Editing Covering lemma

{{see also|Jensen's covering theorem}}
In the [[foundations of mathematics]], a '''covering lemma''' is used to prove that the non-existence of certain [[large cardinal]]s 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. The first such result was proved by [[Ronald Jensen]] for the [[constructible universe]] assuming [[zero sharp|0<sup>#</sup>]] does not exist, which is now known as [[Jensen's covering theorem]].

==Example==
For example, if there is no inner model for a [[measurable cardinal]], then the Dodd–Jensen core model, ''K''<sup>DJ</sup> is the core model and satisfies the '''covering property''', that is for every uncountable set ''x'' of ordinals, there is ''y'' such that ''y''&nbsp;⊃&nbsp;''x'', ''y'' has the same cardinality as ''x'', and ''y''&nbsp;∈&nbsp;''K''<sup>DJ</sup>. (If [[zero sharp|0<sup>#</sup>]] does not exist, then ''K''<sup>DJ</sup>&nbsp;=&nbsp;''L''.)

==Versions==
If the core model K exists (and has no Woodin cardinals), then
# If K has no ω<sub>1</sub>-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.
# If K has no measurable cardinals, then for every uncountable set ''x'' of ordinals, there is ''y''&nbsp;∈&nbsp;K such that x&nbsp;⊂&nbsp;y and |x|&nbsp;=&nbsp;|y|.
#  If K has only one measurable cardinal κ, then for every uncountable set x of ordinals, there is y&nbsp;∈&nbsp;K[C] such that x&nbsp;⊂&nbsp;y  and |x|&nbsp;=&nbsp;|y|. Here C is either empty or Prikry generic over K (so it has order type ω and is cofinal in&nbsp;κ) and unique except up to a finite initial segment.
# 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 κ&nbsp;\&nbsp;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&nbsp;∈&nbsp;K[C] such that x&nbsp;⊂&nbsp;y  and |x|&nbsp;=&nbsp;|y|.
# For every uncountable set x of ordinals, there is a set C of indiscernibles for total extenders on K such that there is y&nbsp;∈&nbsp;K[C] and x &nbsp;⊂&nbsp;y  and |x|&nbsp;=&nbsp;|y|.
# K computes the successors of singular and weakly compact cardinals correctly ('''Weak Covering Property''').  Moreover, if |κ|&nbsp;>&nbsp;ω<sub>1</sub>, then  cofinality((κ<sup>+</sup>)<sup>''K''</sup>)&nbsp;≥&nbsp;|κ|.

==Extenders and indiscernibles==
For core models without overlapping total extenders, the systems of indiscernibles 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 2<sup>κ</sup>&nbsp;=&nbsp;κ<sup>++</sup>, then κ has Mitchell order at least κ<sup>++</sup> in&nbsp;K.  Conversely, a failure of the singular cardinal hypothesis can be obtained (in a generic extension) from κ with o(κ)&nbsp;=&nbsp;κ<sup>++</sup>.

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.

==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&nbsp;K.

Also, if the core model K(X) exists above a set X of ordinals, then it has the above discussed covering properties above&nbsp;X.

==References==
*{{citation | last1=Mitchell|first1=William|chapter=The covering lemma|title=Handbook of Set Theory|publisher=Springer|year=2010|doi=10.1007/978-1-4020-5764-9_19|pages= 1497–1594|  isbn=978-1-4020-4843-2 }}
{{DEFAULTSORT:Covering Lemma}}
[[Category:Inner model theory]]
[[Category:Lemmas]]
[[Category:Covering lemmas]]