- J. D. Hamkins and K. J. Williams, “The $\Sigma_1$-definable universal finite sequence,” ArXiv e-prints, 2019. (under 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 = {under 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}, }`

- J. D. Hamkins and H. W. Woodin, “The universal finite set,” ArXiv e-prints, pp. 1-16, 2017. (manuscript under review)
- 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 = {}, }`

- J. D. Hamkins, “The modal logic of arithmetic potentialism and the universal algorithm,” ArXiv e-prints, pp. 1-35, 2018. (under review)
- A new proof of the Barwise extension theorem
- Kameryn’s blog post about the paper