Forcing and the Universe of Sets: Must We Lose Insight?

  • Neil BartonEmail author


A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions.


Foundations of mathematics Set theory Forcing Potentialism Multiversism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author wishes to thank Carolin Antos, Tim Button, Andrés Eduardo Caicedo, Monroe Eskew, Sy Friedman, Vera Flocke, Victoria Gitman, Bob Hale, Joel Hamkins, Toby Meadows, Sandra Müller, Alex Paseau, Ian Rumfitt, Chris Scambler, Sam Roberts, Jonathan Schilhan, Daniel Soukup, Kameryn Williams, and three anonymous reviewers for insightful and helpful comments, as well as audiences in Cambridge, Konstanz, Vienna, and Toulouse for the opportunity to present and subsequent discussion. He is also very grateful for the generous support of the UK Arts and Humanities Research Council (as a PhD student), FWF (Austrian Science Fund, Project P 28420), and VolkswagenStiftung (through the project Forcing: Conceptual Change in the Foundations of Mathematics).


  1. 1.
    Antos, C. (2018). Class Forcing in Class Theory, (pp. 1–16). Cham: Springer International Publishing.Google Scholar
  2. 2.
    Antos, C., Barton, N., Friedman, S.-D. Universism and extensions of V. Manuscript under review.Google Scholar
  3. 3.
    Antos, C., Friedman, S.-D., Honzik, R., Ternullo, C. (2015). Multiverse conceptions in set theory. Synthese, 192(8), 2463–2488.CrossRefGoogle Scholar
  4. 4.
    Arrigoni, T., & Friedman, S.-D. (2013). The hyperuniverse program. Bulletin of Symbolic Logic, 19, 77–96.CrossRefGoogle Scholar
  5. 5.
    Baumgartner, J., & Hajnal, A. (1973). A proof (involving Martin’s Axiom) of a partition relation. Fundamenta Mathematicae, 78(3), 193–203.CrossRefGoogle Scholar
  6. 6.
    Cohen, P. (1966). Set theory and the continuum hypothesis. W.A. Benjamin, Inc.Google Scholar
  7. 7.
    Cummings, J. (2010). Iterated forcing and elementary embeddings, (pp. 775–883). Netherlands,: Springer.Google Scholar
  8. 8.
    Feferman, S. (1969). Set-theoretical foundations of category theory. In Reports of the midwest category seminar III, lecture notes in mathematics, (Vol. 106 pp. 201–247): Springer.Google Scholar
  9. 9.
    Foreman, M. (1986). Potent axioms. Transactions of the American Mathematical Society, 294(1), 1–28.CrossRefGoogle Scholar
  10. 10.
    Foreman, M. (1998). Generic large cardinals: new axioms for mathematics? Documenta Mathematica, 2, 11–21.Google Scholar
  11. 11.
    Foreman, M. (2010). Ideals and generic elementary embeddings. In Kanamori, A., & Foreman, M. (Eds.) Handbook of set theory (pp. 885–1147): Springer.Google Scholar
  12. 12.
    Friedman, S.-D. (2000). Fine structure and class forcing. de Gruyter. de Gruyter series in logic and its applications (Vol. 3).Google Scholar
  13. 13.
    Friedman, S.-D. (2006). Internal consistency and the inner model hypothesis. Bulletin of Symbolic Logic, 12(4), 591–600.CrossRefGoogle Scholar
  14. 14.
    Friedman, S.-D. (2010). Constructibility and class forcing. In Kanamori, A. (Ed.) Handbook of set theory (pp. 557–604): Springer.Google Scholar
  15. 15.
    Friedman, S.-D. (2016). Evidence for set-theoretic truth and the hyperuniverse programme. IfCoLog Journal of Applied Logics, 4(``Proof, Truth, Computation” 3), 517–555.Google Scholar
  16. 16.
    Gitman, V., Hamkins, J.D., Holy, P., Schlicht, P., Williams, K. (2017). The exact strength of the class forcing theorem.Google Scholar
  17. 17.
    Gitman, V., & Schindler, R. (2018). Virtual large cardinals. Annals of Pure and Applied Logic.Google Scholar
  18. 18.
    Hamkins, J.D. (2003). A simple maximality principle. The Journal of Symbolic Logic, 68(2), 527–550.CrossRefGoogle Scholar
  19. 19.
    Hamkins, J.D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5(3), 416–449.CrossRefGoogle Scholar
  20. 20.
    Hamkins, J.D., & Seabold, D.E. (2012). Well-founded Boolean ultrapowers as large cardinal embeddings. arXiv:1206.6075 [math.LO].
  21. 21.
    Hamkins, J.D., & Yang, R. (2013). Satisfaction is not absolute. arXiv:1312.0670v1 [math.LO].
  22. 22.
    Holy, P., Krapf, R., Lücke, P., Njegomir, A., Schlicht, P. (2016). Class forcing, the forcing theorem, and boolean completions. The Journal of Symbolic Logic, 81(4), 1500–1530.CrossRefGoogle Scholar
  23. 23.
    Holy, P., Krapf, R., Schlicht, P. (2018). Characterizations of pretameness and the ord-cc. Annals of Pure and Applied Logic, 169(8), 775–802.CrossRefGoogle Scholar
  24. 24.
    Jech, T. (2002). Set theory. Springer.Google Scholar
  25. 25.
    Kanamori, A. (2009). The higher infinite: large cardinals in set theory from their beginnings, 2nd edn. Springer.Google Scholar
  26. 26.
    Koellner, P. (2013). Hamkins on the multiverse. In Koellner, P. (Ed.) Exploring the frontiers of incompleteness.Google Scholar
  27. 27.
    Koellner, P. (2014). Large cardinals and determinacy. In Zalta, E.N. (Ed.) The Stanford encyclopedia of philosophy. Metaphysics research lab. Spring 2014 edition: Stanford University.Google Scholar
  28. 28.
    Kunen, K. (1971). Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36, 407–413.CrossRefGoogle Scholar
  29. 29.
    Kunen, K. (2013). Set theory. College Publications.Google Scholar
  30. 30.
    Larson, P. (2004). The stationary tower: notes on a course by W. Hugh Woodin. University lecture series. American Mathematical Society.Google Scholar
  31. 31.
    Malliaris, M., & Shelah, S. (2016). Cofinality spectrum theorems in model theory, set theory, and general topology. Journal of the American Mathematical Society, 29, 237–297.CrossRefGoogle Scholar
  32. 32.
    Painlevé, P. (1900). Analyse des travaux scientifiques. Gauthier-Villars.Google Scholar
  33. 33.
    Schindler, R. (2000). Proper forcing and remarkable cardinals: I. Bulletin of Symbolic Logic, 6(2), 176–184.CrossRefGoogle Scholar
  34. 34.
    Schindler, R. (2001). Proper forcing and remarkable cardinals: II. Journal of Symbolic Logc, 66(3), 1481–1492.CrossRefGoogle Scholar
  35. 35.
    Schindler, R. (2014). Set theory: exploring independence and truth. Springer.Google Scholar
  36. 36.
    Shoenfield, J. (1967). Mathematical logic. Addison-Wesley Publishing Co.Google Scholar
  37. 37.
    Smullyan, R.M., & Fitting, M. (1996). Set theory and the continuum problem. Clarendon Press.Google Scholar
  38. 38.
    Steel, J. (1996). The core model iterability problem. In Lecture notes in logic (Vol. 8). Springer.Google Scholar
  39. 39.
    Todorc̆ević, S., & Farah, I. (1995). Some applications of the method of forcing. Moscow: Yenisei.Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.ZukunftskollegUniversity of KonstanzKonstanzGermany

Personalised recommendations