Semester Award Granted
Spring 2025
Submission Date
May 2025
Document Type
Thesis
Degree Name
Master of Science (MS)
Thesis/Dissertation Advisor [Chair]
Robert Lubarsky
Abstract
This expository paper investigates the equiconsistency of the GCH failing at a measurable cardinal with the existence of a cardinal κ of Mitchell order κ++.
The upper bound of this equiconsistency follows in two parts: Assuming the existence of a model of o(κ) = κ++, one can first use an argument of Gitik to force the existence of an elementary embedding satisfying certain closure conditions, then use a forcing due to Woodin to force the failure of GCH at κ while preserving the measurability of κ. It is this Woodin result which this thesis focuses on in the upper bound.
The lower bound of this equiconsistency is an inner-model-theoretic argument due to Mitchell, where one can show that assuming the GCH fails at a measurable cardinal, then K, the so-called ‘core model below o(κ) = κ++’ exists. This thesis aims to bridge a gap in the literature by providing a much-needed approachable introduction to inner model theory at the level of o(κ) = κ++ for the non-specialist. Mitchell’s argument that the GCH failing at a measurable cardinal implying the existence of a model of o(κ) = κ++ is then given.
Recommended Citation
Watson, Connor, "THE CONSISTENCY STRENGTH OF THE GENERALIZED CONTINUUM HYPOTHESIS FAILING AT A MEASURABLE CARDINAL" (2025). Electronic Theses and Dissertations. 13.
https://digitalcommons.fau.edu/etd_general/13