Pluralism in the ontology of mathematics, MaMuPhi, Paris, February 2022

This will be a talk for the conference L’indépendance mathématique et ses limites logiques, an instance of the MAMUPHI seminar (mathématiques – musique – philosophie), organized by Mirna Džamonja, 12 February 2022. Most talks will be in-person in Paris, but my talk will be on Zoom via https://u-pec-fr.zoom.us/j/86448599486 at 4:30 pm CET (10:30 am EST).


Abstract: What is the nature of mathematical ontology—what does it mean to make existence assertions in mathematics? Is there an ideal mathematical realm, a mathematical universe, that those assertions are about? Perhaps there is more than one. Does every mathematical assertion ultimately have a definitive truth value? I shall lay out some of the back-and-forth in what is currently a vigorous debate taking place in the philosophy of set theory concerning pluralism in the set-theoretic foundations, concerning whether there is just one set-theoretic universe underlying our mathematical claims or whether there is a diversity of possible set-theoretic conceptions.

Diamond (on the regulars) can fail at any strongly unfoldable cardinal

  • [DOI] M. D{u{z}}amonja and J. D. Hamkins, “Diamond (on the regulars) can fail at any strongly unfoldable cardinal,” Ann.~Pure Appl.~Logic, vol. 144, iss. 1-3, p. 83–95, 2006.
    [Bibtex]
    @ARTICLE{DzamonjaHamkins2006:DiamondCanFail,
    AUTHOR = {D{\u{z}}amonja, Mirna and Hamkins, Joel David},
    TITLE = {Diamond (on the regulars) can fail at any strongly unfoldable cardinal},
    JOURNAL = {Ann.~Pure Appl.~Logic},
    FJOURNAL = {Annals of Pure and Applied Logic},
    VOLUME = {144},
    YEAR = {2006},
    NUMBER = {1-3},
    PAGES = {83--95},
    ISSN = {0168-0072},
    CODEN = {APALD7},
    MRCLASS = {03E05 (03E35 03E55)},
    MRNUMBER = {2279655 (2007m:03091)},
    MRREVIEWER = {Andrzej Ros{\l}anowski},
    DOI = {10.1016/j.apal.2006.05.001},
    URL = {http://jdh.hamkins.org/diamondcanfail/},
    month = {December},
    note = {Conference in honor of sixtieth birthday of James E.~Baumgartner},
    eprint = {math/0409304},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    }

If $\kappa$ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which $\Diamond_\kappa(\text{REG})$ fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.