Abstract
Set theory has long been viewed as a foundation of mathematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising mathematics in set theory. We wish to have set theory’s features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Our belief is based on tracing the expansion of uses of transrec3 in Isabelle/ZF.
References
Aczel, P.: Generalised set theory. In: Logic, Language and Computation, vol. 1 of CSLI Lecture Notes (1996)
Bancerek, G., et al.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS (LNAI), vol. 9150, pp. 261–279. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20615-8_17
Brown, C.E., Pak, K.: A tale of two set theories. In: Kaliszyk, C., Brady, E., Kohlhase, A., Sacerdoti Coen, C. (eds.) CICM 2019. LNCS (LNAI), vol. 11617, pp. 44–60. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23250-4_4
Brown, C.E., Smolka, G.: Extended first-order logic. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 164–179. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03359-9_13
Dunne, C., Wells, J.B., Kamareddine, F.: Adding an abstraction barrier to ZF set theory. In: Benzmüller, C., Miller, B. (eds.) CICM 2020. LNCS (LNAI), vol. 12236, pp. 89–104. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53518-6_6
Farmer, W.M.: Formalizing undefinedness arising in calculus. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 475–489. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-25984-8_35
Farmer, W.M., Guttman, J.D., Javier Thayer, F.: Little theories. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 567–581. Springer, Heidelberg (1992). https://doi.org/10.1007/3-540-55602-8_192
Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03545-1_9
Kolodynski, S.: IsarMathLib (2021). https://isarmathlib.org/. Accessed 3 Mar 2021
Krauss, A.: https://www21.in.tum.de/~krauss/publication/2010-soft-types-note/. Adding soft types to Isabelle (2010)
Krauss, A., Chen, J., Kappelmann, K.: Isabelle/Set. https://bitbucket.org/cezaryka/tyset/src/master/
Kunčar, O., Popescu, A.: From types to sets by local type definitions in higher-order logic. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 200–218. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43144-4_13
Kunčar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. J. Autom. Reasoning 62(4), 531–555 (2019)
Maddy, P.: What do we want a foundation to do? In: Centrone, S., Kant, D., Sarikaya, D. (eds.) Reflections on the Foundations of Mathematics. SL, vol. 407, pp. 293–311. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-15655-8_13
Megill, N., Wheeler, D.A.: Metamath: A Computer Language for Mathematical Proofs. LULU Press, Morrisville (2019)
Obua, S.: Partizan games in Isabelle/HOLZF. In: Barkaoui, K., Cavalcanti, A., Cerone, A. (eds.): ICTAC 2006. LNCS, vol. 4281. Springer, Heidelberg (2006). https://doi.org/10.1007/11921240
Paulson, L.C.: The foundation of a generic theorem prover. J. Autom. Reasoning 5(3), 363–397 (1989)
Paulson, L.C.: Set theory for verification: I. From foundations to functions. J. Autom. Reasoning 11(3), 353–389 (1993)
Wiedijk, F., Zwanenburg, J.: First order logic with domain conditions. In: Basin, D., Wolff, B. (eds.) TPHOLs 2003. LNCS, vol. 2758, pp. 221–237. Springer, Heidelberg (2003). https://doi.org/10.1007/10930755_15
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Dunne, C., Wells, J.B., Kamareddine, F. (2021). Generating Custom Set Theories with Non-set Structured Objects. In: Kamareddine, F., Sacerdoti Coen, C. (eds) Intelligent Computer Mathematics. CICM 2021. Lecture Notes in Computer Science(), vol 12833. Springer, Cham. https://doi.org/10.1007/978-3-030-81097-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-030-81097-9_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-81096-2
Online ISBN: 978-3-030-81097-9
eBook Packages: Computer ScienceComputer Science (R0)