A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis

TL;DR

This paper constructs a set-theoretic model demonstrating that ultrafilter structures do not necessarily determine the continuum size at a measurable cardinal, highlighting limitations in the relationship between ultrafilters and cardinal arithmetic.

Contribution

It introduces a novel model of the form $L[A,U]$ with a measurable cardinal where the continuum hypothesis fails, using new methods in embeddings and forcing.

Findings

Ultrafilter structures do not fix the continuum size at large cardinals.

Constructed a model with a measurable cardinal where $2^\kappa = \kappa^{++}$.

Answered Goldberg's question on the failure of CH at a measurable cardinal.

Abstract

We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This result establishes a limitation on the extent to which structural properties of ultrafilters can determine the cardinal arithmetic at large cardinals, and answers a question posed by Goldberg concerning the failure of the Continuum Hypothesis at a measurable cardinal in a model of the Ultrapower Axiom. The construction introduces several methods in extensions of embeddings theory and fine-structure-based forcing, designed to control the behavior of non-normal ultrafilters in generic extensions.