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Set-theoretic blockchains

  • Miha E. Habič
  • Joel David Hamkins
  • Lukas Daniel Klausner
  • Jonathan Verner
  • Kameryn J. WilliamsEmail author
Article

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.

Keywords

Generic multiverse Amalgamability Blockchains Mutual genericity Surgery Exact pair 

Mathematics Subject Classification

Primary 03E40 Secondary 03E35 

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References

  1. 1.
    Arrigoni, T., Friedman, S.-D.: The hyperuniverse program. Bull. Symb. Log. 19(1), 77–96 (2013).  https://doi.org/10.2178/BSL.1901030 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antos, C., Friedman, S.-D., Honzík, R., Ternullo, C. (eds.): The hyperuniverse project and maximality. Birkhäuser, Basel (2018).  https://doi.org/10.1007/978-3-319-62935-3
  3. 3.
    Balcar, B., Hájek, P.: On sequences of degrees of constructibility. Z. Math. Logik Grundlag. Math. 24(4), 291–296 (1978).  https://doi.org/10.1002/MALQ.19780241903 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bukovský, L.: Characterization of generic extensions of models of set theory. Fundam. Math. 83(1), 35–46 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Friedman, S.D., Hathaway, D.: Generic coding with help and amalgamation failure. arXiv:1808.10304 [math.LO]
  6. 6.
    Friedman, S.D.: Fine structure and class forcing. De Gruyter Series in Logic and Its Applications. De Gruyter, Berlin (2000).  https://doi.org/10.1515/9783110809114
  7. 7.
    Friedman, S.D.: The hyperuniverse. Tutorial at the University of Münster (2012), http://www.logic.univie.ac.at/~sdf/papers/muenster.2012.pdf
  8. 8.
    Fuchs, G., Hamkins, J.D., Reitz, J.: Set-theoretic geology. Ann. Pure Appl. Logic 166(4), 464–501, (2015).  https://doi.org/10.1016/J.APAL.2014.11.004; arXiv:1107.4776 [math.LO], comments and discussion: http://jdh.hamkins.org/set-theoreticgeology/
  9. 9.
    Fujimoto, K.: Classes and truths in set theory. Ann. Pure Appl. Logic 163(11), 1484–1523 (2012).  https://doi.org/10.1016/J.APAL.2011.12.006 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gitman, V., Hamkins, J.D.: Kelley–Morse set theory and choice principles for classes (unpublished)Google Scholar
  11. 11.
    Gitman, V., Hamkins, J.D.: Open determinacy for class games. In: Andrés E.C., James, C., Peter, K., Larson, P. B. (eds.) Foundations of Mathematics: Logic at Harvard. Essays in Honor of W. Hugh Woodin’s 60th Birthday. Contemporary Mathematics, vol. 690, pp. 121–143. American Mathematical Society, Providence, RI (2017).  https://doi.org/10.1090/CONM/690; arXiv:1509.01099 [math.LO]; comments and discussion: http://jdh.hamkins.org/open-determinacy-for-class-games/
  12. 12.
    Gitman, V., Hamkins, J.D., Johnstone, T.A.: What is the theory ZFC without power set?, MLQ Math. Log. Q. 62(4–5), 391–406 (2016).  https://doi.org/10.1002/MALQ.201500019; arXiv:1110.2430 [math.LO], comments and discussion: http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/
  13. 13.
    Gitman, V.: Kelley–Morse set theory and choice principles for classes, blog post with slides (2014). https://victoriagitman.github.io/talks/2014/12/31/kelley-morse-set-theory-and-choice-principles-for-classes.html
  14. 14.
    Hamkins, J.D.: Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity? MathOverflow question (2015). https://mathoverflow.net/q/222602
  15. 15.
    Hamkins, J.D.: Upward Closure and amalgamation in the generic multiverse of a countable model of set theory, RIMS Kōkyūroku 1988 Recent Developments in Axiomatic Set Theory, pp. 17–31 (2016) hdl: 2433/224551, arXiv:1511.01074 [math.LO], comments and discussion: http://jdh.hamkins.org/upward-closure-and-amalgamation-in-the-generic-multiverse-of-a-countable-model-of-set-theory/
  16. 16.
    Hamkins, J.D., Löwe, B.: The modal logic of forcing. Trans. Amer. Math. Soc. 360(4), 1793–1817 (2008).  https://doi.org/10.1090/S0002-9947-07-04297-3; arXiv:MATH/0509616 [math.LO], comments and discussion: http://jdh.hamkins.org/themodallogicofforcing/
  17. 17.
    Hamkins, J.D., Löwe, B.: Moving up and down in the generic multiverse. In: Logic and Its Applications, Lecture Notes in Computer Science, vol. 7750, pp. 139–147. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36039-8_13; arXiv:1208.5061 [math.LO], comments and discussion: http://jdh.hamkins.org/up-and-down-in-the-generic-multiverse/
  18. 18.
    Holy, P., Krapf, R., Lücke, P., Njegomir, A., Schlicht, P.: Class forcing, the forcing theorem and Boolean completions. J. Symb. Log. 81(4), 1500–1530 (2016).  https://doi.org/10.1017/JSL.2016.4; arXiv:1710.10820 [math.LO]
  19. 19.
    Holy, P., Krapf, R., Schlicht, P.: Characterizations of pretameness and the Ord-cc. Ann. Pure Appl. Logic 169(8), 775–802 (2018).  https://doi.org/10.1016/J.APAL.2018.04.002; arXiv:1710.10825 [math.LO]
  20. 20.
    Jech, T.: Set theory, Springer Monographs in Mathematics. Springer, Berlin (2003).  https://doi.org/10.1007/3-540-44761-X
  21. 21.
    Jensen, R.: Definable sets of minimal degree. In: Mathematical Logic and Foundations of Set Theory, Proceedings of an International Colloquium Held Under the Auspices of The Israel Academy of Sciences and Humanities, 11–14 November 1968, Jerusalem, pp. 122–128. North-Holland, Amsterdam (1970).  https://doi.org/10.1016/S0049-237X(08)71934-7
  22. 22.
    Lerman, M.: Degrees of Unsolvability: Local and Global Theory, Perspectives in Logic. Cambridge University Press, Cambridge (2017).  https://doi.org/10.1017/9781316717059
  23. 23.
    Mostowski, A.: A remark on Models of the Gödel–Bernays axioms for set theory. In: Gert, H.M (ed.) Sets and Classes: On the Work by Paul Bernays. Studies in Logic and the Foundations of Mathematics, vol. 84, pp. 325–340. North-Holland, Amsterdam (1976).  https://doi.org/10.1016/S0049-237X(09)70288-5
  24. 24.
    Reitz, J.: The ground axiom. J. Symb. Log. 72(4), 1299–1317 (2007).  https://doi.org/10.2178/JSL/1203350787; arXiv:MATH/0609064 [math.LO]
  25. 25.
    Shore, R.A.: Degree structures: local and global investigations. Bull. Symb. Log. 12(3), 369–389 (2006).  https://doi.org/10.1515/9783110809114 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stanley, M.C.: A unique generic real. Ph.D. thesis, University of California, Berkeley (1984)Google Scholar
  27. 27.
    Truss, J.K.: A Note on increasing sequences of constructibility degrees. In: Müller, G.H., Stewart Scott D (eds.) Higher Set Theory, Proceedings, Oberwolfach, Germany 13–23 April, 1977. Lecture Notes in Mathematics, vol. 669, pp. 473–476. Springer, Berlin (1978).  https://doi.org/10.1007/BFB0103096
  28. 28.
    Usuba, T.: The downward directed grounds hypothesis and very large cardinals. J. Math. Log. 17(2), 1750009 (2017).  https://doi.org/10.1142/S021906131750009X; arXiv:1707.05132 [math.LO]
  29. 29.
    Williams, K.J.: The structure of models of second-order set theories. Ph.D. thesis, The Graduate Center of the City University of New York (2018). https://academicworks.cuny.edu/gc_etds/2678/
  30. 30.
    Woodin, W.H.: The continuum hypothesis, the generic-multiverse of sets, and the \(\Omega \) conjecture. In: Juliette, K., Roman, K. (eds.) Set Theory, Arithmetic, and Foundations of Mathematics: Theorems, Philosophies. Lecture Notes Log., vol. 36, pp. 13–42. Assoc. Symbol. Logic, La Jolla, CA (2011)Google Scholar
  31. 31.
    Zarach, A.M.: Replacement \(\nrightarrow \) Collection. In: Gödel 96: Logical Foundations of Mathematics, Computer Science, and Physics–Kurt Gödel’s Legacy, Lecture Notes Log, vol. 6, pp. 307–322. Springer, Berlin/Heidelberg (1996).  https://doi.org/10.1007/978-3-662-21963-8_22

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Information TechnologyCzech Technical University in PraguePraha 6Czech Republic
  2. 2.Department of Logic, Faculty of ArtsCharles UniversityPraha 1Czech Republic
  3. 3.Faculty of PhilosophyUniversity of OxfordOxfordUK
  4. 4.Sir Peter Strawson Fellow in PhilosophyUniversity CollegeOxfordUK
  5. 5.Institute of Discrete Mathematics and GeometryTU WienWienAustria
  6. 6.Department of MathematicsUniversity of Hawai‘i at MānoaHonoluluUSA

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