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Gap forcing

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Abstract

In this paper, I generalize the landmark Lévy-Solovay Theorem [LévSol67], which limits the kind of large cardinal embeddings that can exist in a small forcing extension, to a broad new class of forcing notions, a class that includes many of the forcing iterations most commonly found in the large cardinal literature. After such forcing, the fact is that every embedding satisfying a mild closure requirement lifts an embedding from the ground model. Such forcing, consequently, can create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, or huge cardinals, and so on.

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Authors and Affiliations

  1. The College of Staten Island, The City University of New York, USA

    Joel David Hamkins

  2. The Ph.D. Program in Mathematics, The Graduate Center of The City University of New York, 365 Fifth Avenue, 10016, New York, NY, USA

    Joel David Hamkins

Authors
  1. Joel David Hamkins
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Correspondence to Joel David Hamkins.

Additional information

My research has been supported in part by grants from the PSC-CUNY Research Foundation and from the Japan Society for the Promotion of Science. I would like to thank my gracious hosts at Kobe University in Japan for their generous hospitality. This paper follows up an earlier announcement of the main theorem appearing, without technical details, in [Ham99].

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Hamkins, J.D. Gap forcing. Isr. J. Math. 125, 237–252 (2001). https://doi.org/10.1007/BF02773382

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  • DOI: https://doi.org/10.1007/BF02773382

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