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Representing sets of ordinals as countable unions of sets in the core model. (English) Zbl 0714.03045

The author proves two ‘duals’ to the well-known covering theorems for L and K, due to Jensen et al. His first result says that if \(0^{\#}\) does not exist, then every set of ordinals that is closed under primitive recursive set functions is a countable union of sets from L.
A similar result is given for the core model K, albeit with stronger assumptions and a somewhat weaker conclusion: the assumption is that there is no inner model with an Erdős cardinal; the conclusion is “for every ordinal \(\beta\) there is an algebra in K with countably many operators such that every subset of \(\beta\) which is closed under these operators is a countable union of sets from K.”
In this way one can build a ‘small’ closed and unbounded subset of \({\mathcal P}_{\omega_ 2}(\kappa)\), on each element of which is a countable union of sets from K. This complements a result of Baumgartner who used an Erdős cardinal to produce a model in which every closed and unbounded subset of \({\mathcal P}_{\omega_ 2}(\kappa)\) is of maximum size \(\kappa^{\omega_ 1}\).
Reviewer: K.P.Hart

MSC:

03E45 Inner models, including constructibility, ordinal definability, and core models
03E35 Consistency and independence results
03E05 Other combinatorial set theory
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