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.

Example

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.

Versions

If the core model K exists, then
  1. If K has no ω1-Erdős cardinals, then for a particular countable and definable in K sequence of functions from ordinals to ordinals, every set of ordinals closed under these functions is a union of a countable number of sets in K. If L=K, these are simply the primitive recursive functions.
  2. If K has no measurable cardinals, then for every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|.
  3. If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|. Here C is either empty or Prikry generic over K and unique except up to a finite initial segment.
  4. If K has no inaccessible limit of measurable cardinals and no proper class of measurable cardinals, then there is a maximal and unique set C for K such that for every sequence S in K of measure one sets consisting of one set for each measurable cardinal, C minus ∪S is finite. Note that every κ \ C is either finite or Prikry generic for K at κ except for members of C below a measurable cardinal below κ. For every uncountable set x of ordinals, there is y ∈ K such that x ⊂ y and |x| = |y|.
  5. For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such that there is y ∈ K and x ⊂ y and |x| = |y|.
  6. K computes the successors of singular and weakly compact cardinals correctly. Moreover, if |κ| > ω1, then cofinality ≥ |κ|.

    Extenders and indiscernibles

For core models without overlapping total extenders, the systems of indiscernibles are well understood. Although, 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 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 from κ with o = κ++.
For core models with overlapping total extenders, the systems of indiscernibles are poorly understood, and applications tend to avoid rather than analyze the indiscernibles.

Additional properties

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 exists above a set X of ordinals, then it has the above discussed covering properties above X.