Zusammenfassung
We first start by clarifying what axiomatizing everything can mean. We then study a famous case of axiomatization, the axiomatization of natural numbers, where two different aspects of axiomatization show up, the model-theoretical one and the proof-theoretical one. After that we discuss a case of axiomatization in a sense opposed to the one of arithmetic, the axiomatization of the notion of order, where the idea is not to catch a specific structure, but a notion.
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Notes
- 1.
The notion of structure was promoted as the central notion of mathematics by Bourbaki (cf. Chapter 4 of the book Théorie des ensembles 1970, entitled Structures). See also Bourbaki (1948) and Corry (2004a). And Bourbaki stressed that the structure of natural numbers is not at all the simplest structure, it is a mix/combination of different structures (carrefour de structures in French). See our recent paper (Beziau 2017b), pointing the many aspects of the number 1 according to different structures it is merged in.
- 2.
This definition was given in Tarski (1936), although at this time the notion of model was not yet completely clear. Tarski was also not yet using the symbol “\(\vDash \)”.
- 3.
Tarski was much influenced by Blaise Pascal, in particular when writing “Sur la méthode déductive” (1937).
- 4.
More exactly: from a recursive set of axioms. This means we should be able to identify these axioms, we have to exclude the case where we have any infinite set of axioms, like the extreme case of all formulas true in \(\mathcal {N}\), which trivially is a complete theory for \(\mathcal {N}\).
- 5.
The axiomatization presented here is not independent in the sense that for example the axiom of antisymmetry is a consequence of the axioms of irreflexivity and transitivity.
- 6.
Sometimes a relation of order is defined as a relation also being reflexive. This is not a very good choice, because then the notion of strict order is contradictory.
- 7.
David Hilbert (1918) in his famous paper on axiomatic thought is using the expression “conceptual framework” (in German: Fachwerk von Begriffen), but he identifies it to the notion of theory. We prefer to use the word “theory” as referring to a set of statements that in particular can be considered as axioms.
- 8.
Rolando Chuaqui and Patrick Suppes (1995) have shown that classical mechanics can be axiomatized by formulas with only universal quantifiers.
- 9.
Rougier (1889–1992) was a good friend of Moritz Schlick and one of the main promoters of the Vienna Circle. He organized in 1935 at the Sorbonne in Paris a big congress on scientific philosophy with the participation of Schlick, Carnap, Neurath, Russell, Tarski, Lindenbaum, etc. The result was a 8-volume book: Actes du Congrès International de Philosophie Scientifique – Sorbonne, Paris, 1935.
References
Actes du Congrès International de Philosophie Scientifique – Sorbonne, Paris, 1935 (1936).
Andréka, H., Madarsz, J., & Németi, I. (2007). Logic of space-time and relativity theory. In M. Aiello, I. Pratt-Hartmann, & J. van Benthem (Eds.), Handbook of spatial logics (pp. 607–711). Heidelberg: Springer.
Arnauld, A., & Nicole, P. (1662). La logique ou l’art de penser. Paris: Savreux.
Barnett, L. (1948). The Universe and Dr Einstein. New York: William Morrow.
Beziau, J.-Y. (1994). Théorie législative de la négation pure. Logique et Analyse, 147–148, 209–225.
Beziau, J.-Y. (2002). La théorie des ensembles et la théorie des catégories: Présentation de deux soeurs ennemies du point de vue de leurs relations avec les fondements des mathématiques. Boletín de la Asociación Matemática Venezolana, 9, 45–53.
Beziau, J.-Y. (2006). Les axiomes de Tarski. In R. Pouivet & M. Rebuschi (Eds.), La philosophie en Pologne 1918–1939 (pp. 135–149). Paris: Vrin.
Beziau, J.-Y. (2010). Logic is not logic. Abstracta, 6, 73–102.
Beziau, J.-Y. (2015a). Modelling causality. Conceptual clarifications tributes to Patrick Suppes (1922–2014) (pp. 187–205). London: College Publications.
Beziau, J.-Y. (2015b). Panorama de l’identité. Al Mukhatabat, 14,, 205–219.
Beziau, J.-Y. (2017a). Being aware of rational animals. In G. Dodig-Crnkovic & R. Giovagnoli (Eds.), Representation and reality: Humans, animals and machines (pp. 319–331). Cham: Springer.
Beziau, J.-Y. (2017b). MANY 1 – A transversal imaginative journey across the realm of mathematics. In: M. Chakraborty & M. Friend (Eds.), Special issue on Mathematical Pluralism of the Journal of Indian Council of Philosophical Research, 34(2), 259–287.
Boole, G. (1854). An Investigation of the laws of thought, on which are founded the mathematical theories of logic and probabilities. London: Macmillan & Co.
Bourbaki, N. (1948). L’architecture des mathématiques -La mathématique ou les mathématiques. In F. le Lionnais (Ed.), Les grands courants de la pensée mathématique (pp. 35–47). Cahier du Sud.
Bourbaki, N. (1970). Théorie des ensembles. Paris: Hermann.
Chuaqui, R., & Suppes, P. (1995). Free-variable axiomatic foundations of infinitesimal analysis: A fragment with finitary consistency proof. The Journal of Symbolic Logic, 60, 122–159.
Corry, L. (2004a). Modern algebra and the rise of mathematical structures (2. edn.). Basel: Birkhäuser.
Corry, L. (2004b). David Hilbert and the axiomatization of physics (1898–1918). Dordrecht: Springer.
Dedekind, R. (1888). Was sind und was sollen die Zahlen?. Braunschweig: Vieweg.
Gödel, K. (1930). DieVollständigkeit der Axiome des logischen Funktionenkalküls. Monatshefte für Mathematik und Physik, 37, 349–60.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. Monatshefte für Mathematik und Physik, 38, 173–98.
Gödel, K. (1949). An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 21, 447–450.
Gonseth, F. (1939). La méthode axiomatique. Bulletin de la Société Mathématique de France, 67, 43–63.
Halbach, V. (2011). Axiomatic theories of truth. Cambridge: Cambridge University Press.
Hilbert, D. (1899). Grundlagen der Geometrie. Leipzig: Teubner.
Hilbert, D. (1918). Axiomatisches Denken. Mathematische Annalen, 78, 405–415.
Hintikka, J. (2011). What is the axiomatic method? Synthese, 183, 69–85.
Hodges, W. (1983). Elementary predicate logic. In D. Gabbay & F. Guenthner (Eds.), Handbook of Philosophical Logic I. Dordrecht: Reidel.
Manzano, M. (1996). Extensions of first order logic. Cambridge: Cambridge University Press.
Pascal, B. (1657). De l’esprit géométrique et de l’art de persuader. Œuvres complètes, J. Chevalier (Ed.), 1962. Paris: Bibliothèque de la Plèiade.
Peano, G. (1889). Arithmetices Principia, Novo Methodo Exposita. Rome: Fratres Bocca.
Peirce, C. (1881). On the logic of number. American Journal of Mathematics, 4(1), 85–95.
Rougier, L. (1920). Les paralogismes du rationalisme. Paris: Félix Alcan.
Rubin, H., & Rubin, J. (1963). Equivalents of the axiom of choice I. Amsterdam: North Holland.
Rubin, H., & Rubin, J. (1985). Equivalents of the axiom of choice II. Amsterdam: North Holland.
Skolem, T. (1934). Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundamenta Mathematicae, 23, 150–161.
Stone, M. (1935). Subsumption of the theory of Boolean algebras under the theory of rings. Proceedings of National Academy of Sciences, 21, 103–105.
Tarski, A. (1928). Remarques sur les notions fondamentales de la méthodologie des mathématiques. Annales de la Société Polonaise de Mathématique, 7, 270–272.
Tarski, A. (1936). O pojȩciu wynikania logicznego. Przegad Filozoficzny, 39, 58–68.
Tarski, A. (1937). Sur la méthode déductive. Travaux du IXe Congrès International de Philosophie VI (pp. 95–103). Paris: Hermann.
Tarski, A. (1938). Ein Beitrag zur Axiomatik der Abelschen Gruppen. Fundamenta Mathematicae, 30, 253–256.
Tarski, A. (1944). The Semantic conception of truth and the foundations of semantics. Philosophy and Phenomenological Research, 4, 341–376.
Tarski, A. (1954a). Contributions to the theory of models. I. Indagationes Mathematicae, 16, 572–581.
Tarski, A. (1954b). Contributions to the theory of models. II. Indagationes Mathematicae, 16, 582–588.
Tarski, A. (1955). Contributions to the theory of models. III. Indagationes Mathematicae, 17, 56–64.
Tarski, A. (1969). Truth and proof. Scientific American, 220(6), 63–77.
Vaugth, R. (1954). Applications of the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability. Indagationes Mathematicae, 16, 467–472.
Veblen, O. (1904). A system of axioms for geometry. Transactions of the American Mathematical Society, 5, 343–384.
Wang, H. (1997). A logical journey – From Gödel to philosophy. Cambridge: MIT Press.
Woodger, J. (1937). The axiomatic method in biology, with appendices by Alfred Tarski & W.J. Floyd. Cambridge: Cambridge University Press.
Acknowledgements
The origin of this paper is a talk presented at the meeting Kurt Gödel’s Legacy: Does Future lie in the Past? which took place July 25–27, 2019 in Vienna, Austria. Thanks to the organizers of the event and also thanks to Jan Zygmunt, Robert Purdy and Jorge Petrucio Viana for useful comments and information.
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Béziau, JY. (2021). Is there an Axiom for Everything?. In: Passon, O., Benzmüller, C. (eds) Wider den Reduktionismus. Springer Spektrum, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63187-4_8
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