# 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.
[Bibtex]
@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, p. 1–16, 2017.
[Bibtex]
@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, p. 1–35, 2018.
[Bibtex]
@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