Element contents
Iterative Conceptions of Set
Published online by Cambridge University Press: 15 May 2024
Summary
Many philosophers are aware of the paradoxes of set theory (e.g. Russell's paradox). For many people, these were solved by the iterative conception of set which holds that sets are formed in stages by collecting sets available at previous stages. This Element will examine possibilities for articulating this solution. In particular, the author argues that there are different kinds of iterative conception, and it's open which of them (if any) is the best. Along the way, the author hopes to make some of the underlying mathematical and philosophical ideas behind tricky bits of the philosophy of set theory clear for philosophers more widely and make their relationships to some other questions in philosophy perspicuous.
- Type
- Element
- Information
- Online ISBN: 9781009227223Publisher: Cambridge University PressPrint publication: 13 June 2024
References
Antos, C., Barton, N., and Friedman, S.-D. (2021). Universism and extensions of V. The Review of Symbolic Logic, 14(1):112–54.CrossRefGoogle Scholar
Arrigoni, T., and Friedman, S.-D. (2013). The Hyperuniverse Program. Bulletin of Symbolic Logic, 19(1):77–96.CrossRefGoogle Scholar
Bagaria, J. (2005). Natural axioms of set theory and the continuum problem. In Proceedings of the 12th International Congress of Logic, Methodology, and Philosophy of Science, pages 43–64. King’s College London Publications.Google Scholar
Barton, N. (2017). Executing Gödel’s Programme in Set Theory. PhD thesis, Birkbeck, University of London.Google Scholar
Barton, N. (2021). Indeterminateness and ‘the’ universe of sets: Multiversism, potentialism, and pluralism. In Fitting, M., editor, Selected Topics from Contemporary Logics, Landscapes in Logic 2, pages 105–82. College Publications.Google Scholar
Barton, N. (2022). Structural relativity and informal rigour. In Oliveri, G., Ternullo, C., and Boscolo, S., editors, Objects, Structures, and Logics: FilMat Studies in the Philosophy of Mathematics, pages 133–74. Springer International Publishing.Google Scholar
Barton, N. (MSa). Is (un)countabilism restrictive? To appear in the Journal of Philosophical Logic. Preprint: https://philpapers.org/rec/BARIUR.Google Scholar
Barton, N. (MSb). What makes a ‘good’ modal theory of sets? Manuscript under review. Preprint: https://philpapers.org/rec/BARWMA-9.Google Scholar
Barton, N., and Friedman, S. (2019). Set theory and structures. In Sarikaya, D., Kant, D., and Centrone, S., editors, Reflections on the Foundations of Mathematics, pages 223–53. Springer.Google Scholar
Barton, N., and Friedman, S. (MS). Countabilism and maximality principles. Manuscript under review. Preprint: https://philpapers.org/rec/BARCAM-5.Google Scholar
Barton, N., Müller, M., and Prunescu, M. (2022). On representations of intended structures in foundational theories. Journal of Philosophical Logic, 51(2):283–96.CrossRefGoogle Scholar
Bell, J. (2011). Set Theory: Boolean-Valued Models and Independence Proofs. Oxford University Press.Google Scholar
Bell, J. (2014). Intuitionistic Set Theory. Studies in Logic 50. College Publications.Google Scholar
Boolos, G. (1984). To be is to be a value of a variable (or to be some values of some variables). The Journal of Philosophy, 81(8):430–49.CrossRefGoogle Scholar
Boolos, G. (1998). Reply to Charles Parsons’ ‘Sets and Classes’. In Jeffrey, Richard, editor, Logic, Logic, and Logic, pages 30–6. Harvard University Press.Google Scholar
Brauer, E. (MS). What is forcing potentialism? Manuscript under review.Google Scholar
Builes, D., and Wilson, J. M. (2022). In defense of countabilism. Philosophical Studies, 179(7):2199–236.CrossRefGoogle Scholar
Button, T. (2021a). Level Theory Part 1: Axiomatizing the bare idea of a cumulative hierarchy of sets. The Bulletin of Symbolic Logic, 27(4):436–60.Google Scholar
Button, T. (2021b). Level Theory, Part 2: Axiomatizing the bare idea of a potential hierarchy. Bulletin of Symbolic Logic, 27(4):461–84.Google Scholar
Button, T. (2022). Level Theory, Part 3: A Boolean algebra of sets arranged in well-ordered levels. Bulletin of Symbolic Logic, 28(1):1–26.CrossRefGoogle Scholar
Caicedo, A. E., Cummings, J., Koellner, P., and Larson, P. B., editors (2017). Foundations of Mathematics: Logic at Harvard Essays in Honor of W. Hugh Woodin’s 60th Birthday, volume 690 of Contemporary Mathematics. American Mathematical Society.CrossRefGoogle Scholar
Cain, J. (1995). Infinite utility. Australasian Journal of Philosophy, 73(3):401–4.CrossRefGoogle Scholar
Cappelen, H. (2018). Fixing Language: An Essay on Conceptual Engineering. Oxford University Press.CrossRefGoogle Scholar
Chalmers, D. J. (2020). What is conceptual engineering and what should it be? Inquiry. DOI: https://doi.org/10.1080/0020174X.2020.1817141CrossRefGoogle Scholar
Cohen, P. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, 50(6):1143–8.Google ScholarPubMed
Cohen, P. (2002). The discovery of forcing. Rocky Mountain Journal of Mathematics, 32(4):1071–100.CrossRefGoogle Scholar
Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals. North Holland Publishing Co.Google Scholar
Feferman, S. (2010). On the strength of some semi-constructive set theories. In Berger, U., Diener, H., Schuster, P., and Seisenberger, M., editors, Logic, Construction, Computation, pp. 201–226. De Gruyter.Google Scholar
Feferman, S., Friedman, H., Maddy, P., and Steel, J. (2000). Does mathematics need new axioms? Bulletin of Symbolic Logic, 6(4):401–46.CrossRefGoogle Scholar
Feferman, S., and Hellman, G. (1995). Predicative foundations of arithmetic. Journal of Philosophical Logic, 24(1):1–17.CrossRefGoogle Scholar
Ferreirós, J. (2007). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics (2nd Edition). Springer, Birkhäuser.Google Scholar
Florio, S., and Leach-Krouse, G. (2017). What Russell should have said to Burali-Forti. Review of Symbolic Logic, 10(4):682–718.CrossRefGoogle Scholar
Florio, S., and Linnebo, O. (2021). The Many and the One: A Philosophical Study of Plural Logic. Oxford University Press.CrossRefGoogle Scholar
Forster, T. (2008). The iterative conception of set. The Review of Symbolic Logic, 1(1):97–110.CrossRefGoogle Scholar
Geschke, S., and Quickert, S. (2004). On Sacks forcing and the Sacks property. In Löwe, B., Piwinger, B., and Räsch, T., editors, Classical and New Paradigms of Computation and their Complexity Hierarchies, pages 95–139, Springer Netherlands.CrossRefGoogle Scholar
Giaquinto, M. (2002). The Search for Certainty: A Philosophical Account of Foundations of Mathematics. Oxford University Press.CrossRefGoogle Scholar
Gitman, V., Hamkins, J. D., and Johnstone, T. A. (2016). What is the theory ZFC without power set? Mathematical Logic Quarterly, 62(4–5):391–406.CrossRefGoogle Scholar
Gödel, K. (1940). The Consistency of the Continuum Hypothesis. Princeton University Press.CrossRefGoogle Scholar
Gödel, K. (1947). What is Cantor’s continuum problem? In [Gödel, 1990], pages 176–87. Oxford University Press.Google Scholar
Gödel, K. (1964). What is Cantor’s continuum problem? In [Gödel, 1990], pages 254–70. Oxford University Press.Google Scholar
Gödel, K. (1990). Collected Works, Volume II: Publications 1938–1974. Oxford University Press. Edited by: Solomon Feferman (Editor-in-chief), Dawson, John W., Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., Heijenoort, Jean van.Google Scholar
Goldstein, L. (2012). Paradoxical partners: Semantical brides and set-theoretical grooms. Analysis, 73(1):33–7.Google Scholar
Hamkins, J. D. (2012). The set-theoretic multiverse. The Review of Symbolic Logic, 5(3):416–49.CrossRefGoogle Scholar
Hamkins, J. D. and Lewis, A. (2000). Infinite time Turing machines. Journal of Symbolic Logic, 65(2):567–604.CrossRefGoogle Scholar
Hamkins, J. D. and Loewe, B. (2008). The modal logic of forcing. Transactions of the American Mathematical Society, 360(4):1793–817.Google Scholar
Hamkins, J. D. and Montero, B. (2000). With infinite utility, more needn’t be better. Australasian Journal of Philosophy, 78(2):231–40.CrossRefGoogle Scholar
Holmes, M., Forster, T., and Libert, T. (2012). Alternative set theories. In Sets and Extensions in the Twentieth Century, volume 6 of Handbook of the History of Logic, pages 559–632. Elsevier/North-Holland, Amsterdam.CrossRefGoogle Scholar
Incurvati, L. (2017). Maximality principles in set theory. Philosophia Mathematica, 25(2):159–93.Google Scholar
Incurvati, L. (2020). Conceptions of Set and the Foundations of Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Incurvati, L. and Murzi, J. (2017). Maximally consistent sets of instances of naive comprehension. Mind, 126(502):371–84.Google Scholar
Jockwich, S., Tarafder, S., and Venturi, G. (2022). Ideal objects for set theory. Journal of Philosophical Logic, 51(3):583–602.CrossRefGoogle Scholar
Kanamori, A. (2007). Gödel and set theory. The Bulletin of Symbolic Logic, 13(2):153–88.CrossRefGoogle Scholar
Kanamori, A. (2009). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer, 2nd edition.Google Scholar
Koellner, P. (2009). On reflection principles. Annals of Pure and Applied Logic, 157(2–3):206–19.CrossRefGoogle Scholar
Koellner, P. (2014). Large cardinals and determinacy. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, spring 2014 edition.Google Scholar
Krapf, R. (2017). Class forcing and second-order arithmetic. PhD thesis, The University of Bonn. Universitäts- und Landesbibliothek Bonn.Google Scholar
Linnebo, Ø. (2006). Sets, properties, and unrestricted quantification. In Rayo, A., and Uzquiano, G., editors, Absolute Generality, pages 149–78, Oxford University Press.Google Scholar
Linnebo, Ø. (2010). Pluralities and sets. Journal of Philosophy, 107(3):144–64.CrossRefGoogle Scholar
Linnebo, Ø. (2013). The potential hierarchy of sets. The Review of Symbolic Logic, 6(2):205–28.CrossRefGoogle Scholar
Linnebo, Ø. (2014). Plural quantification. In Zalta, E. N., editor, The Stanford Encyclopedia of Philosophy. Fall 2014 edition.Google Scholar
Linnebo, Ø. (2018). Thin Objects: An Abstractionist Account. Oxford University Press.CrossRefGoogle Scholar
Linnebo, Ø. , and Shapiro, S. (2023). Predicativism as a form of potentialism. Review of Symbolic Logic, 16(1):1–32.CrossRefGoogle Scholar
Maddy, P. (1988a). Believing the axioms I. The Journal of Symbolic Logic, 53(2):481–511.CrossRefGoogle Scholar
Maddy, P. (1988b). Believing the axioms II. The Journal of Symbolic Logic, 53(3):736–64.Google Scholar
Maddy, P. (2017). Set-theoretic foundations. In [Caicedo et al., 2017], pages 289–322. American Mathematical Society.Google Scholar
Maddy, P. (2019). What Do We Want a Foundation to Do?, pages 293–311, in Centrone, S., Kant, D., and Sarikaya, D., editors, Reflections on the Foundations of Mathematics. Synthese Library, vol. 407. Springer International Publishing, Cham.CrossRefGoogle Scholar
Maddy, P., and Meadows, T. (2020). A reconstruction of Steel’s multiverse project. The Bulletin of Symbolic Logic, 26(2):118–69.CrossRefGoogle Scholar
Mancosu, P. (2009). Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? Review of Symbolic Logic, 2(4):612–46.CrossRefGoogle Scholar
Meadows, T. (2015). Naive infinitism. Notre Dame Journal of Formal Logic, 56(1):191–212.CrossRefGoogle Scholar
Menzel, C. (1986). On the iterative explanation of the paradoxes. Philosophical Studies, 49(1):37–61.CrossRefGoogle Scholar
Menzel, C. (2014). ZFCU, wide sets, and the iterative conception. Journal of Philosophy, 111(2):57–83.CrossRefGoogle Scholar
Menzel, C. (2021). Modal set theory. In Bueno, O. and Shalkowski, S. A., editors, The Routledge Handbook of Modality, pages 292–307. Routledge.Google Scholar
Moore, J. T. (2010). The proper forcing axiom. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), pages 3–29.Google Scholar
Muller, F. A. (2001). Sets, classes and categories. British Journal for the Philosophy of Science, 52:539–73.CrossRefGoogle Scholar
Parker, M. W. (2019). Gödel’s argument for Cantorian Cardinality. Noûs, 53(2):375–93.CrossRefGoogle Scholar
Parsons, C. (1983). Sets and modality. In Mathematics in Philosophy: Selected Essays, pages 298–342. Cornell University Press.Google Scholar
Posy, C. J. (2020). Mathematical Intuitionism. Elements in the Philosophy of Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Potter, M. (2004). Set Theory and Is Philosophy: A Critical Introduction. Oxford University Press.CrossRefGoogle Scholar
Quine, W. (1937). New foundations for mathematical logic. The American Mathematical Monthly, 44(2):70–80.CrossRefGoogle Scholar
Rayo, A., and Uzquiano, G., editors (2006). Absolute Generality. Clarendon Press.CrossRefGoogle Scholar
Reinhardt, W. (1974). Remarks on reflection principles, large cardinals, and elementary embeddings. Proceedings of Symposia in Pure Mathematics, 13(2):189–205.CrossRefGoogle Scholar
Rittberg, C. J. (2020). Mathematical Practices Can Be Metaphysically Laden, pages 1–26. Springer International Publishing. DOI: https://doi.org/10.1007/978-3-030-19071-2_22-1Google Scholar
Roberts, S. (2016). Reflection and Potentialism. PhD thesis, Birkbeck, University of London.Google Scholar
Roberts, S. (2019). Modal structuralism and reflection. The Review of Symbolic Logic, 12(4):823–60.CrossRefGoogle Scholar
Roberts, S. (2022). Pluralities as nothing over and above. Journal of Philosophy, 119(8):405–24.CrossRefGoogle Scholar
Roberts, S. (MSa). The iterative conception of properties. Unpublished manuscript.Google Scholar
Roberts, S. (MSb). Ultimate V. Manuscript under review. Preprint: https://philarchive.org/rec/ROBUVS.Google Scholar
Scambler, C. (2020). An indeterminate universe of sets. Synthese, 197(2):545–73.CrossRefGoogle Scholar
Scambler, C. (2021). Can all things be counted? Journal of Philosophical Logic, 50: 1079–106.CrossRefGoogle Scholar
Scambler, C. (MS). On the consistency of height and width potentialism. Manuscript under review.Google Scholar
Scott, D. (1977). Foreword to Boolean-Valued Models and Independence Proofs. In [Bell, 2011], pages xiii–xviii. Oxford University Press.Google Scholar
Shapiro, S. and Wright, C. (2006). All things indefinitely extensible. In Rayo, A. and Uzquiano, G., editors, Absolute Generality, pages 255–304. Oxford University Press.CrossRefGoogle Scholar
Simmons, K. (1990). The diagonal argument and the liar. Journal of Philosophical Logic, 19(3):277–303.CrossRefGoogle Scholar
Skolem, T. (1922). Some remarks on axiomitized set theory. In A Source Book in Mathematical Logic, 1879–1931, van Heijenoort 1967, pages 290–301. Harvard University Press.Google Scholar
Solovay, R. M. (1974). On the cardinality of
sets of reals. Journal of Symbolic Logic, 39(2):330–330.Google Scholar
Steel, J. (2014). Gödel’s program. In Kennedy, J., editor, Interpreting Gödel, pp. 153–179. Cambridge University Press.CrossRefGoogle Scholar
Studd, J. P. (2013). The iterative conception of set: A (bi-) modal axiomatisation. Journal of Philosophical Logic, 42(5):697–725.CrossRefGoogle Scholar
Studd, J. P. (2019). Everything, More or Less: A Defence of Generality Relativism. Oxford University Press.CrossRefGoogle Scholar
Taranovsky, D. (2004). Perfect subset property for co-analytic sets in
. Online article: https://web.mit.edu/dmytro/www/other/PerfectSubsetsAndZFC.htm.Google Scholar
Uzquiano, G. (2006). Unrestricted unrestricted quantification: The cardinal problem of absolute generality. In Absolute Generality, pages 305–32. Oxford University Press.Google Scholar
Wilson, M. (2006). Wandering Significance: An Essay on Conceptual Behaviour. Oxford University Press.CrossRefGoogle Scholar
Woodin, W. H. (2017). In search of Ultimate-L: The 19th Midrasha Mathematicae lectures. The Bulletin of Symbolic Logic, 23(1):1–109.CrossRefGoogle Scholar
Zarach, A. M. (1996). Replacement 
collection, volume 6 of Lecture Notes in Logic, pages 307–22. Springer.CrossRefGoogle Scholar
collection, volume 6 of Lecture Notes in Logic, pages 307–22. Springer.CrossRefGoogle Scholar