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

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  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.

  3. 3.

    Antos, C., Friedman, S.-D., Honzik, R., Ternullo, C. (2015). Multiverse conceptions in set theory. Synthese, 192(8), 2463–2488.

    Article  Google Scholar 

  4. 4.

    Arrigoni, T., & Friedman, S.-D. (2013). The hyperuniverse program. Bulletin of Symbolic Logic, 19, 77–96.

    Article  Google Scholar 

  5. 5.

    Baumgartner, J., & Hajnal, A. (1973). A proof (involving Martin’s Axiom) of a partition relation. Fundamenta Mathematicae, 78(3), 193–203.

    Article  Google Scholar 

  6. 6.

    Cohen, P. (1966). Set theory and the continuum hypothesis. W.A. Benjamin, Inc.

  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.

  9. 9.

    Foreman, M. (1986). Potent axioms. Transactions of the American Mathematical Society, 294(1), 1–28.

    Article  Google 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.

  12. 12.

    Friedman, S.-D. (2000). Fine structure and class forcing. de Gruyter. de Gruyter series in logic and its applications (Vol. 3).

  13. 13.

    Friedman, S.-D. (2006). Internal consistency and the inner model hypothesis. Bulletin of Symbolic Logic, 12(4), 591–600.

    Article  Google Scholar 

  14. 14.

    Friedman, S.-D. (2010). Constructibility and class forcing. In Kanamori, A. (Ed.) Handbook of set theory (pp. 557–604): Springer.

  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.

  17. 17.

    Gitman, V., & Schindler, R. (2018). Virtual large cardinals. Annals of Pure and Applied Logic.

  18. 18.

    Hamkins, J.D. (2003). A simple maximality principle. The Journal of Symbolic Logic, 68(2), 527–550.

    Article  Google Scholar 

  19. 19.

    Hamkins, J.D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5(3), 416–449.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  24. 24.

    Jech, T. (2002). Set theory. Springer.

  25. 25.

    Kanamori, A. (2009). The higher infinite: large cardinals in set theory from their beginnings, 2nd edn. Springer.

  26. 26.

    Koellner, P. (2013). Hamkins on the multiverse. In Koellner, P. (Ed.) Exploring the frontiers of incompleteness.

  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.

  28. 28.

    Kunen, K. (1971). Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36, 407–413.

    Article  Google Scholar 

  29. 29.

    Kunen, K. (2013). Set theory. College Publications.

  30. 30.

    Larson, P. (2004). The stationary tower: notes on a course by W. Hugh Woodin. University lecture series. American Mathematical Society.

  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.

    Article  Google Scholar 

  32. 32.

    Painlevé, P. (1900). Analyse des travaux scientifiques. Gauthier-Villars.

  33. 33.

    Schindler, R. (2000). Proper forcing and remarkable cardinals: I. Bulletin of Symbolic Logic, 6(2), 176–184.

    Article  Google Scholar 

  34. 34.

    Schindler, R. (2001). Proper forcing and remarkable cardinals: II. Journal of Symbolic Logc, 66(3), 1481–1492.

    Article  Google Scholar 

  35. 35.

    Schindler, R. (2014). Set theory: exploring independence and truth. Springer.

  36. 36.

    Shoenfield, J. (1967). Mathematical logic. Addison-Wesley Publishing Co.

  37. 37.

    Smullyan, R.M., & Fitting, M. (1996). Set theory and the continuum problem. Clarendon Press.

  38. 38.

    Steel, J. (1996). The core model iterability problem. In Lecture notes in logic (Vol. 8). Springer.

  39. 39.

    Todorc̆ević, S., & Farah, I. (1995). Some applications of the method of forcing. Moscow: Yenisei.

    Google Scholar 

Download references

Acknowledgments

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).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Neil Barton.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Barton, N. Forcing and the Universe of Sets: Must We Lose Insight?. J Philos Logic 49, 575–612 (2020). https://doi.org/10.1007/s10992-019-09530-y

Download citation

Keywords

  • Foundations of mathematics
  • Set theory
  • Forcing
  • Potentialism
  • Multiversism