Abstract

Publisher Summary This chapter focusses on the generalized continuum hypothesis (GCH) at measurable cardinals. The chapter discusses a theorem, in which it is supposed that there is a measurable cardinal, κ , such that at least one of the following holds: (1) 2κ > κ+, (2) Every κ-complete filter over κ can be extended to a κ-complete ultra-filter, and (3) there is a uniform κ-complete ultrafilter over κ+. The statement of the theorem and the proof are formalized within Morse-Kelley set theory with the axiom of choice. It is also assumed that κ is a measurable cardinal satisfying the theorem.

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