Abstract
In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.
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Acknowledgments
I am grateful to Lev Beklemishev and Pavel Pudlák for their advice on matters of definition. I am grateful to Rosalie Iemhoff and Ali Enayat for stimulating conversations. I thank Rosalie for her comments on the manuscript. Finally, I thank the anonymous referees for their corrections and comments.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Visser, A. Pairs, sets and sequences in first-order theories. Arch. Math. Logic 47, 299–326 (2008). https://doi.org/10.1007/s00153-008-0087-1
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DOI: https://doi.org/10.1007/s00153-008-0087-1