Abstract
The non-well-founded set theories described by Aczel (1988) have received attention from category theorists and computer scientists, but have been largely ignored by philosophers. At the root of this neglect might lie the impression that these theories do not embody a conception of set, but are rather of mere technical interest. This paper attempts to dispel this impression. I present a conception of set which may be taken as lying behind a non-well-founded set theory. I argue that the axiom AFA is justified on the conception, which provides, contra Rieger (Mind 109:241–253, 2000), a rationale for restricting attention to the system based on this axiom. By making use of formal and informal considerations, I then make a case that most of the other axioms of this system are also justified on the conception. I conclude by commenting on the significance of the conception for the debate about the justification of the Axiom of Foundation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aczel, P. (1988). Non-well-founded sets. CSLI, Stanford.
Aczel, P., & Mendler, N. (1989). A final coalgebra theorem. In D. Pitt, D. Rydeheard, P. Dybjer, A. Pitts, A. Poigné (Eds.), Category theory and computer science. Lecture notes in computer science (Vol. 389, pp. 357–365). Berlin: Springer.
Baltag, A. (1999). STS: A structural theory of sets. Logic Journal of the IGPL, 7, 481–515.
Barr, M. (1993). Terminal coalgebras in well-founded set theory. Theoretical Computer Science, 114, 299–315.
Barwise, J. (1986). Situations, sets, and the axiom of foundation. In J. Paris, A. Wilkie, G. Wilmers (Eds.), Logic colloquium ’84 (pp. 21–36). North-Holland, New York.
Barwise, J., & Etchemendy, J. (1987). The liar: An essay on truth and circularity. Oxford: Oxford University Press.
Barwise, J., & Moss, L. (1991). Hypersets. Mathematical Intelligencer, 13, 31–41.
Barwise, J., & Moss, L. (1996). Vicious circles. CSLI, Stanford.
Boffa, M. (1969). Sur la théorie des ensembles sans axiome de Fondement. Bulletin de la Société Mathématique de Belgique, 31, 16–56.
Boolos, G. (1971). The iterative conception of set. Journal of Philosophy, 68, 215–231. Reprinted in Boolos, G. (1998). Logic, logic, and logic (pp. 13–29). Cambridge, Massachusetts: Harvard University Press.
Boolos, G. (1989). Iteration again. Philosophical Topics, 17, 5–21. Reprinted in Boolos, G. (1998). Logic, logic, and LOGIC (pp. 88–104). Cambridge, Massachusetts: Harvard University Press.
Boolos, G. (1998). Logic, logic, and logic. Cambridge, Massachusetts: Harvard University Press.
Booth, D., & Ziegler, R. (1996). Finsler set theory: Platonism and circularity. Birkhäuser Verlag, Basel. Translation of Paul Finsler’s papers with introductory comments.
Devlin, K. (1993). The joy of sets. Fundamentals of contemporary set theory (2nd edn). New York: Springer.
Finsler, P. (1926). Über die Grundlagen der Mengenlehre, I. Mathematische Zeitschrift, 25, 683–713. Reprinted and translated in Booth, D., & Ziegler, R. (1996). Finsler set theory: Platonism and circularity (pp. 103–132). Birkhäuser Verlag, Basel. Translation of Paul Finsler’s papers with introductory comments.
Forster, T. (1995). Set theory with a universal set (2nd edn). Oxford: University Press.
Forti, M., & Honsell, F. (1983). Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 10, 493–522.
Gödel, K. (1944). Russell’s mathematical logic. In P.A. Schilpp (Ed.), The philosophy of bertrand Russell (pp. 123–153). Evanston and Chicago: Northwestern University. Reprinted in Gödel, K. (1990). Collected Works II (pp. 119–141). Oxford: University Press.
Gödel, K. (1990). Collected works II. Oxford: University Press.
Incurvati, L. (2012). How to be a minimalist about sets. Philosophical Studies, 159, 69–87.
Johnstone, P., Power, J., Tsujishita, T., Watanabe, H., Worrell, J. (2001). On the structure of categories of coalgebras. Theoretical Computer Science, 260, 87–117.
Moschovakis, Y.N. (2006). Notes on set theory (2nd edn). New York: Springer.
Moss, L. (2009). Non-wellfounded set theory. In E.N. Zalta (Ed.), Stanford encyclopedia of philosophy (Fall 2009 edition). Available at http://plato.stanford.edu/archives/fall2009/entries/nonwellfounded-set-theory/.
Paseau, A. (2007). Boolos on the justification of set theory. Philosophia Mathematica, 15, 30–53.
Potter, M. (2004). Set theory and its philosophy. Oxford: University Press.
Quine, W.V.O. (1937). New foundations for mathematical logic. American Mathematical Monthly, 44, 70–80.
Ramsey, F.P. (1925). The foundations of mathematics. Proceedings of the London Mathematical Society, 25, 338–384. Reprinted in Ramsey, F.P. (1990). Philosophical Papers (pp. 164–224). Cambridge: Cambridge University Press. Edited by D.H. Mellor.
Ramsey, F.P. (1990). Philosophical papers. Cambridge: Cambridge University Press. Edited by D.H. Mellor.
Rieger, A. (2000). An argument for Finsler-Aczel set theory. Mind, 109, 241–253.
Rutten, J. (2000). Universal coalgebra: A theory of systems. Theoretical Computer Science, 249, 3–80.
Scott, D. (1960). A different kind of model for set theory. Stanford Congress of Logic, Methodology and Philosophy of Science (unpublished paper).
Turi, D., & Rutten, J. (1998). On the foundations of final coalgebra semantics: Non-well-founded sets, partial orders, metric spaces. Mathematical Structures in Computer Science, 8, 481–540.
Tutte, W.T. (2001). Graph theory (paperback edn). Cambridge: Cambridge University Press.
van den Berg, B., & De Marchi, F. (2007). Non-well-founded trees in categories. Annals of Pure and Applied Logic, 146, 40–59.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Incurvati, L. The Graph Conception of Set. J Philos Logic 43, 181–208 (2014). https://doi.org/10.1007/s10992-012-9259-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-012-9259-x