Date of Award
Spring 2025
Document Type
Thesis
Submission Date
May 2025
Degree Name
Master of Science (MS)
Department
Mathematical Sciences
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.
Thesis/Dissertation Advisor [Chair]
Robert Lubarsky
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