Abstract
Krull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations of the Axiom of Choice (AC) in ring theory. For many a consequence of AC, constructive counterparts are well within reach, provided attention is turned to the syntactical underpinning of the problem at hand. This is one of the viewpoints of the revised Hilbert Programme in commutative algebra, which will here be carried out for MIT and several related classical principles.
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References
Aczel, P. (1978). The type theoretic interpretation of constructive set theory. In Logic Colloquium ’77 (Proceedings of the Conference on Wrocław, 1977), volume 96 of Studies in logic and the foundations of mathematics (pp. 55–66). Amsterdam: North-Holland.
Aczel, P. (1982). The type theoretic interpretation of constructive set theory: Choice principles. In The L. E. J. Brouwer centenary symposium (Noordwijkerhout, 1981) of studies in logic and the foundations of mathematics (Vol. 110, pp. 1–40). Amsterdam: North-Holland.
Aczel, P. (1986). The type theoretic interpretation of constructive set theory: Inductive definitions. In Logic, methodology and philosophy of science,vii (Salzburg, 1983), volume 114 of Studies in logic and the foundations of mathematics (pp. 17–49). Amsterdam: North-Holland.
Aczel, P. (2006). Aspects of general topology in constructive set theory. Annals of Pure and Applied Logic, 137(1–3), 3–29.
Aczel, P., & Rathjen, M. (2000). Notes on constructive set theory (p. 40). Technical report, Institut Mittag-Leffler.
Aczel, P., & M. Rathjen. (2010). Constructive set theory. Book draft. https://www1.maths.leeds.ac.uk/~rathjen/book.pdf.
Arapović, M. (1983). On the embedding of a commutative ring into a \(0\)-dimensional commutative ring. Glasnik Matematički Ser. iii, 18(1), 53–59.
Atiyah, M. F., & Macdonald, I. G. (1969). Introduction to commutative algebra. Reading, MA: Addison-Wesley.
Banaschewski, B. (1983). The power of the ultrafilter theorem. Journal of the London Mathematical Society, 27(2), 193–202.
Banaschewski, B. (1994). A new proof that Krull implies Zorn. Mathematical Logic Quarterly, 40, 478–480.
Banaschewski, B., & Vermeulen, J. J. C. (1996). Polynomials and radical ideals. Journal of Pure and Applied Algebra, 113(3), 219–227.
Berger, U. (2004). A computational interpretation of open induction. In F. Titsworth (Ed.), Proceedings of the 19th annual IEEE symposium on logic in computer science (pp. 326–334). Washington, D.C.: IEEE Computer Society.
Bishop, E. (1967). Foundations of constructive analysis. New York: McGraw-Hill.
Bishop, E., & Bridges, D. (1985). Constructive analysis. Berlin: Springer.
Blechschmidt, I. (2017). Using the internal language of toposes in algebraic geometry. Doctoral dissertation, University of Augsburg.
Boileau, A., & Joyal, A. (1981). La logique des topos. Journal of Symbolic Logic, 46(1), 6–16.
Brewer, J., & Richman, F. (2006). Subrings of zero-dimensional rings. In J. W. Brewer, S. Glaz, W. J. Heinzer, & B. M. Olberding (Eds.), Multiplicative ideal theory in commutative algebra: A tribute to the work of Robert Gilmer (pp. 73–88). New York: Springer Science+Business Media.
Campbell, P. J. (1978). The origin of Zorn’s Lemma. Historia Mathematica, 5, 77–89.
Cederquist, J., & Thierry C. (2000). Entailment relations and distributive lattices. In S.R. Buss, P. Hájek, & P. Pudlák (Eds.), Logic Colloquium ’98: Proceedings of the annual European summer meeting of the Association for Symbolic Logic, Prague, Czech Republic, August 9–15, 1998, volume 13 of Lecture notes in logic (pp. 127–139). Natick, MA: A. K. Peters.
Coquand, T. (1997). A note on the open induction principle. Technical report, University of Gothenburg. www.cse.chalmers.se/~coquand/open.ps.
Coquand, T. (2009). Space of valuations. Annals of Pure and Applied Logic, 157, 97–109.
Coquand, T., Ducos, L., Lombardi, H., & Quitté, C. (2009). Constructive Krull dimension. i: Integral extensions. Journal of Algebra and Its Applications, 8 (1), 129–138.
Coquand, T., & Lombardi, H. (2002). Hidden constructions in abstract algebra: Krull dimension of distributive lattices and commutative rings. In M. Fontana, S. -E. Kabbaj, & S. Wiegand (Eds.), Commutative ring theory and applications, volume 231 of Lecture notes in pure and applied mathematics (pp. 477–499). Reading, MA: Addison-Wesley.
Coquand, T., & Lombardi, H. (2006). A logical approach to abstract algebra. Mathematical Structures in Computer Science, 16, 885–900.
Coquand, T., Lombardi, H., & Neuwirth, S. (2019). Lattice-ordered groups generated by an ordered group and regular systems of ideals. Rocky Mountain Journal of Mathematics, 49(5), 1449–1489.
Coquand, T., Lombardi, H., & Roy, M. -F. (2005). An elementary characterisation of Krull dimension. In L. Crosilla & P. Schuster (Eds.), From sets and types to topology and analysis, volume 48 of Oxford logic guides (pp. 239–244). Oxford: Oxford University Press.
Coquand, T., & Neuwirth, S. (2017). An introduction to Lorenzen’s Algebraic and logistic investigations on free lattices (1951). Preprint. https://arxiv.org/abs/1711.06139.
Coquand, T., & Persson, H. (1998a). Gröbner bases in type theory. In T. Altenkirch, B. Reus, & W. Naraschewski (Eds.), Types for proofs and programs (pp. 33–46). Berlin: Springer.
Coquand, T., & Persson, H. (1998b). Integrated development of algebra in type theory. Calculemus and Types Workshop, 98.
Coquand, T., Sambin, G., Smith, J., & Valentini, S. (2003). Inductively generated formal topologies. Annals of Pure and Applied Logic, 124, 71–106.
Coste, M., Lombardi, H., & Roy, M. -F. (2001). Dynamical method in algebra: Effective Nullstellensätze. Annals of Pure and Applied Logic, 111(3), 203–256.
Crosilla, L., & Schuster, P. (2014). Finite methods in mathematical practice. In G. Link (Ed.), Formalism and beyond: On the nature of mathematical discourse, volume 23 of Logos (pp. 351–410). Boston, MA, and Berlin: Walter de Gruyter.
Curi, G. (2010). On some peculiar aspects of the constructive theory of point-free spaces. Mathematical Logic Quarterly, 56(4), 375–387.
Eisenbud, D. (1995). Commutative algebra with a view toward algebraic geometry, volume 150 of Graduate texts in mathematics. New York: Springer.
Español, L. (1982). Constructive Krull dimension of lattices. Revista de la Academia de Ciencias Exactas ... Zaragoza (2) 37, 5–9.
Fine, A. (1993). Fictionalism. Midwest Studies in Philosophy, 18, 1–18.
Fourman, M., & Grayson, R. (1982). Formal spaces. In The L. E. J. Brouwer Centenary Symposium (Noordwijkerhout, 1981). Studies in logic and the foundations of mathematics (Vol. 110, pp. 107–122). Amsterdam: North-Holland.
Goldman, O. (1951). Hilbert rings and the Hilbert Nullstellensatz. Mathematische Zeitschrift, 54(2), 136–140.
Hertz, P. (1922). Über Axiomensysteme für beliebige Satzsysteme. i. Teil: Sätze ersten Grades. Mathematische Annalen, 87(3), 246–269.
Hertz, P. (1923). Über Axiomensysteme für beliebige Satzsysteme. ii. Teil: Sätze höheren Grades. Mathematische Annalen, 89(1), 76–102.
Hertz, P. (1929). Über Axiomensysteme für beliebige Satzsysteme. Mathematische Annalen, 101(1), 457–514.
Hodges, W. (1979). Krull implies Zorn. Journal of the London Mathematical Society, 19, 285–287.
Howard, P., & Rubin, J. (1998). Consequences of the Axiom of Choice. Providence, RI: American Mathematical Society.
Jacobsson, C., & Löfwall, C. (1991). Standard bases for general coefficient rings and a new constructive proof of Hilbert’s basis theorem. Journal of Symbolic Computation, 12(3), 337–372.
Johnstone, P.T. (1982). Stone spaces (Cambridge studies in advanced mathematics 3). Cambridge: Cambridge University Press.
Johnstone, P.T. (2002). Sketches of an elephant: A topos theory compendium, volume 44 of Oxford logic guides (Vol. 2). Oxford: Clarendon Press.
Kemper, G., & Yengui, I. (2019). Valuative dimension and monomial orders. https://arxiv.org/abs/1906.12067
Krull, W. (1929). Idealtheorie in Ringen ohne Endlichkeitsbedingung. Mathematische Annalen, 101(1), 729–744.
Kunz, E. (1991). Algebra. Braunschweig: Vieweg.
Lal, H. (1971). A remark on rings with primary ideals as maximal ideals. Mathematica Scandinavica, 29, 72.
Legris, J. (2012). Paul Hertz and the origins of structural reasoning. In J. -Y. Béziau (Ed.), Universal logic: An anthology: From Paul Hertz to Dov Gabbay, Studies in universal logic (pp. 3–10). Basel: Birkhäuser.
Lombardi, H. (2002). Dimension de Krull, Nullstellensätze et évaluation dynamique. Mathematische Zeitschrift, 242, 23–46.
Lombardi, H., & Quitté, C. (2015). Commutative algebra: Constructive methods: Finite projective modules, volume 20 of Algebra and applications. Dordrecht: Springer Netherlands.
Lorenzen, P. (1950). Über halbgeordnete Gruppen. Mathematische Zeitschrift, 52(1), 483–526.
Lorenzen, P. (1951). Algebraische und logistische Untersuchungen über freie Verbände. Journal of Symbolic Logic, 16(2), 81–106.
Lorenzen, P. (1952). Teilbarkeitstheorie in Bereichen. Mathematische Zeitschrift, 55(3), 269–275.
Lorenzen, P. (1953). Die Erweiterung halbgeordneter Gruppen zu Verbandsgruppen. Mathematische Zeitschrift, 58(1), 15–24.
Lorenzen, P. (1953). Eine Bemerkung über die Abzählbarkeitsvoraussetzung in der Algebra. Mathematische Zeitschrift, 57, 241–243.
Lorenzen, P. (2017). Algebraic and logistic investigations on free lattices (S. Neuwirth of Lorenzen 1951, Trans.). https://arxiv.org/abs/1710.08138
Mines, R., Fred R., & Ruitenburg, W. (1988). A course in constructive algebra. Universitext. New York: Springer.
Moore, G.H. (1982). Zermelo’s axiom of choice: Its origins, development, & influence. Mineola, NY: Dover Publications 2013. Unabridged republication of the work originally published as volume 8 in the series “Studies in the history of mathematics and physical sciences” by Springer-Verlag, New York.
Negri, S. (2014). Proof analysis beyond geometric theories: from rule systems to systems of rules. Journal of Logic and Computation, 26(2), 513–537.
Negri, S., & von Plato, J. (1998). Cut elimination in the presence of axioms. Bulletin of Symbolic Logic, 4(4), 418–435.
Perdry, H. (2004). Strongly Noetherian rings and constructive ideal theory. Journal of Symbolic Computation, 37(4), 511–535.
Perdry, H., & Schuster, P. (2011). Noetherian orders. Mathematical Structures in Computer Science, 21, 111–124.
Perdry, H., & Schuster, P. (2014). Constructing Gröbner bases for Noetherian rings. Mathematical Structures in Computer Science, 24, e240206.
Raoult, J.-C. (1988). Proving open properties by induction. Information Processing Letters, 29(1), 19–23.
Rathjen, M. (2005). Generalized inductive definitions in constructive set theory. In L. Crosilla & P. Schuster (Eds.) From sets and types to topology and analysis: Towards practicable foundations for constructive mathematics, volume 48 of Oxford logic guides (pp. 23–40). Oxford: Clarendon Press.
Richman, F. (1974). Constructive aspects of Noetherian rings. Proceedings of the American Mathematical Society, 44, 436–441.
Rinaldi, D. (2014). Formal methods in the theories of rings and domains. Doctoral dissertation, University of Munich.
Rinaldi, D., & Schuster, P. (2016). A universal Krull–Lindenbaum theorem. Journal of Pure and Applied Algebra, 220, 3207–3232.
Rinaldi, D., Schuster, P., & Wessel, D. (2017). Eliminating disjunctions by disjunction elimination. Bulletin of Symbolic Logic, 23(2), 181–200.
Rinaldi, D., Schuster, P., & Wessel, D. (2018). Eliminating disjunctions by disjunction elimination. Indagationes Mathematicae (N.S.) 29(1), 226–259.
Rinaldi, D., & Wessel, D. (2018). Extension by conservation. Sikorski’s theorem. Logical Methods in Computer Science, 14(4:8), 1–17.
Rinaldi, D., & Wessel, D. (2019). Cut elimination for entailment relations. Archive for Mathematical Logic, 58(5–6), 605–625.
Sambin, G. (2003). Some points in formal topology. Theoretical Computer Science, 305(1–3), 347–408.
Satyanarayana, M. (1967). Rings with primary ideals as maximal ideals. Mathematica Scandinavica, 20, 52–54.
Scholz, H. (1919). Die Religionsphilosophie des Als-ob. Annalen der Philosophie, 1(1), 27–113.
Schuster, P. (2012). Induction in algebra: A first case study. In Proceedings, LICS 2012 27th annual ACM/IEEE symposium on logic in computer science (pp. 581–585). Dubrovnik, Croatia: IEEE Computer Society Publications.
Schuster, P. (2013). Induction in algebra: A first case study. Logical Methods in Computer Science, 9(3), 20.
Scott, D. (1954). Prime ideal theorems for rings, lattices, and Boolean algebras. Bulletin of the American Mathematical Society, 60(4), 390.
Scott, D. (1971). On engendering an illusion of understanding. Journal of Philosophy, 68, 787–807.
Scott, D. (1974). Completeness and axiomatizability in many-valued logic. In L. Henkin, J. Addison, C. C. Chang, W. Craig, D. Scott, & R. Vaught (Eds.), Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. xxv, Univ. California, Berkeley, Calif., 1971) (pp. 411–435). Providence, RI: American Mathematical Society.
Scott, D. S. (1973). Background to formalization. In Truth, syntax and modality (Proc. Conf. Alternative Semantics, Temple Univ., Philadelphia, Pa., 1970), volume 68 of Studies in logic and the foundations of mathematics, edited by Hugues Leblanc (pp. 244–273). Amsterdam: North-Holland.
Seidenberg, A. (1974). What is Noetherian? Rendiconti del Seminario Matematico e Fisico di Milano, 44, 55–61.
Simpson, S. G. (2009). Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.). Cambridge: Cambridge University Press.
Steinitz, E. (1910). Algebraische Theorie der Körper. Journal für die reine und angewandte Mathematik, 137, 167–309.
Vaihinger, H. (1922). Die Philosophie des Als Ob: System der theoretischen, praktischen und religiösen Fiktionen der Menschheit auf Grund eines idealistischen Positivismus. 7. u. 8. Aufl. Leipzig: Verlag von Felix Meiner.
Vaihinger, H. (1924). The philosophy of ‘as if’: A system of the theoretical, practical and religious fictions of mankind (C. K. Ogden, Trans.). London: Routledge & Kegan Paul.
Wessel, D. (2018). Points, ideals, and geometric sequents. Technical report, University of Verona.
Wessel, D. (2020). A note on connected reduced rings. Journal of Commutative Algebra. Forthcoming. https://projecteuclid.org/euclid.jca/1561363253
Yengui, I. (2008). Making the use of maximal ideals constructive. Theoretical Computer Science, 392, 174–178.
Yengui, I. (2015). Constructive commutative algebra: Projective modules over polynomial rings and dynamical Gröbner bases. Lecture notes in mathematics (Vol. 2138). Cham: Springer.
Zermelo, E. (1904). Beweis, daß jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59, 514–516.
Zermelo, E. (1908). Neuer Beweis für die Möglichkeit einer Wohlordnung. Mathematische Annalen, 65, 107–128.
Zorn, M. (1935). A remark on method in transfinite algebra. Bulletin of the American Mathematical Society, 41, 667–670.
Acknowledgements
The research that has led to this paper was carried out within the project “ A New Dawn of Intuitionism: Mathematical and Philosophical Advances” (ID 60842) funded by the John Templeton Foundation. Partial support has come from the programme “Dipartimenti di Eccellenza 2018–2022” of the Italian Ministry of Education, Universities and Research (miur), and the project “Categorical localisation: methods and foundations” (catloc) funded by the Università degli Studi di Verona within the programme “Ricerca di Base 2015”. The final version of this paper was prepared within the project “Reducing complexity in algebra, logic, combinatorics—redcom” belonging to the programme “Ricerca Scientifica di Eccellenza 2018” of the Fondazione Cariverona. (The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of those foundations. ) Essential parts of this paper were conceived when both authors visited the Hausdorff Research Institute for Mathematics (him), University of Bonn, on the occasion of the trimester program “Types, Sets, and Constructions”, May–August 2018. The related financial support is gratefully acknowledged. The authors express their gratitude for interesting discussions and valuable hints to Ingo Blechschmidt, who was so kind as to have a look at the manuscript, as well as to Thierry Coquand, Henri Lombardi, Stefan Neuwirth, Davide Rinaldi, Ihsen Yengui, and, last but not least, to the referee for expertly remarks and an appreciative reading of the manuscript.
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Schuster, P., Wessel, D. (2021). Syntax for Semantics: Krull’s Maximal Ideal Theorem. In: Heinzmann, G., Wolters, G. (eds) Paul Lorenzen -- Mathematician and Logician. Logic, Epistemology, and the Unity of Science, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-030-65824-3_6
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