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Creators/Authors contains: "Goldberg, Gabriel"

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  1. ; ; (, Advances in Mathematics)
    Free, publicly-accessible full text available May 1, 2026
  2. (, The Journal of Symbolic Logic)
    Abstract 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 answer the question of whether the inner model uniquely determines the elementary embedding. 
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  3. ; (, Canadian Journal of Mathematics)
    Abstract We continue the study of the Galvin property from Benhamou, Garti, and Shelah (2023,Proceedings of the American Mathematical Society151, 1301–1309) and Benhamou (2023,Saturation properties in canonical inner models, submitted). In particular, we deepen the connection between certain diamond-like principles and non-Galvin ultrafilters. We also show that any Dodd sound nonp-point ultrafilter is non-Galvin. We use these ideas to formulate what appears to be the optimal large cardinal hypothesis implying the existence of a non-Galvin ultrafilter, improving on a result from Benhamou and Dobrinen (2023,Journal of Symbolic Logic, 1–34). Finally, we use a strengthening of the Ultrapower Axiom to prove that in all the known canonical inner models, a$$\kappa $$-complete ultrafilter has the Galvin property if and only if it is an iterated sum ofp-points. 
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  4. (, Annals of Pure and Applied Logic)
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  5. (, Journal of Mathematical Logic)
    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 strongly compact cardinal. We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin. We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy. We prove that the HOD hypothesis holds if and only if every regular cardinal above the first strongly compact cardinal carries an ordinal definable omega Jónsson algebra. We show that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then every supercompact cardinal is supercompact in HOD. 
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  6. ; (, Journal of the European Mathematical Society)
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  7. (, Journal of Mathematical Logic)
    The Ultrapower Axiom is a combinatorial principle concerning the structure of large cardinals that is true in all known canonical inner models of set theory. A longstanding test question for inner model theory is the equiconsistency of strongly compact and supercompact cardinals. In this paper, it is shown that under the Ultrapower Axiom, the least strongly compact cardinal is supercompact. A number of stronger results are established, setting the stage for a complete analysis of strong compactness and supercompactness under UA that will be carried out in the sequel to this paper. 
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