Bi-interpretation of weak set theories, Oberwolfach, April 2020

This will be a talk for the workshop in Set Theory at the Mathematisches Forschungsinstitute Oberwolfach, April 5-11, 2020. 

Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above.  Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.

Bi-interpretation in weak set theories

    • J. D. Hamkins and A. R. Freire, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020.  
      @ARTICLE{HamkinsFreire:Bi-interpretation-in-weak-set-theories,
      author = {Joel David Hamkins and Alfredo Roque Freire},
      title = {Bi-interpretation in weak set theories},
      journal = {Mathematics arXiv},
      year = {2020},
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      abstract = {},
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      url = {http://jdh.hamkins.org/bi-interpretation-in-weak-set-theories},
      eprint = {2001.05262},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      }

The Ground Axiom

  • J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, pp. 3160-3162, 2005.  
    @ARTICLE{Hamkins2005:TheGroundAxiom,
    AUTHOR = "Joel David Hamkins",
    TITLE = "The {Ground Axiom}",
    JOURNAL = "Mathematisches Forschungsinstitut Oberwolfach Report",
    YEAR = "2005",
    volume = "55",
    number = "",
    pages = "3160--3162",
    month = "",
    note = "",
    abstract = "",
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    eprint = {1607.00723},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    url = {http://jdh.hamkins.org/thegroundaxiom/},
    file = F,
    }

This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.

Oberwolfach Research Report 55/2005 | Ground Axiom on Wikipedia