6 citations · 16 across the 22 of their papers we have counts for
24 papers
The Galvin property under the Ultrapower Axiom
Tom Benhamou, Gabriel Goldberg
We continue the study of the Galvin property from \cite{bgs} and \cite{Benhamou2}. In particular, we deepen the connection between certain diamond-like principles and non-Galvin ul…
Choiceless cardinals and the continuum problem
Gabriel Goldberg
Under large cardinal hypotheses beyond the Kunen inconsistency -- hypotheses so strong as to contradict the Axiom of Choice -- we solve several variants of the generalized continuu…
Reinhardt cardinals in inner models
Gabriel Goldberg
A cardinal is weakly Reinhardt if it is the critical point of an elementary embedding from the universe of sets into a model that contains the double powerset of every ordinal. Thi…
Strongly compact cardinals and ordinal definability
Gabriel Goldberg
This paper explores several topics related to Woodin's HOD conjecture. We improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an extendible cardinal to a s…
Measurable cardinals and choiceless axioms
Gabriel Goldberg
We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.
The uniqueness of elementary embeddings
Gabriel Goldberg
Much of the theory of large cardinals beyond a measurable cardinal concerns the structure of elementary embeddings of the universe of sets into inner models. This paper seeks to an…
A note on cardinal preserving embeddings
Gabriel Goldberg
We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The p…
A note on Woodin's HOD dichotomy
Gabriel Goldberg
A version of Woodin's HOD dichotomy is proved assuming the existence of just one strongly compact cardinal.
On products of ultrafilters
Gabriel Goldberg
Assuming the Generalized Continuum Hypothesis, this paper answers the question: when is the tensor product of two ultrafilters equal to their Cartesian product? It is necessary and…
Some combinatorial properties of Ultimate L and V
Gabriel Goldberg
This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrap…
The Ultrapower Axiom
Gabriel Goldberg
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact…
Periodicity in the cumulative hierarchy
Gabriel Goldberg, Farmer Schlutzenberg
We investigate the structure of rank-to-rank elementary embeddings, working in ZF set theory without the Axiom of Choice. Recall that the levels of the cumulative hierarchy a…
Even ordinals and the Kunen inconsistency
Gabriel Goldberg
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed…
Structure theorems above a strongly compact cardinal
Gabriel Goldberg
While many inner model theoretic combinatorial principles are incompatible with large cardinal axioms, on some rare occasions, large cardinals actually imply that the structure of…
The internal relation
Gabriel Goldberg
This paper explores various generalizations of the Mitchell order focusing mostly on a generalization called the internal relation. The internal relation lacks the implicit strengt…
The Ultrapower Axiom and the equivalence between strong compactness and supercompactness
Gabriel Goldberg
The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis…
The Ultrapower Axiom implies GCH above a supercompact cardinal
Gabriel Goldberg
We prove that the Generalized Continuum Hypothesis holds above a supercompact cardinal assuming the Ultrapower Axiom, an abstract comparison principle motivated by inner model theo…
Fréchet cardinals
Gabriel Goldberg
An infinite cardinal is called Fréchet if the Fréchet filter on extends to a countably complete ultrafilter. We investigate the relationship between Fréchet cardinals and s…
The Rudin-Frolik order and the Ultrapower Axiom
Gabriel Goldberg
We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom,…
The seed order
Gabriel Goldberg
We study various orders on countably complete ultrafilters on ordinals that coincide and are wellorders under a hypothesis called the Ultrapower Axiom. Our main focus is on the rel…
Compactness and Comparison
Gabriel Goldberg
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mit…
Strong Compactness and the Ultrapower Axiom
Gabriel Goldberg
We announce some consequences of an abstract comparison principle.
The Linearity of the Mitchell Order
Gabriel Goldberg
We show from a weak comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on certain kinds of ultrafilters: normal ultrafilters, Dodd solid ultrafilters, an…
The Seed Order
Gabriel Goldberg
This paper introduces the seed order, a partial order of the class of uniform countably complete ultrafilters that generalizes the Mitchell order on normal measures. Like that orde…