, Volume 116, Issue 1, pp 285-321

The generalized continuum hypothesis revisited

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Abstract

We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering the known independence results, is “for most pairs κ<λ of regular cardinals, λ to the revised power of κ is equal to λ”. What is the revised power? λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of λ of cardinality κ is included in the union of strictly less than κ members of the family. We still have to say what “for most” means. The interpretation we choose is: for every λ, for every large enoughK < ℶ w . Under this reinterpretation, we prove the Generalized Continuum Hypothesis.

Partially supported by the Israeli Basic Research Fund and the BSF. The author wishes to thank Alice Leonhardt for the beautiful typing. Publication No. 460