Existence of a Model of $o(κ)=κ^{++}$ from Failure of GCH at a Measurable Cardinal

TL;DR

The paper provides an expository, detail-filled proof of the lower bound for the equiconsistency between the failure of the Generalized Continuum Hypothesis at a measurable cardinal and the existence of a measurable cardinal with Mitchell order $o(\lambda)=\lambda^{++}$. It clarifies and fills gaps in the literature by elaborating on a core-model framework (up to $o(\kappa)=\kappa^{++}$) and the use of iterated ultrapowers, normal ultrafilters, and Fodor-type arguments. The result shows that if $2^{\kappa}>\kappa^{+}$ for some measurable $\kappa$, then there is an inner model containing a measurable cardinal $\lambda$ with $o(\lambda)=\lambda^{++}$. This strengthens the accessibility of Mitchell-style core model arguments and connects GCH failure at a measurable cardinal to precise large-cardinal strength statements.

Abstract

It is well-known that the consistency strength of the GCH failing at a measurable cardinal is the existence of a cardinal $κ$ with $o(κ)=κ^{++}$. As the literature does not contain more than a proof sketch of the lower bound of this equiconsistency, we give an expository proof which fills in the details in order to fill this gap in the literature.

Theorems & Definitions (10)

• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Claim 2.1
• proof
• Claim 2.2
• proof
• Claim 2.3
• proof