematics, a covering lemma is used to prove that the non-existence of certain large cardinals leads to the existence of a canonical inner model, called the core model, that is, in a sense, maximal and approximates the structure of the von Neumann universe V. A covering lemma asserts that under some particular anti-large cardinal assumption, the core model exists and is maximal in a sense that depends on the chosen large cardinal.
For example, if there is no inner model for a measurable cardinal, then the Dodd–Jensen core model, KDJ is the core model and 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.)
If the core model K exists (and has no Woodin cardinals), then
For core models without overlapping total extenders, the systems of indescernibles are well understood. Although (if K has an inaccessible limit of measurable cardinals), the system may depend on the set to be covered, it is well-determined and unique in a weaker sense. One application of the covering is counting the number of (sequences of) indiscernibles, which gives optimal lower bounds for various failures of the singular cardinals hypothesis. For example, if K does not have overlapping total extenders, and κ is singular strong limit, and 2κ=κ++, then κ has Mitchell order at least κ++ in K. Conversely, a failure of the singular cardinal hypothesis can be obtained (in a generic extension) from κ with o(κ)=κ++.
For core models with overlapping total extenders (that is with a cardinal strong up to a measurable one), the systems of indiscernibles are poorly understood, and applications (such as the weak covering) tend to avoid rather than analyze the indiscernibles.
If K exists, then every regular Jónsson cardinal is Ramsey in K. Every singular cardinal that is regular in K is measurable in K.
Also, if the core model K(X) exists above a set X of ordinals, then it has the above discussed covering properties above X.
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