The $\Sigma_1$-definable universal finite sequence

  • J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” ArXiv e-prints, 2019. (undeer review)  
    @ARTICLE{HamkinsWilliams:The-universal-finite-sequence,
    author = {Joel David Hamkins and Kameryn J. Williams},
    title = {The $\Sigma_1$-definable universal finite sequence},
    journal = {ArXiv e-prints},
    year = {2019},
    volume = {},
    number = {},
    pages = {},
    month = {},
    note = {undeer review},
    abstract = {},
    keywords = {under-review},
    eprint = {1909.09100},
    archivePrefix = {arXiv},
    primaryClass = {math.LO},
    source = {},
    doi = {},
    }

Abstract. We introduce the $\Sigma_1$-definable universal finite sequence and prove that it exhibits the universal extension property amongst the countable models of set theory under end-extension. That is, (i) the sequence is $\Sigma_1$-definable and provably finite; (ii) the sequence is empty in transitive models; and (iii) if $M$ is a countable model of set theory in which the sequence is $s$ and $t$ is any finite extension of $s$ in this model, then there is an end extension of $M$ to a model in which the sequence is $t$. Our proof method grows out of a new infinitary-logic-free proof of the Barwise extension theorem, by which any countable model of set theory is end-extended to a model of $V=L$ or indeed any theory true in a suitable submodel of the original model. The main theorem settles the modal logic of end-extensional potentialism, showing that the potentialist validities of the models of set theory under end-extensions are exactly the assertions of S4. Finally, we introduce the end-extensional maximality principle, which asserts that every possibly necessary sentence is already true, and show that every countable model extends to a model satisfying it.

  • The universal algorithm,
    • J. D. Hamkins and H. W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)  
      @ARTICLE{HamkinsWoodin:The-universal-finite-set,
      author = {Joel David Hamkins and W. Hugh Woodin},
      title = {The universal finite set},
      journal = {ArXiv e-prints},
      year = {2017},
      volume = {},
      number = {},
      pages = {1--16},
      month = {},
      note = {manuscript under review},
      abstract = {},
      keywords = {under-review},
      source = {},
      doi = {},
      eprint = {1711.07952},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      url = {http://jdh.hamkins.org/the-universal-finite-set},
      }
  • The modal logic of arithmetic potentialism,
    • J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (under review)  
      @ARTICLE{Hamkins:The-modal-logic-of-arithmetic-potentialism,
      author = {Joel David Hamkins},
      title = {The modal logic of arithmetic potentialism and the universal algorithm},
      journal = {ArXiv e-prints},
      year = {2018},
      volume = {},
      number = {},
      pages = {1--35},
      month = {},
      eprint = {1801.04599},
      archivePrefix = {arXiv},
      primaryClass = {math.LO},
      note = {under review},
      url = {http://wp.me/p5M0LV-1Dh},
      abstract = {},
      keywords = {under-review},
      source = {},
      doi = {},
      }
  • A new proof of the Barwise extension theorem
  • Kameryn’s blog post about the paper


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