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NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

Published online by Cambridge University Press:  18 November 2019

WILFRIED SIEG
Affiliation:
DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITYPITTSBURGH, PA15213, USAE-mail: [email protected]
PATRICK WALSH
Affiliation:
DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITY QATARDOHA, QATARE-mail: [email protected]

Abstract

Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.

To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.

We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.

Hilbert 1918

Type
Research Article
Copyright
© Association for Symbolic Logic, 2019

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References

BIBLIOGRAPHY

Alcock, L., Hodds, M., Roy, S., & Inglis, M. (2015). Investigating and improving undergraduate proof comprehension. Notice of the American Mathematical Society, 62(7), 742753.CrossRefGoogle Scholar
Banach, S. (1924). Un théorème sur les transformations biunivoques. Fundamenta Mathematicae, 1(6), 236239.CrossRefGoogle Scholar
Barendregt, H. & Wiedijk, F. (2005). The challenge of computer mathematics. Philosophical Transactions of the Royal Society A, 363(1835), 23512375.CrossRefGoogle ScholarPubMed
Blaine, L. H. (1981). Programs for structured proofs. In (Suppes, 1981, pp. 81120).Google Scholar
Borel, É. (1921). Leçons sur la théorie des fonctions. Paris: Gauthier-Villars. First edition was published in 1898.Google Scholar
Bourbaki, N. (2004). Theory of Sets. Berlin: Springer.CrossRefGoogle Scholar
Bundy, A. (1991). A science of reasoning. In Lassez, J. and Plotkin, G., editors. Computational Logic: Essays in Honor of Alan Robinson. Cambridge, MA: MIT Press, pp. 178198.Google Scholar
Cantor, G. (1932). Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts. Berlin: Springer.Google Scholar
de Bruijn, N. G. (1970). The mathematical language AUTOMATH, its usage, and some of its extensions. In Laudet, M., Lacombe, D., Nolin, L., and Schützenberger, M., editors. Symposium on Automatic Demonstration. Lecture Notes in Mathematics. Berlin: Springer, pp. 2961.CrossRefGoogle Scholar
de Bruijn, N. G. (1973). AUTOMATH: A Language for Mathematics. Les Presses de l’Université de Montréal.Google Scholar
Dedekind, R. (1888). Was sind und was sollen die Zahlen? Braunschweig: Vieweg. Translated in (Ewald, 1996, pp. 787833).Google Scholar
Dedekind, R. (1932). Gesammelte Mathematische Werke, Vol. 3. Braunschweig: Vieweg.Google Scholar
Deiser, O. (2010). Introductory note to 1901. In (Zermelo, 2010, pp. 5270).Google Scholar
Diaz-Lopez, A. (2016). Interview with Sir Timothy Gowers. Notices of the American Mathematical Society, 63(9), 10261028.CrossRefGoogle Scholar
Ewald, W. B. (editor) (1996). From Kant to Hilbert: Readings in the Foundations of Mathematics. Oxford: Oxford University Press.Google Scholar
Ferreirós, J. (1993). On the relations between Georg Cantor and Richard Dedekind. Historia Mathematica, 20(4), 343363.CrossRefGoogle Scholar
Frege, G. (1893). Grundgesetze der Arithmetik. Jena: Pohle Verlag.Google Scholar
Ganesalingam, M. & Gowers, W. T. (2013). A fully automatic problem solver with human-style output. arXiv:1309.4501.Google Scholar
Ganesalingam, M. & Gowers, W. T. (2017). A fully automatic theorem prover with human-style output. Journal of Automated Reasoning, 58(2), 253291.CrossRefGoogle ScholarPubMed
Gentzen, G. (1936). Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112(1), 493565.CrossRefGoogle Scholar
Gibson, T. (2006). Proof Search in First-Order Logic with Equality. Master’s Thesis, Carnegie Mellon University.Google Scholar
Gowers, W. T. (2007). Mathematics, memory, and mental arithmetic. In Leng, M., Paseau, A., and Potter, M., editors. Mathematical Knowledge. Oxford: Oxford University Press, pp. 3358.Google Scholar
Grattan-Guinness, I. (2000). The Search for Mathematical Roots, 1870–1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel. Princeton: Princeton University Press.Google Scholar
Hallett, M. (1988). Cantorian Set Theory and Limitation of Size. Oxford: Oxford University Press.Google Scholar
Hamami, Y. (2018). Mathematical inference and logical inference. The Review of Symbolic Logic, 11(4), 665704.CrossRefGoogle Scholar
Hamami, Y. (2019). Mathematical rigor and proof. Manuscript.CrossRefGoogle Scholar
Harrison, J. (2008). Formal proof–theory and practice. Notices of the American Mathematical Society, 55(11), 13951406.Google Scholar
Harrison, J. (2009). Handbook of Practical Logic and Automated Reasoning. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Harrison, J., Urban, J., & Wiedijk, F. (2014). History of interactive theorem proving. In Gabbay, D., Siekmann, J., and Woods, J., editors. Computational Logic: Handbook of the History of Logic, Vol. 9. Amsterdam: North-Holland, pp. 135214.CrossRefGoogle Scholar
Hewitt, E. & Stromberg, K. (1969). Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable. Berlin: Springer-Verlag.Google Scholar
Hilbert, D. (1918). Axiomatisches Denken. Mathematische Annalen, 78, 405415.CrossRefGoogle Scholar
Hilbert, D. (1927). Die Grundlagen der Mathematik. Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, (6), 6585.Google Scholar
Hilbert, D. (2013). David Hilbert’s Lectures on the Foundations of Arithmetic and Logic: 1917–1933, Vol. 3, edited by Ewald, W. B. and Sieg, W.. Berlin: Springer-Verlag.Google Scholar
Hinkis, A. (2013). Proofs of the Cantor-Bernstein Theorem: A Mathematical Excursion. Science Networks, Historical Studies, Vol. 45. Basel: Birkhäuser Verlag.CrossRefGoogle Scholar
Jaśkowski, S. (1934). On the rules of suppositions in formal logic. Studia Logica, 1, 432.Google Scholar
Jutting, L.S. (1973). The development of a text in AUT-QE. In Braffort, P., editor. APLASM’73, Symposium d’Orsay sur la Manipulation des Symboles et l’Utilisation d’APL, Volume 1. Paris: Université Paris Sud, Chapter 4.Google Scholar
Kanamori, A. (2004). Zermelo and set theory. Bulletin of Symbolic Logic, 10(4), 487553.CrossRefGoogle Scholar
Kash, I. (2004). A Partially Automated Proof of the Cantor-Bernstein Theorem. Senior Thesis, Carnegie Mellon University.Google Scholar
Knaster, B. & Tarski, A. (1928). Un théorème sur les fonctions d’ensembles. Annales de la Société Polonaise des Mathématiques, 6, 133134.Google Scholar
König, J. (1906). Sur la théorie des ensembles. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, 143, 110112.Google Scholar
Korselt, A. (1911). Über einen Beweis des Äquivalenzsatzes. Mathematische Annalen, 70(2), 294296.CrossRefGoogle Scholar
Lamport, L. (1995). How to write a proof. The American Mathematical Monthly, 102(7), 600608.CrossRefGoogle Scholar
Lamport, L. (2012). How to write a 21 st century proof. Journal of Fixed Point Theory and Applications, 11(1), 4363.CrossRefGoogle Scholar
Livnat, E. (2011). The Cantor-Bernstein Theorem in AProS. Master’s Thesis, Carnegie Mellon University.Google Scholar
Mac Lane, S. (1934). Abgekürzte Beweise im Logikkalkül. Ph.D. Thesis, Göttingen.Google Scholar
Mac Lane, S. (1935). A logical analysis of mathematical structure. The Monist, 45(1), 118130.CrossRefGoogle Scholar
Mac Lane, S. (1979). A late return to a thesis in logic. In Kaplansky, I., editor. Saunders MacLane — Selected Papers. New York: Springer, pp. 6366.CrossRefGoogle Scholar
Mac Lane, S. (2005). Saunders Mac Lane: A Mathematical Autobiography. Wellesley, MA: AK Peters/CRC Press.CrossRefGoogle Scholar
McDonald, J. & Suppes, P. (1984). Student use of an interactive theorem prover. Contemporary Mathematics, 29, 315360.CrossRefGoogle Scholar
Nederpelt, R. (1977). Presentation of natural deduction. Symposium: Set Theory, Foundations of Mathematics. Nouvelle Série, Vol. 2(10), Recueil des travaux de l’Institut Mathématique, pp. 115126.Google Scholar
Omodeo, E. G. & Schwartz, J. T. (2002). A ‘Theory’ mechanism for a proof-verifier based on first-order set theory. In Kakas, A. and Sadri, F., editors. Computational Logic: Logic Programming and Beyond - Essays in Honour of Bob Kowalski, Part II. Heidelberg: Springer, pp. 214230.CrossRefGoogle Scholar
Pastre, D. (2002). Strong and weak points of the MUSCADET theorem prover–examples from CASC-JC. AI Communications, 15(2, 3), 147160.Google Scholar
Paulson, L. C. (1993). Set theory for verification I: From foundations to functions. Journal of Automated Reasoning, 11(3), 353389.CrossRefGoogle Scholar
Paulson, L. C. (1994). A fixedpoint approach to implementing (co)inductive definitions. In Bundy, A., editor. Automated Deduction CADE-12. Lecture Notes in Computer Science, Vol. 814. Berlin: Springer, pp. 148181.CrossRefGoogle Scholar
Paulson, L. C. (1995). Set theory for verification II: Induction and recursion. Journal of Automated Reasoning, 15(2), 167215.CrossRefGoogle Scholar
Paulson, L. C. (2014). A machine-assisted proof of Gödel’s incompleteness theorems for the theory of hereditarily finite sets. The Review of Symbolic Logic, 7(3), 484498.CrossRefGoogle Scholar
Peano, G. (1906a). Super theorema de Cantor-Bernstein. Rendiconti del Circolo Matematico di Palermo (1884–1940), 21, 360366.CrossRefGoogle Scholar
Peano, G. (1906b). Super theorema de Cantor-Bernstein. Revisita di Matematica, 8, 136143.Google Scholar
Poincaré, H. (1906). Les mathématiques et la logique. Revue de métaphysique et de morale, 14(3), 294317.Google Scholar
Prawitz, D. (1965). Natural Deduction: A Proof-Theoretical Study. Stockholm: Almqvist & Wiksell.Google Scholar
Schröder, E. (1898). Über zwei Definitionen der Endlichkeit und G. Cantorsche Sätze. Nova Acta Academiae Caesareae Leopoldino-Carolinae, 71, 303362.Google Scholar
Schwartz, J. T., Cantone, D., & Omodeo, E. G. (2011). Computational Logic and Set Theory. London: Springer.CrossRefGoogle Scholar
Sieg, W. (1992). Mechanisms and Search – Aspects of Proof Theory, Vol. 14. Associazione Italiana di Logica e sue Applicazioni.Google Scholar
Sieg, W. (1997). Aspects of mathematical experience. Reprinted in (Sieg, 2013, pp. 329343).CrossRefGoogle Scholar
Sieg, W. (2010). Searching for proofs (and uncovering capacities of the mathematical mind). Reprinted in (Sieg, 2013, pp. 377401).Google Scholar
Sieg, W. (2013). Hilbert’s Programs and Beyond. Oxford: Oxford University Press.Google Scholar
Sieg, W. (2019a). The Cantor–Bernstein theorem: How many proofs? Philosophical Transactions of the Royal Society A, 377(2140), 20180031.CrossRefGoogle Scholar
Sieg, W. (2019b). Methodological frames: Paul Bernays, mathematical structuralism, and proof theory. In Reck, E. & Schiemer, G., editors. The Pre-history of Mathematical Structuralism, to appear. Oxford: Oxford University Press.Google Scholar
Sieg, W., & Byrnes, J. (1998). Normal natural deduction proofs (in classical logic). Studia Logica, 60(1), 67106.CrossRefGoogle Scholar
Sieg, W. & Cittadini, S. (2005). Normal natural deduction proofs (in non-classical logics). In Hutter, D. & Stephan, W., editors. Mechanizing Mathematical Reasoning. Lecture Notes in Computer Science, Vol. 2605. Heidelberg: Springer, pp. 169191.CrossRefGoogle Scholar
Sieg, W. & Derakshan, F. (2020). Human-oriented automated proof search. Manuscript.Google Scholar
Sieg, W. & Field, C. (2005). Automated search for Gödel’s proofs. Annals of Pure and Applied Logic, 133, 319338.CrossRefGoogle Scholar
Sieg, W. & Schlimm, D. (2014). Dedekind’s abstract concepts: Models and mappings. Philosophia Mathematica, 25(3), 292317.Google Scholar
Stegmüller, W. (1979). The Structuralist View of Theories: A Possible Analogue of the Bourbaki Programme in Physical Science. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Suppes, P. (editor) (1981). University-level Computer-assisted Instruction at Stanford: 1968–1980. Stanford, CA: Institute for Mathematical Studies in the Social Sciences.Google Scholar
Suppes, P. (2002). Representation and Invariance of Scientific Structures. Stanford, CA: CSLI Publications.Google Scholar
Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5(2), 285309.CrossRefGoogle Scholar
Thiele, R. (2003). Hilbert’s twenty-fourth problem. The American Mathematical Monthly, 110(1), 124.CrossRefGoogle Scholar
Thiele, R. (2005). Hilbert and his twenty-four problems. In van Brummeln, G. and Kinyon, M., editors. Mathematics and the Historians’ Craft. Springer, pp. 243295.CrossRefGoogle Scholar
Troelstra, A. S. & Schwichtenberg, H. (2000). Basic Proof Theory (second edition). Cambridge University Press.CrossRefGoogle Scholar
Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230265.Google Scholar
Turing, A. M. (1948a). Intelligent machinery. In Ince, D., editor. Collected Works of A.M. Turing, Mechanical Intelligence. Amsterdam: North Holland, pp. 107127. Originally written as a report for the national physical labratory, 1992.Google Scholar
Turing, A. M. (1948b). Practical forms of type theory. The Journal of Symbolic Logic, 13(2), 8094.CrossRefGoogle Scholar
van Heijenoort, J. (editor) (1967). From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge: Harvard University Press.Google Scholar
Whitehead, A. N. (1902). On cardinal numbers. American Journal of Mathematics, 24(4), 367394.CrossRefGoogle Scholar
Whitehead, A. N., & Russell, B. (1912). Principia Mathematica, Vol. 2. Cambridge: Cambridge University Press.Google Scholar
Wiedijk, F. (2004). Formal proof sketches. In Berardi, S., Coppo, M., and Damiani, F., editors. Types for Proofs and Programs; International Workshop, Types 2003, Turin. Lecture Notes in Computer Science, Vol. 3085. Berlin: Springer, pp. 378393.Google Scholar
Wiedijk, F. (2008). Formal proof: Getting started. Notices of the American Mathematical Society, 55(11), 14081414.Google Scholar
Windsteiger, W. (2006). An automated prover for Zermelo-Fraenkel set theory in Theorema. Journal of Symbolic Computation, 41(3–4), 435470.CrossRefGoogle Scholar
Wu, S. (2017). Logic Translation Algorithm. Senior Thesis, Carnegie Mellon University.Google Scholar
Zermelo, E. (1901). Über die Addition transfiniter Kardinalzahlen. Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse aus dem Jahre 1901, 1901, 3438. Reprinted in (Zermelo, 2010).Google Scholar
Zermelo, E. (1908). Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen, 65(2), 261281. Translated in (van Heijenoort, 1967, pp. 199–215).CrossRefGoogle Scholar
Zermelo, E. (1930). Über Grenzzahlen und Mengenbereiche. Fundamenta Mathematicae, 16, 2947. Translated in (Ewald, 1996, pp. 1219–1233).CrossRefGoogle Scholar
Zermelo, E. (2010). Collected Works, Gesammelte Werke, 1: Set theory, Miscellanea, Mengenlehre, Varia, edited by Ebbinghaus, H.-D., Kanamori, A., and Fraser, C. G.. Berlin: Springer Verlag.Google Scholar
Zipperer, A. (2016). A Formalization of Elementary Group Theory in the Proof Assistant Lean. Master’s Thesis, Carnegie Mellon University.Google Scholar