The σ1-definable universal finite sequence

Journal of Symbolic Logic 87 (2):783-801 (2022)
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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, the sequence is $\Sigma _1$ -definable and provably finite; the sequence is empty in transitive models; and 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.



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Joel David Hamkins
Oxford University

Citations of this work

Varieties of Class-Theoretic Potentialism.Neil Barton & Kameryn J. Williams - 2024 - Review of Symbolic Logic 17 (1):272-304.
Modal Model Theory.Joel David Hamkins & Wojciech Aleksander Wołoszyn - 2024 - Notre Dame Journal of Formal Logic 65 (1):1-37.

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