×

Comparing material and structural set theories. (English) Zbl 1412.18004

ZFC (Zermelo-Fraenkel set theory with the axiom of choice) and BZC (bounded Zermelo set theory) are material set theories in the sense that they are based upon a global membership predicate (the word “material” was suggested in [S. Awodey, Philos. Math., III. Ser. 4, No. 3, 209–237 (1996; Zbl 0874.00012)]), while ETCS (the theory of a well-pointed topos with a natural number object abiding by the axiom of choice) and its ilk are structural set theories which take functions as primitive entities.
It is well known [J. C. Cole, in: The proceedings of the Bertrand Russell memorial logic conference, Uldum, Denmark, August 4–16, 1971. 351–399 (1973; Zbl 1416.03019); W. Mitchell, J. Pure Appl. Algebra 2, 261–274 (1972; Zbl 0245.18001); W. Mitchell, J. Pure Appl. Algebra 3, 193–201 (1973; Zbl 0272.18005)] that ETCS is equiconsistent with BZC.
The principal objective in this paper is to obtain similar results for stronger and weaker theories in a unified manner. By removing such axioms as choice, classical logic and power set from BZC, one gets intuitionistic and predicative set theories corresponding to more general toposes and pretoposes, while the author proposes category-theoretic properties corresponding to such set-theoretic axioms as separation, replacement and collection.
This paper is concerned entirely with (pre)toposes corresponding directly to set theories in the sense that the set-theoretic elements of a set \(X\) are in bijection with its category-theoretic global emements \(1\rightarrow X\), that is to say, the category is well-pointed. By dropping well-pointedness, one can interpret BZC in any Boolean topos with an NNO and choice with recourse to the internal logic.
The paper is on the lines of [F. W. Lawvere, Proc. Natl. Acad. Sci. USA 52, 1506–1511 (1964; Zbl 0141.00603)]; F. W. Lawvere, Repr. Theory Appl. Categ. 2005, No. 11, 1–35 (2005; Zbl 1072.18005); C. McLarty, Philos. Math. (3) 12, No. 1, 37–53 (2004; Zbl 1051.18004); G. Osius, J. Pure Appl. Algebra 4, 79–119 (1974; Zbl 0282.02027)], but generalizes readily to the intuitionistic case. The author is now preparing a paper extending the usual internal logic that admits unbounded quantifiers [“Stack semantics and unbounded quantifiers in topos theory”].

MSC:

18B25 Topoi
03F65 Other constructive mathematics
03G30 Categorical logic, topoi
18B05 Categories of sets, characterizations
03E70 Nonclassical and second-order set theories
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aczel, Peter, Non-well-founded Sets, CSLI Lecture Notes, vol. 14 (1988), Stanford University Center for the Study of Language and Information: Stanford University Center for the Study of Language and Information Stanford, CA, available at · Zbl 0668.04001
[2] Aczel, Peter; Rathjen, Michael, Notes on Constructive Set Theory, Reports on Mathematical Logic, vol. 40 (2000/2001), Institut Mittag-Leffler, available at
[3] Awodey, S., Structure in mathematics and logic: a categorical perspective, Philos. Math. (3), 4, 3, 209-237 (1996) · Zbl 0874.00012
[4] Awodey, Steven, Logic in Topoi: Functorial Semantics for Higher-order Logic (1997), University of Chicago, PhD thesis
[5] Awodey, Steve, An answer to G. Hellman’s question: “Does category theory provide a framework for mathematical structuralism?” [Philos. Math. (3) 11 (2) (2003) 129-157; mr1980559], Philos. Math. (3), 12, 1, 54-64 (2004) · Zbl 1065.18002
[6] Awodey, Steve; Butz, Carsten; Simpson, Alex; Streicher, Thomas, Relating first-order set theories and elementary toposes, Bull. Symbolic Logic, 13, 3, 340-358 (2007) · Zbl 1152.03043
[7] Awodey, Steve; Butz, Carsten; Simpson, Alex; Streicher, Thomas, Relating first-order set theories, toposes and categories of classes, Ann. Pure Appl. Logic, 165, 2, 428-502 (2014) · Zbl 1323.03073
[8] Awodey, S.; Forssell, H., Algebraic models of intuitionistic theories of sets and classes, Theory Appl. Categ., 15, 5, 147-163 (2005/2006), (electronic) · Zbl 1085.18004
[9] Awodey, Steve; Forssell, Henrik; Warren, Michael A., Algebraic models of sets and classes in categories of ideals (2006), available online at
[10] Awodey, Steve; Warren, Michael A., Predicative algebraic set theory, Theory Appl. Categ., 15, 1, 1-39 (2005/2006), (electronic) · Zbl 1072.18004
[11] Blanc, Georges, Équivalence naturelle et formules logiques en théorie des catégories, Arch. Math. Logik Grundlag., 19, 3-4, 131-137 (1978/1979) · Zbl 0407.03035
[12] Bourbaki, Nicolas, Elements of Mathematics. Theory of Sets (1968), Hermann, Publishers in Arts and Science: Hermann, Publishers in Arts and Science Paris, translated from the French · Zbl 1061.03001
[13] Cole, J. C., Categories of sets and models of set theory, (The Proceedings of the Bertrand Russell Memorial Conference. The Proceedings of the Bertrand Russell Memorial Conference, Uldum, 1971 (1973), Bertrand Russell Memorial Logic Conf.: Bertrand Russell Memorial Logic Conf. Leeds), 351-399 · Zbl 1416.03019
[14] Diaconescu, Radu, Axiom of choice and complementation, Proc. Amer. Math. Soc., 51, 176-178 (1975) · Zbl 0317.02077
[15] Freyd, Peter, Properties invariant within equivalence types of categories, (Algebra, Topology, and Category Theory (a Collection of Papers in Honor of Samuel Eilenberg) (1976), Academic Press: Academic Press New York), 55-61 · Zbl 0342.18001
[16] Gitman, Victoria; Hamkins, Joel David; Johnstone, Thomas A., What is the theory ZFC without power set? (2011) · Zbl 1375.03059
[17] Johnstone, Peter T., Topos Theory, London Mathematical Society Monographs, vol. 10 (1977), Academic Press [Harcourt Brace Jovanovich Publishers]: Academic Press [Harcourt Brace Jovanovich Publishers] London · Zbl 0368.18001
[18] Johnstone, Peter T., Partial products, bagdomains and hyperlocal toposes, (Applications of Categories in Computer Science. Applications of Categories in Computer Science, Durham, 1991. Applications of Categories in Computer Science. Applications of Categories in Computer Science, Durham, 1991, London Math. Soc. Lecture Note Ser., vol. 177 (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 315-339 · Zbl 0785.18002
[19] Johnstone, Peter T., Sketches of an Elephant: A Topos Theory Compendium, vols. 1 and 2, Oxford Logic Guides, vol. 43 (2002), Oxford Science Publications · Zbl 1071.18001
[20] Johnstone, Peter T.; Wraith, Gavin C., Algebraic theories in toposes, (Indexed Categories and Their Applications. Indexed Categories and Their Applications, Lecture Notes in Math., vol. 661 (1978), Springer: Springer Berlin), 141-242 · Zbl 0392.18006
[21] Joyal, A.; Moerdijk, I., Algebraic Set Theory, London Mathematical Society Lecture Note Series, vol. 220 (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0847.03025
[22] Lawvere, F. William, An elementary theory of the category of sets, Proc. Natl. Acad. Sci. USA. Proc. Natl. Acad. Sci. USA, Repr. Theory Appl. Categ., 11, 1506-1511 (2005), with comments by the author and Colin McLarty · Zbl 0141.00603
[23] Lawvere, William; Rosebrugh, Robert, Sets for Mathematics (2003), Cambridge University Press · Zbl 1031.18001
[24] Leinster, Tom, Rethinking set theory, Amer. Math. Monthly, 121, 5, 403-415 (2014) · Zbl 1345.03100
[25] Mac Lane, Saunders, Mathematics, Form and Function (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0675.00001
[26] Mac Lane, Saunders; Moerdijk, Ieke, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext (1994), Springer-Verlag: Springer-Verlag New York, corrected reprint of the 1992 edition · Zbl 0822.18001
[27] Makkai, Michael, First order logic with dependent sorts, with applications to category theory (1995), available at
[28] Makkai, Michael, On comparing definitions of weak \(n\)-category (August 2001), available at
[29] Mathias, A. R.D., The strength of Mac Lane set theory, Ann. Pure Appl. Logic, 110, 1-3, 107-234 (2001) · Zbl 1002.03045
[30] McLarty, Colin, Numbers can be just what they have to, Noûs, 27, 4, 487-498 (1993) · Zbl 1366.03098
[31] McLarty, Colin, Exploring categorical structuralism, Philos. Math. (3), 12, 1, 37-53 (2004) · Zbl 1051.18004
[32] Mitchell, William, Boolean topoi and the theory of sets, J. Pure Appl. Algebra, 2, 261-274 (1972) · Zbl 0245.18001
[33] Mitchell, William, Categories of Boolean topoi, J. Pure Appl. Algebra, 3, 193-201 (1973) · Zbl 0272.18005
[34] Osius, Gerhard, Categorical set theory: a characterization of the category of sets, J. Pure Appl. Algebra, 4, 79-119 (1974) · Zbl 0282.02027
[35] Palmgren, Erik, Constructivist and structuralist foundations: Bishop’s and Lawvere’s theories of sets, Ann. Pure Appl. Logic, 163, 1384-1399 (2012) · Zbl 1257.03095
[36] Schreiber, Urs; Shulman, Michael; Trimble, Todd, Local geometric morphism (2017)
[37] Michael Shulman, Stack semantics and unbounded quantifiers in topos theory, 2018, in preparation.; Michael Shulman, Stack semantics and unbounded quantifiers in topos theory, 2018, in preparation.
[38] Shulman, Michael, SEAR (2009)
[39] van den Berg, Benno; De Marchi, Federico, Models of non-well-founded sets via an indexed final coalgebra theorem, J. Symbolic Logic, 72, 3, 767-791 (2007) · Zbl 1124.03049
[40] van den Berg, Benno; Moerdijk, Ieke, Aspects of predicative algebraic set theory I: exact completion (2007) · Zbl 1165.03045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.