6 papers
Isomorphism Classes of Generating Sets
Tom Benhamou, James Cummings, Gabriel Goldberg +2
We introduce a new class of ultrafilters which generalizes the well-known class of simple -point ultrafilters. We prove that for any well-founded -directed partial order $\m…
Ultrapowers of determinacy models as iteration trees on HOD
Gabriel Goldberg, Grigor Sargsyan, Benjamin Siskind
In the 1990s, Steel and Woodin showed that under large cardinal hypotheses, the HOD of admits a fine-structural analysis. Although this theorem sheds light on variou…
Measures that violate the Generalized Continuum Hypothesis
Tom Benhamou, Gabriel Goldberg
A simple \(P_λ\)-point on a regular cardinal \(κ\) is a uniform ultrafilter on \(κ\) with a mod-bounded decreasing generating sequence of length \(λ\). We prove that if there i…
Applications of the Magidor Iteration to Ultrafilter Theory
Tom Benhamou, Gabriel Goldberg
We characterize sums of normal ultrafilters after the Magidor iteration (product) of Prikry forcings over a discrete set of measurable cardinals. We apply this to show that the wea…
On Absoluteness Between V and HOD
Gabriel Goldberg, Dan Hathaway
We put together Woodin's basis theorem of AD and VopÄnka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(Σ^2_1)^{\m…
On the optimality of the HOD dichotomy
Gabriel Goldberg, Jonathan Osinski, Alejandro Poveda
In the first part of the manuscript, we establish several consistency results concerning Woodin's $\HOD$ hypothesis and large cardinals around the level of extendibility. First, we…