THE Σ1-DEFINABLE UNIVERSAL FINITE SEQUENCE

作     者：Hamkins, Joel David Williams, Kameryn J. 

作者机构：University of Oxford University College High Street OxfordOX1 4BH United Kingdom University of Hawai‘i at Mānoa Department of Mathematics 2565 McCarthy Mall Keller 401A HonoluluHI96822 United States 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2019年

核心收录：

主　　题：Computer circuits 

摘      要：We introduce the Σ1-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 Σ1-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.03H05, 03E40, 03E45 Copyright © 2019, The Authors. All rights reserved.