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Set-theoretic blockchains

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Abstract

Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.

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Authors and Affiliations

  1. Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00, Praha 6, Czech Republic

    Miha E. Habič

  2. Department of Logic, Faculty of Arts, Charles University, Nám. Jana Palacha 2, 116 38, Praha 1, Czech Republic

    Miha E. Habič & Jonathan Verner

  3. Faculty of Philosophy, University of Oxford, Radcliffe Observatory Quarter 555, Woodstock Road, Oxford, OX2 6GG, UK

    Joel David Hamkins

  4. Sir Peter Strawson Fellow in Philosophy, University College, Oxford, OX1 4BH, UK

    Joel David Hamkins

  5. Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10/104, 1040, Wien, Austria

    Lukas Daniel Klausner

  6. Department of Mathematics, University of Hawai‘i at Mānoa, 2565 McCarthy Mall, Keller 401A, Honolulu, HI, 96822, USA

    Kameryn J. Williams

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  1. Miha E. Habič
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Correspondence to Kameryn J. Williams.

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We thank the anonymous referee for their helpful comments. Miha E. Habič was supported by the ESIF, EU Operational Programme Research, Development and Education, the International Mobility of Researchers in CTU project no. (CZ.02.2.69/0.0/0.0/16_027/0008465) at the Czech Technical University in Prague, the joint FWF–GAČR Grant No. 17-33849L: Filters, Ultrafilters and Connections with Forcing and by the Progres grant Q14. Krize racionality a moderní myšlení. Joel David Hamkins is grateful for the financial support provided by the Czech Academy of Sciences and the Winter School in Abstract Analysis 2018 in connection with his visit to Prague and Hejnice in January and February 2018. This work is truly a collaboration between Prague, New York and Vienna, with various combinations of the authors interacting significantly in each of these cities. Lukas Daniel Klausner was supported by the Austrian Science Fund (FWF) Project P29575 “Forcing Methods: Creatures, Products and Iterations”. Jonathan Verner was supported by the joint FWF–GAČR Grant No. 17-33849L: Filters, Ultrafilters and Connections with Forcing, by the Progres grant Q14. Krize racionality a moderní myšlení and by Charles University Research Centre Program No. UNCE/SCI/022.

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Habič, M.E., Hamkins, J.D., Klausner, L.D. et al. Set-theoretic blockchains. Arch. Math. Logic 58, 965–997 (2019). https://doi.org/10.1007/s00153-019-00672-z

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