Gap forcing

  • J. D. Hamkins, “Gap forcing,” Israel J.~Math., vol. 125, pp. 237-252, 2001.  
    AUTHOR = {Hamkins, Joel David},
    TITLE = {Gap forcing},
    JOURNAL = {Israel J.~Math.},
    FJOURNAL = {Israel Journal of Mathematics},
    VOLUME = {125},
    YEAR = {2001},
    PAGES = {237--252},
    ISSN = {0021-2172},
    MRCLASS = {03E40 (03E55)},
    MRNUMBER = {1853813 (2002h:03111)},
    MRREVIEWER = {Renling Jin},
    DOI = {10.1007/BF02773382},
    URL = {},
    eprint = {math/9808011},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},

Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as $P*Q$, where $P$ has size less than $\delta$ and $Q$ is forced to be $\delta$-strategically closed. In this paper, generalizing the Levy-Solovay theorem, I show that after such forcing, every embedding $j:V[G]\to M[j(G)]$ in the extension which satisfies a mild closure condition is the lift of an embedding $j:V\to M$ in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground model and every measure in the extension which concentrates on a set in the ground model extends a measure in the ground model. It follows that gap forcing cannot create new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.