Covering lemma

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 is the core model and satisfies the Covering Property, that is for every uncountable set x of ordinals, there is y such that yx, y has the same cardinality as x, and yKDJ. (If 0# does not exist, then KDJ=L.)

If the Core Model K exists (and has no Woodin cardinals), then

  1. If K has no ω1-Erdős cardinals, then for a particular countable (in K) 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[C] such that x ⊂ y and |x|=|y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal in κ) 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 (except for a finite set of ordinals) set C (called a system of indiscernibles) 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 κC 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[C] 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[C] and x ⊂ y and |x|=|y|.
  6. K computes the successors of singular and weakly compact cardinals correctly (Weak Covering Property). Moreover, if |κ|>ω1, then cofinality((κ+)K) ≥ |κ|.

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++, 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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