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On Bourbaki’s axiomatic system for set theory

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Abstract

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Éléments de Mathématique. We begin by examining the role played by the sign \(\uptau \) in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group.

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Notes

  1. In this paper we employ the French version from 1970 (Bourbaki 1970) for the study of the axiomatic system. This edition is the one with the widest circulation. Hermann editorial published several reproductions during the decade of the 1970s. The different editions in French and English published by Springer between 2004 and 2008 are textual reprints of this edition. However, for convenience, we employ the English version from 1968 (Bourbaki 1968) to quote paragraphs of the Introduction or the Historical Notes since there are no changes in the two editions.

  2. Bourbaki utilizes the term assemblage that means an assembly or union (of signs). We have preferred the term formula for being one of the most currently employed in Logic texts.

  3. A formula is a term if it starts with the sign \(\uptau \) or if it is reduced to a single letter. A formula is a relation if it starts with: \(\vee , \lnot , =, \in \). This notion is from the Polish logician Jan Lukasiewicz. Although Bourbaki employs it at the start of the first chapter, he quickly abandons it.

  4. Note that Bourbaki employs the abbreviating symbol \(\Rightarrow \) instead of the formula \(\vee \lnot \). These four schemes correspond to the axioms considered by Hilbert and Ackermann for propositional logic in the 1928 book, Principles of Mathematical Logic (Hilbert and Ackermann 1950, p. 27). They are essentially the same as those of Russell and Whitehead in the Principia Mathematica (1910–1913).

  5. Frege in 1893 was the first one to use a descriptive operator; later, Peano did so in 1899, then Hilbert in 1923, and afterwards Russell and Whitehead in 1925, among others. The different semantics are analyzed by Wirth (2008, p. 289) when searching for a semantic of his own for Hilbert’s \(\upvarepsilon \)-operator.

  6. Hilbert utilizes the letter \(a\) to identify the variable in predicate A. We use the letter \(x\) for simple convenience.

  7. Zach (2004), in his second footnote, comments that during the winter of 1922–1923 Hilbert and Bernays introduced the \(\upvarepsilon \)-operator for the first time. Initially, with the same interpretation as the \(\uptau \) operator; then in the notes attached to the typed copy of the course, it appeared with the final interpretation of the \(\upvarepsilon \)-operator.

  8. Two years earlier, in his article “On the Infinite”, Hilbert presented it in his demonstration theory as a very singular axiom because all transfinite axioms derive from it and it is at the heart of the axiom of choice (Hilbert 1925, p. 382).

  9. Using the \(\upvarepsilon \)-terms to replace the existential and universal quantifiers is known as the epsilon substitution method or the \(\upvarepsilon \)-substitution method.

  10. Mathematics of content or real mathematics is composed of propositions that express a particular content or a given reality. Formal mathematics or ideal mathematics, besides the propositions of content mathematics, includes the ideal propositions (with no interpretation or reference to any particular content). They have been introduced for the purpose of preserving mathematical reasoning in a simple and efficient manner (Detlefsen 2005, p. 291).

  11. Mathias (2012) criticizes this definition and its possible interpretations. He considers that the \(\uptau \)-operator perverts the natural order of mental acts. It specifically says: “to interpret \(\exists \) you look for witnesses and must first check whether a witness exist before you can pick an interpretation for \({\uptau }_{x}\)(R); and then you define \((\exists x)\hbox {R}\) to mean that the witness witnesses that \(\hbox {R}({\uptau }_{x}(\hbox {R}))\): a strange way to justify what Bourbaki, if pressed, would claim to be a meaningless text” (Mathias 2012, p. 13). This interpretation goes against the way Hilbert understood the logical function of this operator. From the syntactic point of view, says Hilbert, the set of symbols is not arbitrary. To the contrary, “it reflects the technique of our thinking” when it permits “the expression of mathematical content in a uniform manner and, at the same time, guides its development in a direction that clarifies the interconnections between individual propositions and facts” (Arboleda 2009, p. 256).

  12. See some of the relations (that imply equivalences) that can be established between the Hilbert’s transfinite axiom, the \(\upvarepsilon \)-axiom and the axiom of choice in (Moore 1982, pp. 253–255).

  13. Although Bourbaki does not speak explicitly of classes, they are formulas of the formal logical system that represent collections of objects that verify a certain given property.

  14. The conjunction of the two schemes may be considered as a particular interpretation of Leibniz’s Law that states that “two things are identical if and only if they have the same properties”.

  15. Particularly, in the second chapter of the book, utilizes \(\uptau \) to justify the generalized product of a non-empty family, of non-empty sets, is non-empty.

  16. In the third chapter, based on the properties of ordered sets, of the sign \(\uptau \) and of S5, Bourbaki proves Zermelo’s Theorem which states that all sets can be well-ordered.

  17. Une objection que l’on peut faire à l’introduction du symbole \(\upvarepsilon \), c’est que l’axiome de choix devient un théorème, de sorte qu’il n’y a plus de distinction entre démonstrations avec ou sans l’axiome de choix. Mais je dois dire que cet “inconvénient” me paraît plutôt un avantage. Il me semble que les distinctions entre démonstrations avec ou sans l’axiome de choix n’ont d’intérêt que dans des systèmes beaucoup plus pauvres que celui de Bourbaki. Je rappelle à ce sujet que Gödel a montré que la mathématique sans axiome de choix possède une interprétation dans laquelle l’axiome du choix est un théorème: c’est-à-dire qu’on y peut définir une classe d’ensembles pour lesquels tous les autres axiomes de la théorie des ensembles sont vrais ainsi que l’axiome du choix (Bourbaki 1950a, b, p. 3-4). The italic font is ours.

  18. José Grimm (INRIA-France) shows, using the COQ Proof Assistant, that it is possible to implement the work of Bourbaki into a computer. In (Grimm 2013a) is the implementation of the first two chapters of the book Set Theory, and in (Grimm 2013b) the implementation of the third chapter.

  19. In the first theorem of chapter II, the existence of the empty set is proved (Bourbaki 1970, E II.6).

  20. Bourbaki supposes that the relation \(x \not \in x\) is collectivizing, then \((\exists y)(\forall x)((x\in y) \Leftrightarrow x\not \in x)\). Let \(a\) be an auxiliary constant, then we have: \((\forall x) (x\in a \Leftrightarrow x\not \in x)\). Therefore, the relation “\(a \in a \Leftrightarrow a\not \in a\)” is true, which is a contradiction.

  21. Mathias (2012, p. 13) says that this “notation is highly misleading in that all classes which are not sets have become «equal »”. This would mean that Bourbaki’s formal system is inadequate to describe set theory. Let’s consider Mathias’s argument in detail. Let R and S be two relations of the \({\fancyscript{T}}\) theory such that R: \(x\not \in x\) and S: \(x = x\). Bourbaki proves that R is not collectivizing and analogously demonstrates that S is not either. That is, \(\hbox {notColl}_{x}\hbox {R}\) and \(\hbox {notColl}_{x}\hbox {S}\) are theorems in \({\fancyscript{T}}\). If we write \(\hbox {C}_{\mathrm{R}}:(\forall x)((x\in y)\Leftrightarrow R)\) and \(\hbox {C}_{\mathrm{S}}:(\forall x)((x\in y)\Leftrightarrow \hbox {S})\), as Mathias proposes, we have that \(\lnot (\exists y)\hbox {C}_{\mathrm{R}}\) and \(\lnot (\exists y)\hbox {C}_{\mathrm{S}}\) are theorems of \({\fancyscript{T}}\). In other words, \((\forall y)\lnot \hbox {C}_{\mathrm{R}}\) and \((\forall y)\lnot \hbox {C}_{\mathrm{S}}\) are theorems of \({\fancyscript{T}}\). These relationships can be interpreted as, “for all \(y\), \(\hbox {C}_{\mathrm{R}}\) is false” and “for all \(y\), \(\hbox {C}_{\mathrm{S}}\) is false”. From these two theorems Mathias concludes that \(\hbox {C}_{\mathrm{R}} \Leftrightarrow \hbox {C}_{\mathrm{S}}\). Mathias continues and employs the S7 scheme to prove that \({\uptau }_{y}\hbox {C}_{\mathrm{R}} = {\uptau }_{y}\hbox {C}_{\mathrm{S}}\). This allows him to conclude that the privileged objects are identical. Or that classes \(\hbox {A} = \{x{\vert } x\not \in x\}\) and \(\hbox {B} = \{x{\vert } x= x\}\) would be equal; this would become a serious limitation of the formal language of Bourbaki.

    For us, once one has that \((\forall y)\lnot \hbox {C}_{\mathrm{R}}\) and \((\forall y)\lnot \hbox {C}_{\mathrm{S}}\) are theorems of \({\fancyscript{T}}\), one can only deduce that they are logically equivalent; i.e., \((\forall y)\lnot \hbox {C}_{\mathrm{R}}\Leftrightarrow (\forall y)\lnot \hbox {C}_{\mathrm{S}}\). From this equivalence it cannot be deduced that \(\hbox {C}_{\mathrm{R}}\Leftrightarrow \hbox {C}_{\mathrm{S}}\).

  22. Observe that the implication \((\forall x)(\hbox {R} \Leftrightarrow \hbox {S}) \Rightarrow \{x|\hbox {R}\}=\{x|\hbox {S}\}\) corresponds to a particular version of S7 scheme, where R and S are collectivizing relations and the privileged objects \({\uptau }_{x}\)(R) and \({\uptau }_{x}\)(S) correspond to the sets \(\{x|\hbox {R}\}\) and \(\{x|\hbox {S}\}\).

  23. In some editions of the book “Theory of Sets”, five explicit axioms appear. For example, in the English version from 1968, the Axiom of ordered pairs is enunciated (Bourbaki 1968, p. 72):

    $$\begin{aligned} (\forall x)(\forall x')(\forall y)(\forall y')((x, y) = (x', y') \Rightarrow (x=x' \wedge y=y')) \end{aligned}$$

    In the French version from 1970, Bourbaki defines the ordered pair \((x,\,y)\) as the set \(\{\{x\},\,\{x,\,y\}\}\) and demonstrates that the relation \((x,\,y) = (x',\,y')\) is equivalent to «\(x=x\)’ \(\wedge \,\, y=y\)’ »   (Bourbaki 1970, E II.7). Thus, the axiom becomes a proposition of the theory.

  24. A complete historical analysis of the emergence and consolidation of various axiom systems for set theory, including the Zermelo–Fraenkel system, is found in (Ferreirós 2007, pp. 297–392). See also (Ferreirós 1992).

  25. The notation ZFC is used to make explicit the inclusion of the axiom of choice; since in the “extended ZF system”, Zermelo does not include the axiom of choice because he considers it a universal logical principle and does not admit the axiom of infinity by considering that it is not part of general set theory.

  26. In the historical notes, Bourbaki refers to this system as Zermelo–Fraenkel system.

  27. Recall that two axiomatic systems T and T’ are equivalent if the axioms of T are axioms or theorems of T’ and vice-versa.

  28. There are consistent axiomatic theories that do not include the axiom of foundation and therefore they are axiomatic theories that accept sets that are not members of themselves. In this regard, see (Aczel 1988). In this book, Peter Aczel presents a consistent axiomatic theory that allows the existence of sets which form membership chains with infinite descent. In its proposal is incorporated the anti-foundation axiom.

  29. A complete analysis of the Bourbaki group and their work between 1934 and 1944 can be found at (Beaulieu 1989).

  30. Mathias (1992) extensively analyzes this article. He vehemently criticizes that Dieudonné does not explicitly mention the work of Gödel and fails to note its importance to the foundations of mathematics. He offers several explanations (of the logical, sociological and mathematical order, among others) about the conservative stance of Bourbaki.

  31. Mathias (1992) emphatically criticized the absence of this axiom.

  32. Toute la mathématique, telle qu’elle existe aujourd’hui, peut s’exprimer dans le système que nous venons d’esquisser. (Cartan 1943, p. 10).

  33. Ensembles Chap. 1 (état 3) Chap. II (état 3). Archives de l’Association des Collaborateurs de Nicolas Bourbaki, R057_iecnr_066.

  34. Livre I. Théorie des ensembles. Ch.II (Etat 3 ou 4) Théorie des ensembles abstraits. Archives de l’Association des Collaborateurs de Nicolas Bourbaki, R063_iecnr_072.

  35. Conference presented by André Weil in the name of Bourbaki at the eleventh meeting of the Association of Symbolic Logic held in Columbus, Ohio on December 31, 1948.

  36. Because of this listing, Mathias (1992) does not fail to emphasize that the theory of Bourbaki is Zermelo’s theory and not that of Zermelo–Fraenkel. But, strictly speaking, it does not even correspond to the theory of Zermelo. In another article, Mathias (2010) shows that a model of this system exists, called Bou49 where the set axiom with two elements is false.

  37. Livre I Théorie des ensembles Chapitre II. (état 6) Archives de l’Association des Collaborateurs de Nicolas Bourbaki, R160_nbr_062.

  38. The publication was made 14 years after the first edition of the Fascicule de Résultat and after having published ten chapters of the book of General Topology, seven of the nine chapters of the book of Algebra, seven chapters of the book of Functions of a real variable, the first two chapters of the book on Topological vector spaces and the first four chapters of the book on Integration. The third chapter on Ordered Sets was published in 1956 and in 1957 the fourth and final chapter on Structures was published (Fang 1970, p. 47).

  39. Corry (1992, 2004) illustrates this ambivalence from publications of the group and from the individual documents of some of its members. Examples of this ambivalence are that the group had decided to develop a fifth chapter of set theory on categories and functors; however, it was never published. Similarly, a previously agreed to volume on abelian categories was not published. At the same time, several members of the group, in their individual research-work, used and developed concepts of category theory.

  40. Samuel Eilenberg (joined the group in 1950) worked on homological algebra with Cartan and on group theory and Lie algebras with Chevalley, Jean-Pierre Serre worked on sheaf theory and algebraic geometry, Pierre Samuel on universal functions, Roger Godement on sheaf theory and algebraic topology and André Weil on the foundations of algebraic geometry. All these issues, which are the basis of category theory, were discussed in the seminar Bourbaki, in the seminar Cartan and in various meetings of the Group.

  41. Si une contradiction se rencontre un jour, on modifiera les axiomes en conséquence, et ce sera sans doute une source de grands progrès. (Bourbaki 1948, p. 6).

  42. The relative consistency of ZFC system implies the relative consistency of \(\hbox {ZFC}^{-}\).

  43. Another significant historical example is the choice of the topology of neighborhoods. For Bourbaki was the axiomatic more convenient and productive at the time to generalize the theory of functions and functional analysis in abstract spaces (Arboleda 2012).

  44. This issue is broadly dealt with by Feferman (2013), Malatesta (2011) and Marquis (1995, 2009), among others. Particularly, Marquis (1995) examines the relationships between category theory and set theory and suggests the possibility to regard category theory itself as a foundational theory.

  45. Marquis (2009, pp. 52–53) makes an analysis of the different practical solutions offered by Eilenberg and MacLane in this matter. Among them, the consideration is of \(a\) category of groups rather than the category of groups; but this solution has the difficulty of defining the composition of functors in general. Similarly, it presents the possibility of adopting the theory of types as a foundation for the theory of classes, but it could complicate the study of natural isomorphisms because one would have to consider isomorphims between groups of different types.

  46. This is an immediate consequence of the axioms. The axioms of NBG are: (i) Axioms in common with ZFC: pair, union, infinity, power set; (ii) Axioms both for sets and classes: extensionality, foundation; (iii) Axiom of limitation of size: a class is a set if and only if is not in bijection with the class of all sets V; and (iv) Axiom schema of comprehension: for every property \(\upvarphi (x)\), without quantifiers over classes, there exists the class \(\{x: \upvarphi (x)\}\). Using the axiom of limitation of size, it is shown that in NBG the class of all sets, V, is well ordered. This affirmation is equivalent to the axiom of global choice.

  47. For example, if A is a large category, global choice is required to define the functor product \(\hbox {A} \times \hbox {A} \rightarrow \hbox {A}\).

  48. Grothendieck was present at this congress as “guinea pig”. La Tribu No. 24.

  49. Certains ont bien envie de “gödeliser” pour traiter plus commodément de choses comme l’homologie axiomatique ou les applications universelles, mais se demandent si classes et \(\upvarepsilon \) sans restrictions, mis ensemble, ne vont pas canuler. Enfin Cartan se méfie d’un système “fermé” où tout est donné dès le début. (Bourbaki 1951a, p. 3).

  50. It was in the decade of the 60s that Kripke, Cohen and Solovay, in an independent manner, showed that NBG (with global choice) is a conservative extension of ZFC (Ferreirós 2007, p. 381).This means that, if a theorem about sets can be proved in NBG, a corresponding proposition can also be proved in ZFC. Furthermore, it was known that every theorem of ZFC is a theorem of NBG. Therefore, it can be prove that ZFC is consistent if and only if NBG is consistent.

  51. We use Krömer’s works (2006, 2007) because he consulted several unpublished literature sources. Furthermore, in the pages of the Archives de l’Association des collaborateurs de Nicolas Bourbaki, the only records found were of the congresses carried out between 1934 and 1953, i.e. until La Tribu No. 30 (Bourbaki 1953a).

  52. Cartier propose une méthode métamathématique d’introduire [les catégories et foncteurs] sans modifier notre système logique. Mais ce système est vomi car il tourne résolument le dos au point de vue de l’extension [...]. On décide donc qu’il va mieux élargir le système pour y faire rentrer les catégories; à première vue le système Gödel semble convenir. (Krömer 2006, p. 146).

  53. The proposal is found in the article Formalisation des classes et categories, a copy of which is in the collection of files of Desaltre in Nancy. This paper was particularly consulted by Krömer.

  54. The copy is in the Archives of Desaltre in Nancy. According to the investigation of Krömer, this anonymous document (in the Bourbaki style) was written by Grothendieck between July 1958 and March 1959.

  55. Marquis (2009, p. 53) is surprised that Eilenberg and MacLane (1945) have not seen these limitations of NBG. However, he understands that their main concerns lay in the applications and not in the categories themselves.

  56. As explained by Marquis (2009, p. 53), “the category of all categories” cannot be formed because the objects on being considered themselves to be its own class, cannot belong to another class. Similarly, Marquis (2009, p. 63) shows that if A and B are large categories, the functor F: \(\hbox {A} \rightarrow \hbox {B}\) is a proper class and therefore it is not possible to form the category of all functors from A to B.

  57. This issue is broadly dealt with by both Feferman (2013) and Marquis (2009), among others.

  58. In this article, called Tôhoku paper, Grothendieck (1957) positions the homological algebra in terms of abelian categories. This work is considered a milestone in the history of category theory. A comprehensive study on the relationship of this item with the Bourbaki project is in (Krömer 2006, 2007).

  59. This work was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. It is now considered the cornerstone of modern algebraic geometry.

  60. The complete presentations are available in http://www.numdam.org/numdam-bin/feuilleter?j=SHC&sl=0. See particularly section 13 of the seminar.

  61. The website of the seminar is available in: http://library.msri.org/books/sga/sga/index.html.

  62. A universe is a nonempty set U that has the following properties: \(U_{1}\): If \(x\in \hbox {U}\) and if \(y \in x\) then \(y \in \hbox {U}, U_{2}\): If \(x, y\in \hbox {U}\) then \(\{x, y\}\in \hbox {U}, U_{3}\): If \(x\in \hbox {U}\) then \(\wp (x) \in \hbox {U}\), where \(\wp (x)\) denotes the set of all subsets of \(x, U_{4}\): If (\(x_{i}\), \(i \in \hbox {I})\) is a family of elements of U, and if \(\hbox {I}\in \hbox {U}\), then \(\cup x_{i}\in \hbox {U}\). The axiom of universes says: For every set \(x\), there is a universe U such that \(x\in \hbox {U}\). These definitions are in (Grothendieck and Verdier 1972, pp. 1–2).

  63. A first treatment without reference to Grothendieck universes is located in the article of Tôhoku (Grothendieck 1957, p. 134) in which he positions homological algebra in terms of abelian categories. This work is considered a milestone in the history of category theory. A comprehensive study on the relationship of this item with the Bourbaki project is in (Krömer 2006, 2007).

  64. Pour éviter certaines difficultés logiques, nous admettrons ici la notion d’Univers, qui est un ensemble “assez gros” pour qu’on n’en sorte pas par les opérations habituelles de la théorie des ensembles; un “axiome des Univers” garantit que tout objet se trouve dans un Univers. (Grothendieck 1971, p. 146).

  65. Pour conclure, il me semble donc point qu’on soit obligé de rien changer aux trois premiers chapitres du Livre I [...] Il sera suffisant d’introduire au nouveau chapitre 4 (qui remplacera l’ancien inutilisable de toutes façons) les axiomes supplémentaires de la théorie des ensembles, et y développer la théorie des catégories aussi loin qu’il semble désirable. (Krömer 2006, p. 149).

  66. Tarski’s axiom says (Bourbaki 1972, p. 196): every cardinal is passed strictly for a strongly inaccessible cardinal. That is, for each cardinal \(k\), there is a strongly cardinal \(\lambda \) inaccessible, which is strictly greater than \(k\). The notion of strongly inaccessible cardinal is located the first time in (Sierpiński and Tarski 1930, p. 292).

  67. This is a “universe” particular of artinianos sets, which are sets that are not elements of any universe.

  68. A comprehensive historical study on the inaccessible cardinal is located in (Álvarez 1994).

  69. Thus, the acceptance of the axiom of universes implies a “leap of faith” similar to that required for acceptance of the Axiom of Infinity (Hrbacek and Jech 1999, p. 279).

  70. For the proof proceeds by contradiction: suppose that there exists such finitary proof and using Gödel’s incompleteness theorem in the process concludes that ZFC is inconsistent.

  71. Il serait très intéressant de démontrer que l’axiome (A.6) des univers est inoffensif. Ça paraît difficile et c’est même indémontrable, dit Paul Cohen. (Bourbaki 1972, p. 214).

  72. Numerosos descubrimientos que han probado una amplia evidencia de una estrecha relación entre los grandes cardinales y varios problemas matemáticos. (Jech 2005, p. 374).

  73. Cualquier intento de transgredir este orden conservador es visto por la conciencia como una agresión del exterior. Si se plantea por la necesidad de incorporar un conocimiento nuevo, tomará tiempo a la conciencia asimilar tal necesidad, pues ello implica “reordenar toda una serie de fenómenos e imbricar lo que acabo de aprender en mis propios esquemas”. (Arboleda 2007, p. 216).

  74. Grothendieck himself has produced beautiful images to represent a way of thinking inherent in certain mathematical practices, identified by him as conservative. See (Grothendieck 1986, pp. 38–39).

  75. We now know that inside each topos there is a logical natural intermediate defined by Heyting’s algebras classes and these algebras determine each intermediate logic. See (Santamaría 2008).

  76. The developments in intuitionistic logic have been of great importance for the advancement of computational theory in recent decades. The institutional developments of the logic in connection with computer science in the late twentieth century can be found in (Ferreirós 2010).

  77. To show its radical opposition to intuitionism we note the following citations from Cartan and Bourbaki: (i) “Notre but est de montrer comment la logique peut server de base à tout l’édifice des mathématiques, contrairement à l’opinion des “intuitionnistes”(Cartan 1943, p. 3)”; (ii) “The intuitionist school, whose memory will undoubtedly survive only as a historical curiosity, has at least rendered the service of having obliged its opponents, that is to say the vast majority of mathematicians, to clarify their own positions and to become more consciously aware of the reasons (whether logical or sentimental) for their confidence in mathematics (Bourbaki 1968, p. 336)”.

  78. Zalamea (2012), Zalamea (2011) identifies contemporary mathematics with all the accumulated knowledge from the middle of the twentieth century onwards.

  79. ZF is understood in the work of Zalamea as ZFC.

  80. Zalamea uses this expression to refer to these boundaries of knowledge apparently unmanageable: non-elementary classes, non-commutative geometry, non-linear logic, etc.

  81. Grothendieck’s contributions are especially studied by Zalamea (2012) in the fourth chapter.

  82. Varias invenciones de la época provienen del “espíritu” bourbakista encarnado en sus exponentes singulares: el “buen” acoplamiento de la información local y global sobre una estructura, que originó la teoría de haces (Leray, Cartan); el tránsito natural entre las estructuras de la geometría proyectiva y de la geometría analítica, que impulsó el GAGA de haces sobre variedades algebraicas complejas (Serre); la jerarquización y la clasificación de álgebras diferenciales, que dio lugar a los teoremas de representación de grupos y algebras de Lie (Chevalley, Borel); la conjugación de técnicas de teoría de números y de cálculo armónico abstracto, que concluyó en los adeles e ideles (Weil); la búsqueda de entornos generales para el análisis funcional, que llevó a la paracompacidad (Dieudonné); la eliminación de obstrucciones singulares a favor de convoluciones globales, que propulsó la teoría de distribuciones (Schwartz), etc. (Zalamea 2011, p. 115).

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Acknowledgments

We want to express our thanks to Guillermo Ortiz (Universidad del Valle) for his valuable contributions and comments on the mathematical content of the document; also, Fernando Rambla and Carmen Pérez (Universidad de Cádiz) for their observations on the content and English version; finally, we would like to thank the Vice President of Research of the Universidad del Valle for the linguistic revision. This paper is drawn from the doctoral studies of Maribel Anacona at the Universidad de Cádiz and is part of the project (code 1106-521-28616) funded by COLCIENCIAS and the Universidad del Valle. Special thanks are extended to the anonymous reviewers for their valuable contributions and suggestions to improve the quality of the manuscript.

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  1. Área de Educación Matemática, Instituto de Educación y Pedagogía, Universidad del Valle, Cali, Colombia

    Maribel Anacona

  2. Instituto de Educación y Pedagogía, Universidad del Valle, Cali, Colombia

    Luis Carlos Arboleda

  3. Departamento de Matemáticas, Universidad de Cádiz, Cádiz, Spain

    F. Javier Pérez-Fernández

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  1. Maribel Anacona
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Anacona, M., Arboleda, L.C. & Pérez-Fernández, F.J. On Bourbaki’s axiomatic system for set theory. Synthese 191, 4069–4098 (2014). https://doi.org/10.1007/s11229-014-0515-1

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