2000 character limit reached
The proof-theoretic strength of Constructive Second-order set theories
Published 20 Dec 2023 in math.LO | (2312.12854v2)
Abstract: In this paper, we define constructive analogues of second-order set theories, which we will call $\mathsf{IGB}$, $\mathsf{CGB}$, $\mathsf{IKM}$, and $\mathsf{CKM}$. Each of them can be viewed as $\mathsf{IZF}$- and $\mathsf{CZF}$-analogues of G\"odel-Bernays set theory $\mathsf{GB}$ and Kelley-Morse set theory $\mathsf{KM}$. We also provide their proof-theoretic strengths in terms of classical theories, and we especially prove that $\mathsf{CKM}$ and full Second-Order Arithmetic have the same proof-theoretic strength.
- Peter Aczel “The type theoretic interpretation of constructive set theory” In Logic Colloquium ’77 (Proc. Conf., Wrocław, 1977) 96, Studies in Logic and the Foundations of Mathematics North-Holland, Amsterdam-New York, 1978, pp. 55–66
- Peter Aczel “The type theoretic interpretation of constructive set theory: choice principles” In The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981) 110, Studies in Logic and the Foundations of Mathematics North-Holland, Amsterdam, 1982, pp. 1–40
- Peter Aczel “The type theoretic interpretation of constructive set theory: inductive definitions” In Logic, methodology and philosophy of science, VII (Salzburg, 1983) 114, Studies in Logic and the Foundations of Mathematics North-Holland, Amsterdam, 1986, pp. 17–49
- “Notes on Constructive Set Theory”, 2001
- “CST Book draft”, 2010 URL: https://www1.maths.leeds.ac.uk/~rathjen/book.pdf
- Michael J. Beeson “Foundations of constructive mathematics” Springer-Verlag, Berlin, 1985
- “Constructive analysis” 279, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 1985, pp. xii+477
- Laura Crosilla “Set Theory: Constructive and Intuitionistic ZF” In The Stanford Encyclopedia of Philosophy Metaphysics Research Lab, Stanford University, https://plato.stanford.edu/archives/sum2020/entries/set-theory-constructive/, 2020
- Sy-David Friedman, Victoria Gitman and Vladimir Kanovei “A model of second-order arithmetic satisfying AC but not DC” In Journal of Mathematical Logic 19.01 World Scientific, 2019, pp. 1850013
- Harvey Friedman “Some applications of Kleene’s methods for intuitionistic systems” In Cambridge Summer School in Mathematical Logic (Cambridge, 1971) Vol. 337, Lecture Notes in Math. Springer, Berlin-New York, 1973, pp. 113–170
- Harvey Friedman “The consistency of classical set theory relative to a set theory with intuitionistic logic” In The Journal of Symbolic Logic 38.2 Cambridge University Press, 1973, pp. 315–319
- Harvey M. Friedman and Andrej Ščedrov “The lack of definable witnesses and provably recursive functions in intuitionistic set theories” In Adv. in Math. 57.1, 1985, pp. 1–13
- Nicola Gambino “Types and sets: a study on the jump to full impredicativity”, 1999
- “Very large set axioms over constructive set theories”, 2021 arXiv:2204.05831
- Gerald Leversha “Formal Systems for Constructive Mathematics”, 1976
- Robert S. Lubarsky “CZF and Second Order Arithmetic” In Annals of Pure and Applied Logic 141, 2006, pp. 29–34
- John Myhill “Some properties of intuitionistic Zermelo-Frankel set theory” In Cambridge Summer School in Mathematical Logic (Cambridge, 1971) Vol. 337, Lecture Notes in Math. Springer, Berlin-New York, 1973, pp. 206–231
- John Myhill “Constructive set theory” In The Journal of Symbolic Logic 40.3, 1975, pp. 347–382
- Iosif Petrakis “Families of Sets in Bishop Set Theory”, 2020
- Michael Rathjen “The strength of some Martin-Löf type theories”, Preprint, Department of Mathematics, Ohio State University, 1993 URL: http://www1.maths.leeds.ac.uk/~rathjen/typeOHIO.pdf
- Michael Rathjen “Realizability for constructive Zermelo-Fraenkel set theory” In Logic Colloquium ’03 24, Lecture Notes in Logic La Jolla, CA: Association for Symbolic Logic, 2006, pp. 282–314
- Michael Rathjen “Constructive Zermelo-Fraenkel set theory, power set, and the calculus of constructions” In Epistemology versus Ontology Springer, 2012, pp. 313–349
- Stephen G. Simpson “Subsystems of second order arithmetic” Cambridge University Press, 2009
- Andrew W. Swan “CZF does not have the existence property” In Annals of Pure and Applied Logic 165.5 Elsevier, 2014, pp. 1115–1147
- Andrew Wakelin Swan “Automorphisms of Partial Combinatory Algebras and Realizability Models of Constructive Set Theory”, 2012
- Gaisi Takeuti “Proof theory” 81, Studies in Logic and the Foundations of Mathematics North-Holland Publishing Co., Amsterdam, 1987
- “Constructivism in mathematics. Vol. II” North-Holland Publishing Co., Amsterdam, 1988
- Dominik Wehr “Aczel’s Type-Theoretic Interpretation of Constructive Zermelo-Fraenkel Set Theory”, 2021 URL: https://eprints.illc.uva.nl/id/eprint/1769/1/wehr.pdf
- Kameryn J. Williams “The structure of models of second-order set theories”, 2018
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.