on bourbaki’s axiomatic system for set theory

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SyntheseAn International Journal forEpistemology, Methodology andPhilosophy of Science ISSN 0039-7857Volume 191Number 17 Synthese (2014) 191:4069-4098DOI 10.1007/s11229-014-0515-1

On Bourbaki’s axiomatic system for settheory

Maribel Anacona, Luis Carlos Arboleda& F. Javier Pérez-Fernández

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Synthese (2014) 191:4069–4098DOI 10.1007/s11229-014-0515-1

On Bourbaki’s axiomatic system for set theory

Maribel Anacona · Luis Carlos Arboleda ·F. Javier Pérez-Fernández

Received: 18 August 2013 / Accepted: 23 June 2014 / Published online: 26 July 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this paper we study the axiomatic system proposed by Bourbaki for theTheory of Sets in the Éléments de Mathématique. We begin by examining the roleplayed by the sign τ in the framework of its formal logical theory and then we showthat the system of axioms for set theory is equivalent to Zermelo–Fraenkel systemwith the axiom of choice but without the axiom of foundation. Moreover, we studyGrothendieck’s proposal of adding to Bourbaki’s system the axiom of universes forthe purpose of considering the theory of categories. In this regard, we make somehistorical and epistemological remarks that could explain the conservative attitude ofthe Group.

Keywords Bourbaki · Set theory · Axiomatic system · Grothendieck · Categorytheory

1 Introduction

The influence of Bourbaki’s structuralist program and its rigorous style of communi-cation have been widely recognized both in the development of mathematics as well

M. Anacona (B)Área de Educación Matemática, Instituto de Educación y Pedagogía, Universidad del Valle,Cali, Colombiae-mail: [email protected]

L. C. ArboledaInstituto de Educación y Pedagogía, Universidad del Valle, Cali, Colombiae-mail: [email protected]

F. J. Pérez-FernándezDepartamento de Matemáticas, Universidad de Cádiz, Cádiz, Spaine-mail: [email protected]

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as in mathematical educational processes since the mid-twentieth century. The textsformalized to the Bourbaki style became the standard form of communicating math-ematical thought and his study of structures still prevails in university mathematicalprograms. Despite this influence, some aspects of the axiomatic system, from whichBourbaki’s structuralist proposal arises, have not been amply discussed. However, themathematical literature reveals several papers which judge both the set of axioms andthe conservative attitude of the Group regarding the incorporation of the theory ofcategories.

This paper studies Bourbaki’s axiomatic system for set theory in the Éléments deMathématique and Grothendieck’s proposal to consider category theory in Bourbaki’sproject. Particularly, we analyze the role played by Hilbert’s τ operator in the frame-work of the logical theory of Bourbaki and then we prove that the axiomatic systemfor set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice andwithout the axiom of foundation. Likewise, Grothendieck’s proposal of adding theaxiom of universes to Bourbaki’s system as an alternative of categorical foundationwithout leaving set theory is analyzed. Finally, several epistemological explanationsare explored to understand their negative response.

2 Bourbaki’s axiomatic system

Bourbaki’s formal language considers three classes of signs:1 (i) logical signs: �, τ,∨,¬, (ii) letters: x , y, A, A’, A”,…, and (iii) specific signs: =, ∈ (for the theory of sets).Any succession of these signs will be denominated as a formula2 under the theory T .Grammatically correct formulas are called terms or relations. Terms represent objectsof the theory and relations represent statements about such objects; i.e. propositions.3

Two types of axioms are part of this theory: explicit and implicit axioms. Explicitaxioms are constituted by a certain number of given relations in a particular theory.Implicit axioms are relations made by the application of a scheme of the theory. Ascheme is a rule that is verified throughout mathematics which provides relations(implicit axioms) of a particular theory. The logic theory from Bourbaki has fourschemes:

S1. If A is a relation in T , the relation (A ∨ A) ⇒ A is an axiom of T .S2. If A and B are relations in T , the relation A ⇒ (A ∨ B) is an axiom of T .S3. If A and B are relations in T , the relation (A ∨ B) ⇒ (B ∨ A) is an axiom of T .

1 In this paper we employ the French version from 1970 (Bourbaki 1970) for the study of the axiomaticsystem. This edition is the one with the widest circulation. Hermann editorial published several reproductionsduring the decade of the 1970s. The different editions in French and English published by Springer between2004 and 2008 are textual reprints of this edition. However, for convenience, we employ the English versionfrom 1968 (Bourbaki 1968) to quote paragraphs of the Introduction or the Historical Notes since there areno changes in the two editions.2 Bourbaki utilizes the term assemblage that means an assembly or union (of signs). We have preferred theterm formula for being one of the most currently employed in Logic texts.3 A formula is a term if it starts with the sign τ or if it is reduced to a single letter. A formula is a relationif it starts with: ∨,¬, =, ∈. This notion is from the Polish logician Jan Lukasiewicz. Although Bourbakiemploys it at the start of the first chapter, he quickly abandons it.

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S4. If A, B, and C are relations in T , the relation (A ⇒ B) ⇒ ((C∨A) ⇒ (C∨B))

is an axiom of T .

These schemes provide infinite implicit axioms since they depend on the A, B and Crelations.4 In addition to these four schemes, quantified theory has a very particularscheme that, as we will see, leads to an axiom considered of crucial importance forthe development of mathematics.

2.1 The axiom of choice

Before referring to the axiom of choice in the Éléments de Mathématique, the operatorτ of Hilbert is briefly presented as historical background to Bourbaki’s proposal.

2.1.1 Hilbert’s τ

The operator τ was introduced for the first time by Hilbert in his article “The logicalfoundations of mathematics” (Hilbert 1923, p. 1140).5 He was convinced that thecritical remarks by the intuitionists regarding the infinite application of the principleof the third excluded and the unlimited use of existential quantifiers all had the sameorigin in the end: the axiom of choice (Zach 2004, p. 79). Hilbert then considered itas an indispensable principle for classical mathematical reasoning and incorporated itinto his logical system though the symbol τ and the transfinite axiom.

In this first version, the operator τx A(x) represents an object for which the proposi-tion A(x) is false, if such object exists.6 This means that τx A(x) is a counterexampleof A(x). The transfinite axiom states that: if there exists an object τx A(x) that pos-sesses property A(x) to a lesser extent, then any other object also possesses it. Aclarifying example is offered by Hilbert himself: “…let us take for A the predicate‘bribable’. Then τ(A) would be a definite man with such a firm sense of justice that,if he were to turn out to be bribable, then in fact all men whatsoever are bribable”(Hilbert 1923, p. 1141). Object τx A(x) represents the limit or frontier from whichall objects are endowed with property A(x). In symbols, the axiom is expressed as:A(τx A(x)) → A(x).

Later Hilbert and Bernays changed operator τx A(x) for operator εx A(x),7 whichrepresents an undetermined object for which A(x) is true, if A(x) is true for an object.With this new symbol, the meaning of counter-example was changed, but the logical

4 Note that Bourbaki employs the abbreviating symbol ⇒ instead of the formula ∨¬. These four schemescorrespond 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 asthose 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 in1923, and afterwards Russell and Whitehead in 1925, among others. The different semantics are analyzedby Wirth (2008, p. 289) when searching for a semantic of his own for Hilbert’s ε-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 Bernaysintroduced the ε-operator for the first time. Initially, with the same interpretation as the τ operator; then inthe notes attached to the typed copy of the course, it appeared with the final interpretation of the ε-operator.

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act that guarantees the election of the object εx A(x) is the same. In this final versionof the ε-operator the transfinite axiom is reformulated as:

A(x) → A(εx A(x))

This new angle was presented in his article, entitled “The foundations of mathe-matics”, where he termed it the logical ε-axiom when describing the formal system onwhich his proof theory was based (Hilbert 1927, p. 466).8 Here, εx A(x) represents anobject that satisfies property A(x) amongst all objects that verify it. Hilbert explainsthe role of this function through several observations: if proposition A(x) is satis-fied for one unique object, then εx A(x) is the object that satisfies such proposition.If proposition A(x) is satisfied for several objects, εx A(x) is one of the objects thatcomplies with the property. Therefore, ε carries out the role of a choice function.

Hilbert thus replied to the criticism of the intuitionists. The necessary presence ofthe axiom of choice is guaranteed through formal language. Through the symbol ε

and the transfinite axiom, it can be shown that unlimited quantification is admissiblein mathematics. For such an effect, it is important to keep in mind that in a first-orderlogical system, the ε-axiom is equivalent to:

(∃x)A(x) ↔ A(εx A(x))

From which we conclude that:

(∀x)A(x) ↔ A(εx¬A(x))

Through these definitions, all general statements formulated in the language of first-order logic (whose domain may be an infinite set) become statements about individuals.In this way, Hilbert substituted the questionable “infinitist” elements for the moremanageable “finite” ones (Slater 2009, p. 390).9 The ε-terms perform the roles ofideal elements in these definitions, whose addition to the theory of finite propositionsintroduces the powerful methods of infinite mathematics (Zach 2004, p. 80).

Boniface (2004) shows that Hilbert distinguishes between idea and ideal in a waythat is similar to Kant. The ideal elements for Hilbert constitute a concrete, individualrepresentation of an idea. That is, the formal representation of a rational conceptbeyond all sensory experience and that allows for completion and unification of theknowledge of understanding (Boniface 2004, p. 230). For example, the concept ofinfinity is an idea that becomes mathematically concrete through transfinite numbersof Cantor ω,ω +1, etc., which plays the role of ideal elements.

Therefore, ideal elements and ideal propositions are signs and formulas of formallanguage. Their introduction in mathematics is a response to the necessity of extending

8 Two years earlier, in his article “On the Infinite”, Hilbert presented it in his demonstration theory as avery 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 ε-terms to replace the existential and universal quantifiers is known as the epsilon substitutionmethod or the ε-substitution method.

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a domain of objects for solving problems without solution in the original domain.In general terms under Hilbert’s formalist program, the method of ideal elementsconsists of applying the genetic method to mathematical propositions to overcome therestrictions of the mathematics of content and thus pass onto a formal mathematicswhose body of reasoning is more ample and productive (Arboleda 2009, p. 254).10

Therefore, the ideal elements are objects of formal mathematics governed by the rulesof logic whose admission is only restricted to conditions of non-contradiction.

According to Bell (1993, p. 1), Hilbert introduces the ε-axiom not only to make thesimplicity of formalism more evident, but to reflect better his philosophical convictionthat mathematical practice requires the postulation of ideal elements. This is how idealelements have a logical function and at the same time answer the requirements ofmathematical practice.

Through ε-axiom the ideal object εx A(x) is obtained which verifies the A propertypreviously given. Thanks to the ε-axiom it is known that the meaning of εx A(x) iscompletely determined by the extension of A (Bell 1993, p. 6); i.e., by the class ofobjects that satisfy A. Hence, the equivalence of the properties is a condition underwhich objects that represent these properties are identical. The axiom that assures thisimplication was introduced by Ackermann in 1938 (Wirth 2008, p. 291). This is thesecond ε-axiom and is also known as the principle of extensionality for ideal objects:

∀x(A(x) ↔ B(x)) → εx A(x) = εx B(x)

This axiom states that what really characterizes an ε-term is the property it satisfies.This principle is essential in a first-order logical system; is required for introducethe concept of functional relation. Indeed, Wirth (2002, p. 303) considers that whenHilbert says that “ε performs the role of a function of choice”, the word “function”cannot have the connotation of a functional relation because this would imply thesecond ε-axiom, which cannot be derived from Hilbert’s axioms.

On the other hand, the second ε-axiom is required in the proof of the principle ofthe excluded middle. Bell (1993, pp. 4–8) shows that, having only the first ε-axiom,Markov’s principle and De Morgan’s laws can be derived from intuitionistic proposi-tional logic. Furthermore, it proves that when both axioms are added to intuitionisticpropositional logic the law of the excluded third is obtained. Through these ideal ele-ments the validity of the laws of classical logic is extended, particularly the validityof the principle of the excluded middle (Arboleda 2009, p. 254).

Hilbert and Bernays used the ε-terms to provide a basis for defining the existentialand universal quantifiers and to prove two meta-theorems of predicate calculus: the firstepsilon theorem, from which the development of semantic tables was introduced; andthe second epsilon theorem, that shows that logic with the ε is a conservative extensionof predicate calculus, providing a sort of completeness theorem for predicate calculus

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

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(Slater 2009; Moore 1982). Nevertheless, for Slater (2009) the strict definition of alogical system with ε, structured as a complete logical program comparable to predicatelogic, was first found in Bourbaki’s Éléments de Mathématique.

2.1.2 Bourbaki’s τ

Bourbaki introduced the sign τ in the same sense as Hilbert’s ε: If A is a relation ofa theory T and x is a letter, τx (A) represents a term of T , an object of the theory.This means that, if there exists an object which has the property A, τx (A) represents adistinguished object which has this property; if not, τx (A) represents an object aboutwhich nothing can be said (Bourbaki 1970, E I.18). Consider the following clarifyingexample. Let A be the property A: x ∈ R such that x2 − 6x + 8 = 0. Then τx (A)represents the number 2 or 4. Now, since [τx (A)]2 − 6[τx (A)] + 8 = 0, we can statethat the object τx (A) satisfies in turn the A property which is written as (τx (A) |x) A.That is, it is certain that there is an object τx (A) which has the property A. This isbriefly symbolized by the expression “there is an x such that A” or simply by “(∃x)A”.In this way Bourbaki introduces the existential quantifier.11 The universal quantifieris obtained by negation of the existential one.

However, it’s not enough to have a non-empty class of objects that verify a certainproperty to guarantee the existence of a particular object that verifies A. The guaranteeof such an existence is due to the scheme S5 of Bourbaki, the only axiomatic schemeof quantified theories.

S5. If A is a relation in T , if T is a term in T , and if x is a letter, then the relation(T|x)A ⇒ (∃x)A is an axiom.

The relation can be rewritten as:

(T|x)A ⇒ (τx (A)|x)A

The scheme S5 states that if there is an object T for which the relation A (whichexpresses a property of x) is true, then A is true for the object τx (A). It is thus evidentthat S5 corresponds to Hilbert’s ε-axiom.

In fact, the original sense of Hilbert’s idea is thus captured, since if relation A issatisfied by several objects, τx (A) represents one of the objects that verifies propertyA. Although τx (A) may be an object selected from those that comply with property

11 Mathias (2012) criticizes this definition and its possible interpretations. He considers that the τ-operatorperverts the natural order of mental acts. It specifically says: “to interpret ∃ you look for witnesses andmust first check whether a witness exist before you can pick an interpretation for τx (R); and then youdefine (∃x)R to mean that the witness witnesses that R(τx (R)): a strange way to justify what Bourbaki, ifpressed, would claim to be a meaningless text” (Mathias 2012, p. 13). This interpretation goes against theway 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 developmentin a direction that clarifies the interconnections between individual propositions and facts” (Arboleda 2009,p. 256).

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A, it cannot be affirmed that S5 is equivalent to the axiom of choice.12 As a matter offact, in Bourbaki’s logical system the axiom of choice is not found in any of its forms,although its effects are generated by using the symbol τ and the schemes that rule it(Slater 2009, p. 396).

Note that the notion of choice established by Hilbert through ε and retaken byBourbaki with τ is defined with respect to a predicate or any relation of the theory.This means that the starting point is composed by the class of objects which satisfiesthe given relation.13 The sets, as we will observe in the following section, are obtainedfrom very particular relations that comply with the condition of being collectivizing.This means that in Bourbaki’s logical system one can’t speak of functions in the strictsense, since their definition is part of set theory. Hence, it seems imprecise to attributethe role of a function of choice to τ and to S5. Its specter of influence is much broader.

In Bourbaki’s logical system there is room for a much more general concept thanthat of a function: that of functional relation. Bourbaki does this in the egalitariantheory: a theory where, given the sign =, the schemes S6 and S7 defined here areverified:

S6. Let x be a letter, let T and U be terms in T , and let R{x} be a relation in T ; thenthe relation (T = U) ⇒ (R{T} ⇔ R{U}) is an axiom.

This scheme states that if two objects are equal, they have the same properties. Thisis one of the axioms of equality postulated by Hilbert in his demonstration theory(Hilbert 1923, 1925, 1927).

S7. If R and S are relations in T and if x is a letter, then the relation (∀x)(R ⇔ S) ⇒τx (R) = τx (S) is an axiom.

This scheme corresponds to the second ε-axiom. As we have seen, it states that iftwo properties are equivalent, the privileged objects τx (R) and τx (S) (chosen fromthose that verify R and from those that verify S, if such objects exist) are equal.14 Theschemes S1 to S7 are the axiomatic schemes of the logical, quantified and egalitariantheories.

From the previous points we conclude that the axiomatic system for the Bourbaki’stheory of sets is a system without the axiom of choice. All the effects of the axiom areobtained through τ and the schemes that regulate it.15 The axiom of choice is foundin Bourbaki’s theory of sets as a theorem, and particularly as Zermelo’s theorem.16 In1950, Claude Chevalley refers to this logical consequence in the preliminary comments

12 See some of the relations (that imply equivalences) that can be established between the Hilbert’s trans-finite axiom, the ε-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 systemthat 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 Lawthat 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 τ to justify the generalized product of a non-emptyfamily, of non-empty sets, is non-empty.16 In the third chapter, based on the properties of ordered sets, of the sign τ and of S5, Bourbaki provesZermelo’s Theorem which states that all sets can be well-ordered.

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to the fifth version of the chapter, “Description of formal mathematics”, when heintroduces Hilbert’s ε-symbol for the first time to Bourbaki’s formal system:

An objection that can be made to the introduction of the symbol ε is that the axiomof choice becomes a theorem, so that there is no distinction between proofs withor without the axiom of choice. However, I must say that this “disadvantage”seems to me to be more of an advantage. It seems that the distinctions betweendemonstrations with or without the axiom of choice have no interest but in muchpoorer systems than that of Bourbaki. I remember on this subject that Gödelshowed that mathematics without the axiom of choice has an interpretation inwhich the axiom of choice is a theorem: that is, that a class of sets may be definedfor which all other axioms of set theory are true as well as the axiom of choice.17

Recall that in the decade of the 40s, although Gödel’s result of 1938 on the compati-bility between the system ZF and the axiom of choice was known, the advantages anddisadvantages of choosing a system where the results are proved with or without theaxiom of choice are still being discussed. With the incorporation of τ, the dilemma dis-appears since the axiom of choice is not found explicitly in any of its forms. Therefore,Bourbaki overcomes a discussion of the epoch through syntax.

2.2 Explicit axioms of the theory of sets

The theory of sets is a theory which contains the specific signs = and ∈ and the schemesS1 to S7. Bourbaki warns from the start, that the word “set” must be considered strictlyas a synonym for term; i.e., a well-formed formula which represents an object ofthe theory. There are neither attributed meanings nor particular definitions. They aresimply signs of a formal language.18 If T and U are terms, the well-formed formula∈ TU is a relation (called relation of membership) and it is frequently denoted asT ∈ U.

From the definition of inclusion, Bourbaki presents the first explicit axiom of thetheory of sets (Bourbaki 1970, E II.3):

A1. Axiom of Extensionality

(∀x)(∀y)((x ⊂ y et y ⊂ x) ⇒ (x = y))

Although this axiom states when two sets are equal, there are still no sets. The sets areintroduced thanks to the concept of collectivizing relation. Let R be a relation and let

17 Une objection que l’on peut faire à l’introduction du symbole ε, c’est que l’axiome de choix devientun 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 distinctionsentre démonstrations avec ou sans l’axiome de choix n’ont d’intérêt que dans des systèmes beaucoup pluspauvres que celui de Bourbaki. Je rappelle à ce sujet que Gödel a montré que la mathématique sans axiomede choix possède une interprétation dans laquelle l’axiome du choix est un théorème: c’est-à-dire qu’on ypeut définir une classe d’ensembles pour lesquels tous les autres axiomes de la théorie des ensembles sontvrais 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 thework of Bourbaki into a computer. In (Grimm 2013a) is the implementation of the first two chapters of thebook Set Theory, and in (Grimm 2013b) the implementation of the third chapter.

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x be a letter of a theory T . If y denotes a letter different from x which does not appearin R, then the relation (∃y)(∀x)((x ∈ y) ⇔ R) is denoted by Collx R. If Collx R is atheorem of T , R is said to be collectivizing in x in T . If this happens, an auxiliaryconstant a can be introduced to obtain the relation (∀x)((x ∈ a) ⇔ R). Therefore, tosay that R is collectivizing in x is to say that there exists a set a such that the objectsx which possess the property R are precisely the elements of a.

In this manner, Bourbaki does not require the axiom of the empty set to guaranteethe existence of a first set.19 If a relation R is collectivizing, a set is obtained and itselements are those that verify the relation R. This restriction on relations also preventsthe formation of strange “sets”, such as “the set of sets that don’t belong to themselves”.Specifically, Bourbaki shows that the relation x �∈ x is not collectivizing in x .20 Inothers words, notCollx (x �∈ x) is a theorem in T . Hence, the set A = {x |x �∈ x} cannotbe formed. In this way, Russell’s paradox is avoided.21

Bourbaki proves that if R is collectivizing in x , the relation (∀x)((x ∈ y) ⇔ R)

is functional in y (criterion C49). This means that there only exists a set y such that(∀x)((x ∈ y) ⇔ R). This set is represented by the term τy(∀x)((x ∈ y) ⇔ R),which is substituted by the symbol {x |R}. From axiom A1, Bourbaki later proves thecriterion C50 which asserts that if R and S are collectivizing, the relation (∀x)(R ⇔ S)

is equivalent to {x |R} = {x |S}.22 In this manner, the elements of the sets are completelydetermined by the collectivizing relations.

The other explicit axioms found in the second chapter are (Bourbaki 1970, E II.4-30):

A2 . Axiom of the set of two elements

(∀x)(∀y)Collz(z = x o z = y)

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 �∈ x is collectivizing, then (∃y)(∀x)((x ∈ y) ⇔ x �∈ x). Let a bean auxiliary constant, then we have: (∀x)(x ∈ a ⇔ x �∈ x). Therefore, the relation “a ∈ a ⇔ a �∈ a” istrue, which is a contradiction.21 Mathias (2012, p. 13) says that this “notation is highly misleading in that all classes which are notsets have become «equal »”. This would mean that Bourbaki’s formal system is inadequate to describe settheory. Let’s consider Mathias’s argument in detail. Let R and S be two relations of the T theory such thatR: x �∈ x and S: x = x . Bourbaki proves that R is not collectivizing and analogously demonstrates that S isnot either. That is, notCollx R and notCollx S are theorems in T . If we write CR : (∀x)((x ∈ y) ⇔ R) andCS : (∀x)((x ∈ y) ⇔ S), as Mathias proposes, we have that ¬(∃y)CR and ¬(∃y)CS are theorems of T .In other words, (∀y)¬CR and (∀y)¬CS are theorems of T . These relationships can be interpreted as, “forall y, CR is false” and “for all y, CS is false”. From these two theorems Mathias concludes that CR ⇔ CS.Mathias continues and employs the S7 scheme to prove that τyCR = τyCS. This allows him to concludethat the privileged objects are identical. Or that classes A = {x |x �∈ x} and B = {x |x = x} would be equal;this would become a serious limitation of the formal language of Bourbaki.For us, once one has that (∀y)¬CR and (∀y)¬CS are theorems of T , one can only deduce that theyare logically equivalent; i.e., (∀y)¬CR ⇔ (∀y)¬CS. From this equivalence it cannot be deduced thatCR ⇔ CS.22 Observe that the implication (∀x)(R ⇔ S) ⇒ {x |R} = {x |S} corresponds to a particular version of S7scheme, where R and S are collectivizing relations and the privileged objects τx (R) and τx (S) correspondto the sets {x |R} and {x |S}.

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A3 . Axiom of the set of parts

(∀X)CollY(Y ⊂ X)

The fourth and last axiom23 of the theory is incorporated in the third chapter of the bookdedicated to ordered sets, cardinals and integer numbers (Bourbaki 1970, EIII.45):

A4. Axiom of the infinite set

There exists an infinite set

Through this axiom, he introduces the set N and the principles of induction and recur-sion, which guarantee the arithmetic of the naturals.

2.3 The scheme of selection and union

Let us look in detail at the scheme of selection and union and the implicit axioms itsupplies to the theory of sets (Bourbaki 1970, E II.4):

S8. Let R be a relation, let x and y be distinct letters, and let X and Y be lettersdistinct from x and y which do not appear in R. Then the relation(∀y)(∃X)(∀x)(R ⇒ (x ∈ X)) ⇒ (∀Y)Collx ((∃y)((y ∈ Y) ∧ R)) is an axiom.

Intuitively, this means that for every object y there exists a set X (which may dependon y) such that the objects x which are in the relation R with the given object y areelements of X (but not necessarily the whole of X). The scheme asserts that if this isthe case and if Y is any given set, then there exists a set whose elements are exactlythe objects x which are in the relation R with at least one object y of the set Y.

Here we demonstrate that the scheme provides: the axiom of unions and the axiomof replacement, and that it is required to prove the “axiom” of separation.

(i) Let R be the relation: y R x if and only if x ∈ y. Observe that this relation satisfiesthe premise of the scheme S8: (∀y)(∃X)(∀x)(x ∈ y ⇒ x ∈ X). Indeed, for everyobject y there exists a set X (in this case X = y), such that if x ∈ y then obviouslywe have that x ∈ X. Therefore, it is deduced from S8 that (∀Y)(∃W)(∀x)(x ∈W ⇔ (∃y)(y ∈ Y ∧ x ∈ y)). This means that if Y is a set, then there exists a setwhose elements are the elements of the elements of Y. This set W is called theunion of Y and is denoted by ∪Y. Hence, we have the axiom of unions.

(ii) Let R(y, x) be a functional relation in x . Therefore, there only exists an xsuch that R(y, x). When replacing R(y, x) by R in the premise, it follows that

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

(∀x)(∀x ′)(∀y)(∀y′)((x, y) = (x ′, y′) ⇒ (x = x ′ ∧ y = y′))

In the French version from 1970, Bourbaki defines the ordered pair (x, y) as the set {{x}, {x, y}} anddemonstrates that the relation (x, y) = (x ′, y′) is equivalent to «x = x’ ∧ y = y’ » (Bourbaki 1970, EII.7). Thus, the axiom becomes a proposition of the theory.

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(∀y)(∃X)(∀x)(R(y, x) ⇒ x ∈ X). The premise is true: indeed, for every y thereis a set X (in this case X = {x}), such that if x is in the relation R with y, thenx ∈ {x} = X. Hence, from the scheme of selection and union it can be deducedthat (∀Y)(∃W)(∀x)(x ∈ W ⇔ (∃y)(y ∈ Y ∧ R(x, y))). This means that if Y isany set, there exists a set W whose elements are precisely the objects x which arein the relation R(x, y) for some y ∈ Y. In other words, if Y is a set, the imageof Y under the functional relation R(y, x) is also a set. Therefore, we have theaxiom of replacement.

(iii) Bourbaki proves in criterion C51 that the relation “P and x ∈ A” is collectivizingin x , where P is a given relation, A is a given set and x a letter which does notappear in A (Bourbaki 1970, E II.5). The proof requires of S5 and S8. Underthis criterion, Bourbaki guarantees the existence of a set whose elements areprecisely the elements of A which verify the property P, i.e., the existence of{x : x ∈ A ∧ P} or simply {x ∈ A : P}, a criterion that corresponds to the axiomof separation formulated by Zermelo.

In conclusion, Bourbaki’s axiomatic system “contains” the following axioms:

(A) In explicit manner:A1. Axiom of extensionalityA2. Axiom of the set of two elementsA3. Axiom of the set of subsetsA4. Axiom of infinity

(B) In implicit manner:• Axiom of unions• Axiom of replacement

(C) As a deductive criterion, the axiom of separation.(D) As a theorem (of Zermelo), the axiom of choice.

2.4 Equivalence between Bourbaki’s system and ZFC− system

The system ZFC− contains the seven axioms proposed by Zermelo (1908, pp. 193–201) and the axiom of replacement introduced by Fraenkel and Skolem (1922, p. 297).The axioms of Zermelo are: axiom of extensionality, of elementary sets, of separation,of power sets, of union, of choice and of infinity.24 In modern literature it is common toidentify these axioms and the axiom of foundation or regularity as the system ZFC.25

This last axiom was included years later by Zermelo in his “extended ZF system”(Zermelo 1930, p. 1220). Following Kunen (1980, p. xvi), we use the notation ZFC−to indicate the non-consideration of the axiom of foundation in our list.26

24 A complete historical analysis of the emergence and consolidation of various axiom systems for set the-ory, including the Zermelo–Fraenkel system, is found in (Ferreirós 2007, pp. 297–392). See also (Ferreirós1992).25 The notation ZFC is used to make explicit the inclusion of the axiom of choice; since in the “extended ZFsystem”, Zermelo does not include the axiom of choice because he considers it a universal logical principleand 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.

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In the previous section, we have verified that the axioms of ZFC− are in Bourbakias axioms or as propositions; therefore, all that is left is to show that the axioms ofBourbaki and the scheme S8 are in ZFC−.27 The explicit axioms belong in an evidentmanner, it is only necessary to make evident that S8 is verified in ZFC−.

We suppose then that the hypothesis of the scheme of selection and union is verifiedin ZFC−: (∀y)(∃X)(∀x)(R ⇒ (x ∈ X)). This means that for every object y there existsa set X, which has amongst its elements, all the elements x which are in the relationR with the given object y. Then, in each set X, a set M is formed by those elements xwhich are in the relation R with y. This is possible by the axiom of separation.

Let us observe that this set is unique. We suppose that there exists another set N suchthat N = {x ∈ X : R}. Now, for every x , we have: x ∈ M ↔ x ∈ X ∧ R ↔ x ∈ N.Hence, M = N. Since every y is in relation with a single M, we have a functionalrelation between the objects y and the sets M. Now, using the axiom of replacement,we have that for every set Y there exists the set M* whose elements are precisely theelements M which are in the relation R with y, for some y ∈ Y. From the axiom ofthe unions, we have that there exists a set W whose elements are the elements of theelements of M*; i.e., a set W of those elements x which are in the relation R with y,for some y ∈ Y.

In conclusion we have that for every set Y, there exists a set W whose elementsare precisely the objects x which are in the relation R with at least one object yof the set Y. This is to say that (∀Y)(∃W)(∀x)(x ∈ W ⇔ (∃y)((y ∈ Y) ∧ R));i.e., (∀Y)Collx((∃y)((y ∈ Y) ∧ R)). Thus, we have the conclusion of the scheme ofselection and reunion. It has been proven that S8 is verified as a proposition in ZFC−.This concludes the proof that the axiomatic systems are equivalent.

It is relevant to say that Bourbaki does not incorporate the axiom of foundation inhis proposal. Although this axiom can be considered necessary for the developmentof set theory,28 the theorems of analysis, algebra and topology, can be proven withoutit. Bourbaki decided to include in his work only the essential for the development ofhis proposal. If the axioms of ZFC− satisfy the necessary conditions on which to basehis proposal, Bourbaki does not require additional axioms.

2.5 Brief history of the axioms of set theory in the Bourbaki group

Although the axioms of the final version of the set theory book correspond to theZFC− system, it is important to mention that this was not always the case. As Mathias(1992) highlighted, the group had serious difficulties and controversial positions withrespect to logic and set theory. In order to show broadly this difficult and the extensiveprocess of conceptual elaboration, some of the axioms and schemes are presented

27 Recall that two axiomatic systems T and T’ are equivalent if the axioms of T are axioms or theorems ofT’ and vice-versa.28 There are consistent axiomatic theories that do not include the axiom of foundation and therefore theyare 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 formmembership chains with infinite descent. In its proposal is incorporated the anti-foundation axiom.

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in the following that were previously adopted by Bourbaki before it reached a finalversion in 1954.

The initial version of the chapter on set theory was presented by Henri Cartan inDecember, 1935. In it he presented the basic themes of a naïve set theory as had beenagreed to six months earlier in the first plenary meeting at Besse (Bourbaki 1935).In the second plenary meeting at the Escorial in Chançay in September 1936, thegroup recognized that it must start by enunciating the axioms of set theory, but at thattime there were “insuperable difficulties that could not be overcome which confrontedthe formulation of such axioms”. Therefore, they would provisionally attribute “tothese words the meaning that they had in the common language (Bourbaki 1936,p. 2)”. Despite the difficulties likely related to the lack of clarity on the degree offormality that set theory would have, the group decided at the meeting of March, 1937to present set theory starting with the logic of predicates. This implies the introductionof a grammatical and logical preamble that includes definitions, propositions, logicalconnectors and quantifiers (Bourbaki 1937, p. 3).

Between 1939 and 1942 they had few meetings and with a much reduced numberof members since the majority of them were mobilized for war.29 The group funda-mentally dedicated itself to drawing up the topology, algebra and integration whilethe book of set theory remained “provisionally suspended” (Bourbaki 1942, p. 1). Theonly official expression of the group members regarding the subject of logic and settheory is found in the articles of Dieudonné (1939) and Cartan (1943).

Dieudonné (1939) in his article “Les méthodes axiomatiques modernes et les fonde-ments des mathématiques” argues in favor of Hilbert’s formalist program, as a wayout of the crisis of fundamentals.30 In his presentation of the components of a formallogical system, he refers specifically to the logical schemes, which have the conno-tation of a rule, whether syntactic or deductive. The topic of set theory, however, isnot approached. Later, in the article “Sur le fondement logique des mathématiques”,Cartan (1943) intends to sustain the notion that mathematics, in general, are based onlogic. The axioms that Cartan presents at the beginning of his article are the sevenaxioms of the first proposal of Zermelo (1908). Later he describes the componentsof first-order logic. In the end he returns to the axioms of Zermelo to write themin the formal language described and refers to them as the axioms of the Zermelo–Fraenkel system, but in his list, the axiom of replacement is absent despite him writinga footnote that says that Fraenkel’s contributions have been incorporated.31 Cartancloses this presentation by pointing out that “all of mathematics as it exists today canbe expressed in the system just outlined”.32 This statement is clearly open to criti-

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 notexplicitly mention the work of Gödel and fails to note its importance to the foundations of mathematics. Heoffers several explanations (of the logical, sociological and mathematical order, among others) about theconservative 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 venonsd’esquisser. (Cartan 1943, p. 10).

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cism considering the importance that the axiom of replacement bears for the proofsof transfinite recursion which guarantee the proper definition of ordinal and cardinalarithmetic.

But this murky perspective was not quickly overcome. The list of axioms subse-quently submitted by Dieudonné, in the third and fourth versions of the chapter on settheory is even more reprehensible than that of Cartan (1943). In the third version ofthe second chapter, entitled “Abstract set theory”, Dieudonné considers set theory tobe composed of three types of primitive relations: relations of equality, belonging andpairs.33 The relations of equality as well as in the “Principles of mathematical logic”from 1928 are not a part of logic but of set theory (Hilbert and Ackermann 1950, p.68). The theory of belonging is subordinated to the theory of equality and the theoryof pairs is subordinated to the theory of belonging. These same axioms appear in thefourth version of the chapter on set theory. However, some changes occur: the theoryof relations of belonging remains subordinated to the theory of pairs; and set theorynow belongs both to egalitarian theory as well as to pair theory.34 Therefore, axiomsand schemes are organized in a different order.

In egalitarian theory, the axioms and the scheme of the previous version are main-tained:

E(I) : ∀x(x = x)

S(I) : (∀. . .)((x = y) → (R{x, y, x} → R{x, y, y}))

In pair theory, subordinated to egalitarian theory, the following three axioms are con-sidered:

E(II) : (∀x)(∀y)(∀z)(z = (x, y))

E(III) : (∀x)(∀y)(∀z)(∀z′)((z = (x, y) ∧ z′ = (x, y)) → z = z′)E(IV) : (∀x)(∀x ′)(∀y)(∀y′)(((x, y) = (x ′, y′) → x = x ′ ∧ y = y′))

And in set theory, subordinated to egalitarian theory and pair theory, there are fouraxioms and a scheme:

E(V) : (∀x)(∀y)(x ⊂ y ∧ y ⊂ x → x = y)

S(II) : (∀...)(∀ensE)(∃X)(∀x)((x ∈ X ↔ x ∈ E ∧ R))

E(VI) : (∀ensX)(∃Y)(∀Z)((Z ⊂ X ↔ Z ∈ Y))

E(VII) : (∀ensX)(∀ensY)(∃W)(∀z)((∃x)(∃y)(x ∈ X ∧ y ∈ Y ∧ z = (x, y) ↔ z ∈ W))

E(VIII) : Zermelo′s axiom.

33 Ensembles Chap. 1 (état 3) Chap. II (état 3). Archives de l’Association des Collaborateurs de NicolasBourbaki, R057_iecnr_066.34 Livre I. Théorie des ensembles. Ch.II (Etat 3 ou 4) Théorie des ensembles abstraits. Archives del’Association des Collaborateurs de Nicolas Bourbaki, R063_iecnr_072.

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Precisely, these axioms and schemes distributed in the same three groups of primitiverelations are the ones that Bourbaki presents in his article on “Foundations of math-ematics for the working mathematician” (Bourbaki 1949).35 In this article Bourbakipresents a general view of the central aspects of what he considered his axiomaticsystem. Zermelo’s axiom of choice is enunciated without saying of what it consistedand clarifying that its correct formulation requires the development of the theory basedon the previously presented axioms (Bourbaki 1949, p. 8). The axiom of infinity ismentioned very informally at the end of the article. This list of axioms for set theoryis very weak compared with the ZF system: not only is the axiom of replacementmissing but also theaxiom of a set with two elements and the axiom of unions.36 It isstill surprising that the proposal by Cartan (1943) was not considered, whose list ofaxioms was more complete and that at the beginning of the decade of the 1950s theseconceptual deficiencies were presented.

In the fifth version of Chapter I presented by Chevalley in July of 1950, significantchanges can be observed. The chapter starts with the presentation of the signs of theformal system, amongst which Hilbert’s sign ε appears for the first time (Bourbaki1950a). The egalitarian theory is considered as part of logic and not of set theory.Although the S4 scheme appears corresponding to Hilbert’s transfinite axiom, theS6 and S7 schemes of the final version are not considered, and the definition ofKuratowski is used to introduce the ordered pair. Two months afterwards, Chevalleyhad a fifth version of the chapter on set theory (Bourbaki 1950b). For Chevalley, settheory is a theory without axioms; that is, a generic theory underlying all mathematicaldisciplines. Thus, the mentioning of these is completely nominal. The same axiomsappear in the final version, but presented in terms of schemes.

The wording for the sixth version of the first two chapters of the Set Theory bookwas written by Dixmier. Here, set theory loses the absolute role of the fifth versionand is again considered a theory with axioms. In the first chapter appear the fouraxiomatic schemes (of Russell and Whitehead) and the S5, S6 and S7 axioms of thefinal version (Bourbaki 1951b). In the second chapter, Dixmier introduces an additionalaxiom,37 the axiom of pairs, which appears in the first edition of the book, but latereditions consider Kuratowski’s definition to introduce the ordered pair. The schemeof selection and reunion corresponds to the S8 scheme. This version is very close tothe final version.

In January of 1953, Dieudonné presented the eighth and definitive version of sec-tions §1, §2, §4 and §5 of chapter II without changes in relation to the axioms andschemes of set theory (Bourbaki 1953b). Sections §3 and §6 were delegated to Dixmier.

35 Conference presented by André Weil in the name of Bourbaki at the eleventh meeting of the Associationof 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’stheory and not that of Zermelo–Fraenkel. But, strictly speaking, it does not even correspond to the theory ofZermelo. In another article, Mathias (2010) shows that a model of this system exists, called Bou49 wherethe set axiom with two elements is false.37 Livre I Théorie des ensembles Chapitre II. (état 6) Archives de l’Association des Collaborateurs deNicolas Bourbaki, R160_nbr_062.

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Thus in 1954 Bourbaki finally published Chapters I and II of the book on set theoryfor the first time.38

In this brief history, we seek to appreciate the group efforts toward reaching a defini-tive version. While it is difficult to understand that the group had mistaken conceptionsover axiomatic set theory by the end of the decade of the forties, it is necessary torecognize the laborious search and individual efforts that allowed formulate finally anaxiomatic system equivalent to ZFC−. In this history, academic spirit is highlightedalong with the work capacity of a team that always confronted new intellectual chal-lenges. But just at this time, the group had to confront a theme which jeopardized allthe work performed and questioned set theory as the only source of mathematics: theemergence of the theory of categories.

3 Bourbaki’s system and the theory of categories

The non-acceptance of the theory of categories by Bourbaki is a complex subjectthat Corry (1992, 2004) and Krömer (2006, 2007) have particularly studied. Corryfocuses his analysis on the opposition between the notion of structure and categorytheory. Particularly, it shows Bourbaki’s ambivalence regarding the incorporation ofthe language of category theory in his project.39 Besides this topic, Krömer studies thedifficulties of set theoretical foundations of category theory and the difficult personalrelations among members of the group, which are deemed decisive to his analysis.

The consideration of the theory of categories became a true dilemma for the group.Bourbaki witnessed as this proposal was born from several of its members (among themSamuel Eilenberg and Alexandre Grothendieck).40 He knew well the generalizingpower of categorical language; however, the incorporation of this theory into his projectmeant giving up on his structuralist proposal of mathematical organization.

We stop at a very specific issue of this controversy that is closely related to thecentral topic of this work. Following Krömer’s guidelines (2006), we checked theGroup’s option of adding a new axiom to his axiomatic system for the purpose ofpermitting the entry of the theory of categories.

38 The publication was made 14 years after the first edition of the Fascicule de Résultat and after havingpublished 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 Topologicalvector spaces and the first four chapters of the book on Integration. The third chapter on Ordered Sets waspublished 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 individualdocuments of some of its members. Examples of this ambivalence are that the group had decided to develop afifth chapter of set theory on categories and functors; however, it was never published. Similarly, a previouslyagreed 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 grouptheory 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 AndréWeil 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.

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3.1 Assurance in the system selected

The system selected constitutes an axiomatic safe ground for the Bourbaki group.Safety is related to the conviction that this theory bears no contradictions. In La TribuNo. 15 of 1948, Bourbaki states that “if a contradiction is found one day, the axiomsare modified accordingly and it will certainly be a source of great progress”.41 Thisconfidence in the non-contradiction is identified by Krömer (2006, 2007) as “the«hypothetic-deductive» position”, and originated in the relative consistency of ZFCand in the lack of contradictions during the several decades of its development.42 Bour-baki asserts the following in his well-known article on fundamentals for mathematicalwork:

Absence of contradiction, in mathematics as a whole or in any given branchof it, thus appears as an empirical fact, rather than as a metaphysical principle.The more a given branch has been developed, the less likely it becomes thatcontradictions may be met with in its further development. (Bourbaki 1949,p. 3)

This is the philosophy of Bourbaki’s practice. Adhesion to this or that epistemo-logical principle or to a specific presentation of the theory comes from the empiricalverification of its convenience in mathematical practice. This practice philosophy isthe determinant factor in the selection of research strategies more than the logical orphilosophical considerations of such principle.43 The absence of contradiction throughthe years is an empirical fact that generates complete trust in the set approach as alogical device on which to base the program. This trust is based on the observation ofits sound use in different fields. In the Introduction of the book on set theory, publishedin 1954, Bourbaki states:

Nevertheless, during the half-century since the axioms of this theory were firstprecisely formulated, these axioms have been applied to draw conclusions in themost diverse branches of mathematics without leading to a contradiction, so thatwe have grounds for hope that no contradiction will ever arise […].To sum up,we believe that mathematics is destined to survive, and that the essential partsof this majestic edifice will never collapse as a result of the sudden appearanceof a contradiction […]; but already for two thousand five hundred years math-ematicians have been correcting their errors to the consequent enrichment andnot impoverishment of their science; and this gives them the right to face thefuture with serenity. (Bourbaki 1968, p. 13)

This logical confidence was threatened by the emergence of category theory, whichrequires a different axiomatic system for its foundation.

41 Si une contradiction se rencontre un jour, on modifiera les axiomes en conséquence, et ce sera sans douteune source de grands progrès. (Bourbaki 1948, p. 6).42 The relative consistency of ZFC system implies the relative consistency of ZFC−.43 Another significant historical example is the choice of the topology of neighborhoods. For Bourbakiwas the axiomatic more convenient and productive at the time to generalize the theory of functions andfunctional analysis in abstract spaces (Arboleda 2012).

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3.2 The foundation problem of category theory

With the birth of the theory of categories the question of what is the most appropriatefoundational framework simultaneously arises. A standard problem to provide a foun-dation for category theory, from a set theoretical framework, is related to the size of thenew objects.44 The new collection of objects and morphisms overflows the concept ofsets. For example, the “category of all sets” cannot be a set because if the collection ofall sets V was a set, then the collection of all its subsets ℘(V), would be a set includedin V, contradicting Cantor’s theorem.

Eilenberg and MacLane in their celebrated and groundbreaking article on a “Generaltheory of natural equivalences” (1945), while proposing the basic concepts of categorytheory, they stated their early reflections noted in relation to the problem of foundation.They were convinced that it was the same problem that led to the paradoxes of thenaive set theory “the difficulties and antinomies here involved are exactly those ofordinary intuitive Mengenlehre; no essentially new paradoxes are apparently involved”(Eilenberg and MacLane 1945, p. 246). Similarly, they considered category theorycould be developed in a set theoretical framework: “Any rigorous foundation capableof supporting the ordinary theory of classes would equally well support our theory”(Eilenberg and MacLane, p. 246). They even thought the problem of foundation wasindependent from the development of category theory: “Hence we have chosen toadopt the intuitive standpoint, leaving the reader free to insert whatever type of logicalfoundation (or absence thereof) he may prefer” (Eilenberg and MacLane, p. 246).

Starting with this conception, they presented some technical solutions related tothe restriction of the domain.45 Finally they proposed the system of von Neumann–Bernays–Gödel (NBG) as a solution to the problem of foundation:

One can also choose a set of axioms for classes as in the Fraenkel-von Neumann-Bernays system. A category is then any (legitimate) class in the sense of thisaxiomatic. Another device would be that of restricting the cardinal number,considering the category of all denumerable groups, of all groups of cardinal atmost the cardinal of the continuum, and so on. The subsequent developmentsmay be suitable interpreted under any one of these viewpoints. (Eilenberg andMacLane 1945, p. 247).

The NBG system is a foundational conceptual framework that ensures the con-ceptual legitimacy of categories like “the category of all sets” or “the category ofall denumerable groups”. This response, as expressed by Marquis (2009, p. 53) is

44 This issue is broadly dealt with by Feferman (2013), Malatesta (2011) and Marquis (1995, 2009), amongothers. Particularly, Marquis (1995) examines the relationships between category theory and set theory andsuggests 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 Eilenbergand MacLane in this matter. Among them, the consideration is of a category of groups rather than thecategory 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 isomorphimsbetween groups of different types.

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satisfactory for the specific field of applications in which Eilenberg and MacLaneworked.

Recall that in NBG primitive notion is that of class. Classes are in general terms“collections of objects”. Sets are special types of classes: a set is a class that is containedin another class. The classes that are not sets are called proper classes. The notion declass is used to interpret the notion of size arising in category theory. Therefore, in thissystem, a category is any legitimate class of the system. Categories are classified aslarge and small. A large category is a category whose class of morphisms is a properclass of NBG. Otherwise, the category is said to be small.

An advantage of NBG with respect to ZFC is that the classes can be quantified,which is impossible in ZFC. Another interesting aspect of NBG is the possibility ofadopting a form of global choice.46 As shown by Shulman (2008, p. 13), global choiceis fundamental in category theory for choosing representatives of universal construc-tions in large categories.47 However, fundamental issues of category theory cannot beresolved from the foundational framework provided by NBG. These limitations willbe raised in Sect. 3.4.

3.3 The proposal of axiomatic change

The proposal of changing the system ZFC by the system NBG arose at the Congressof February 1951.48 This attempt was registered in La Tribu No. 24:

Some people have wanted to “gödeliser” in order to treat more convenientlythings like the axiomatic homology or universal applications, but they wonderif classes and ε without restrictions, taken together, will not combine. Finally,Cartan is wary of a “closed” system where everything is given at the beginning.49

Let us remember that for the first time Hilbert’s ε was included in July of 1950. It wasa natural thing to be concerned for the functionality of this operator in the new theory.Furthermore, the equivalence between the consistency of ZFC and the consistencyNBG (with global choice) was only shown several years afterwards.50 Despite this,

46 This is an immediate consequence of the axioms. The axioms of NBG are: (i) Axioms in common withZFC: 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 ϕ(x), without quantifiers over classes, there existsthe class {x : ϕ(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 A×A → 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’homologieaxiomatique ou les applications universelles, mais se demandent si classes et ε sans restrictions, mis ensem-ble, 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 thatNBG (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.

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the subject of a foundational framework for the theory of categories continued to bediscussed in the following years. In La Tribu No. 39, the following was recorded inthe minutes of the second congress of 1956:51

Cartier proposes a meta-mathematical method to introduce [the categories andfunctors] without modifying our logic system. But this system is rejected becauseit turns away from the viewpoint of the extension. Therefore, it is decided that itis better to extend the system to make the categories enter there; at first glanceGödel’s system seems to agree.52

For Krömer (2006, p. 147), the expression “turn resolutely on the point of view ofthe extension” would be referring to the possibility of introducing objects (categoriesand functors) that cannot be submitted to the usual operations of set theory. Therefore,to be outside of set theory, it would mean to be outside of mathematics. In this regardone can understand the proposal by Cartier to formulate a meta-mathematical method:a method that does not have set theory as a foundational framework. Therefore, thegroup rejected this proposal and the system NBG remained an option to explore. Inthe historical notes of the book on sets theory, the following is recorded:

In favour of the Zermelo–Fraenkel system it can be said that it limits itself toformulating prohibitions which do no more than sanction current practice in theapplications of the notion of set to various mathematical theories. The systemsof von Newmann and Gödel are more remote from usual conceptions. On theother hand, we cannot exclude the possibility that it may be easier to insert thebasis of some mathematical theories into the framework of such systems thaninto the more rigid framework of the Zermelo–Fraenkel system. (Bourbaki 1968,pp. 330–331)

The group was aware of the advantages of ZFC and it recognized the limitationsit had to provide a basis for the theory of categories; therefore, the system NBG wasan alternative that continued being analyzed. This option was emphasized by logicianDaniel Lacombe, who had been consulted by Serre and Dixmier. Several possibilitiesfor foundations were proposed by Lacombe,53 among these, a more profound distinc-tion between class and set (Krömer 2007, p. 256). But, as we shall see below, thisoption was rejected by Grothendieck who proposed a different alternative.

Footnote 50 continuedFurthermore, it was known that every theorem of ZFC is a theorem of NBG. Therefore, it can be prove thatZFC 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, theonly 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 modifiernotre 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 lesystème 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 thecollection of files of Desaltre in Nancy. This paper was particularly consulted by Krömer.

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3.4 Grothendieck’s Proposal

Fundamental elements to understand the proposal of Alexandre Grothendieck can befound in the article “On the formalization of categories and functors”.54 Refer to thefollowing quote extracted from Krömer (2007):

Thus, Lacombe’s ‘solution’ seems to be totally inadequate. […] if one wantsto introduce a new category of mathematical objects, the classes which wouldbe too large ‘sets’ to call them by this name, the only way to distinguish themformally from the ‘true’ sets seemed to forbid them being themselves elementsof something […]. However, we said that we couldn’t tolerate such a prohibition.Hence we need to be able to consider classes of classes, and it would be naiveto believe that one could stop at this second level. From now on, one doesn’tsee any longer what distinguishes the so-called classes, hyper classes and soon from ordinary sets, both of them being characterized by the collection of itselements and being elements of other collections; the only difference is that inthe mathematical universe there appears a kind of natural filtration. The usualoperations of set theory (i.e., those resulting from the strict application of ourMaster’s axioms) will not force us to leave a given level Ui of the filtration,and one needs new operations like the one corresponding to the intuitive notionof ‘forming the category of all objects’-more accurately, of all objects of Ui -toleave Ui and to enter Ui+1. By virtue of what I just said, such operations couldonly be carried out using a new axiom in set theory which will be formulatedlater. (Krömer 2007, p. 257).

In the first part, Grothendieck argues that if the only way to distinguish classes ofsets is that classes do not belong to another class, it may be impossible to form “the classof all classes” simply because it contradicts the definition of class.55 Therefore, thisdistinction cannot be admitted because they could not form some categories that seemnatural in the development of the theory: “the category Grp of all groups”, “the categoryTop of all topological spaces” or “category Cat of all categories” whose morphismsare precisely the functors F: A → B, where A and B are arbitrary categories.56

Including, for a given pair of arbitrary categories A and B, one could not form the BA

category of all functors from A to B, whose morphisms are all natural transformationsη: F → G; and therefore one could not talk about categories, such as GrpT op andCatCat , considered as objects of Cat.57

54 The copy is in the Archives of Desaltre in Nancy. According to the investigation of Krömer, this anony-mous 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 limitationsof NBG. However, he understands that their main concerns lay in the applications and not in the categoriesthemselves.56 As explained by Marquis (2009, p. 53), “the category of all categories” cannot be formed because theobjects 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: A → B is a proper classand 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.

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All these concepts were keys to the work being undertaken by Grothendieckin algebraic geometry. As can be verified in his famous paper “Sur quelquespoints d’algebre homologique”,58 in his book written with Dieudonné “Éleménts degéométrie algébrique”,59 in the presentations of the Séminaire Henri Cartan60 andof the Séminaire de Géométrie Algébrique du Bois-Marie (SGA),61 Grothendieckrequired extensive use of category theory to develop his theory of schemas. For exam-ple, in the topics studied in the SGA4 such as presheaves, topologies and sheaves,funtoriality of categories of sheaves and topoi (Grothendieck and Verdier 1972), thefunctors and functor categories play a fundamental role. Therefore, the mathematicalpractice of Grothendieck showed the insufficiencies of the NBG system. It was neces-sary to build a foundational framework from which these new mathematical conceptsmight have complete legitimacy.

To resolve this issue, without abandoning the set theoretic framework, Grothendieckproposes his notion of universe and the axiom of universes. The notion of universeis used to make a better distinction between small category and large category; andconstitutes a natural way to accommodate the functor categories. These conceptsare formally presented in the SGA4 (1963–1964).62 Nevertheless, Grothendieck hadalready made an informal presentation of these concepts in the first version of theSeminar SGA1 (1960-1961):63

To avoid some logical difficulties, we will here admit the notion of Universethat is a “large enough” set which is not consisting of the usual operations ofthe theory of sets; an “axiom of universes” ensures that every object is in aUniverse.64

Accepting that every set is contained in a universe U is equivalent to postulatingthe existence of an infinite succession of universes. Then U-categories can be con-

58 In this article, called Tôhoku paper, Grothendieck (1957) positions the homological algebra in terms ofabelian categories. This work is considered a milestone in the history of category theory. A comprehensivestudy 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 conceptof 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: U1: If x ∈ U and if y ∈ x theny ∈ U, U2: If x, y ∈ U then {x, y} ∈ U, U3: If x ∈ U then ℘(x) ∈ U, where ℘(x) denotes the set of allsubsets of x, U4: If (xi , i ∈ I) is a family of elements of U, and if I ∈ U, then ∪xi ∈ U. The axiom ofuniverses says: For every set x , there is a universe U such that x ∈ U. These definitions are in (Grothendieckand 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 therelationship 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).

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structed and functors can be defined amongst these categories; this resolves most ofthe problems since the strong closure properties of U makes it a model of ZFC. Thus,the method of Grothendieck universes constitutes a foundation for category theorystarting from set theory. In this sense, Grothendieck adds:

Thus, the formalization of categories, contrary to what one might have thought,in reality is done in a stronger theory than [usual] set theory. In this theory eachUi could be considered as a model of the ‘weakened’ set theory. (Krömer 2007,p. 257)

Finally, he says:

To conclude, it appears to me that there is no need to change anything in the firstthree chapters of book I […]. It will be enough to include in the new Chapter 4(that replaces the previous unusable one anyways) the complementary axiomsof set theory and develops there the theory of categories as much as is desired.65

This is fundamental for the Bourbaki’s project, since set theory continues at thebase of the proposal. In this manner, Grothendieck’s proposal pointed in the samedirection as the mathematical work of the group: it allowed keeping what had beencreated in the first three chapters of Book I and only some complementary axiomswould have to be added in order to develop category theory.

One could think of making a comparison between Hilbert’s transfinite axiom andthe axiom of Grothendieck universes as ideal elements that allow for extension of thetheory. However, this comparison does not work because the axiom of Grothendieckuniverses does not play the same role that Hilbert’s transfinite axiom. As we haveseen, universes essentially respond to Grothendieck’s practical necessity of providingconceptual legitimacy to functor categories; an essential concept in his theory that onhis part fulfills a genuine role of extension. They are at the heart of the theory and arethe cornerstone of Grothendieck’s mathematics. We owe this clarification to one ofthe anonymous reviewers of the manuscript.

3.5 Bourbaki and the Grothendieck universes

The Bourbaki group studied this issue thoroughly and in depth. In fact, he wrote apaper entitled Universes (Bourbaki 1972, pp. 185–217), which was first published as anappendix to the chapter “Presheaves” in SGA 4. In this article, Bourbaki reviewed therelationship of Grothendieck universes with some types of structures with categoriesand with strongly inaccessible cardinals.

Bourbaki presents the definition of Grothendieck universes, the axiom of universes,and among other things shows that the axiom of universes is equivalent to Tarski’s

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

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axiom (Bourbaki 1972, pp. 196–199).66 In the last item dedicated to “murky meta-mathematical Observations”, in item b, Bourbaki shows that the axiom of universes(or its equivalent of the strongly inaccessible cardinals) is not a consequence of the restof set theory (Bourbaki 1972, p. 214). This is to say, that the existence of inaccessiblecardinals or Grothendieck universes cannot be proved in ZFC. In his proof, Bour-baki proceeds by contradiction: he supposes the existence of the first non-numerablestrongly inaccessible cardinal and presents a model of set theory where the axiomof universes fails.67 This contradicts the initial hypothesis. Hence, the existence ofstrongly inaccessible cardinals68 is not be derived from the axioms of ZFC. Thismeans that the existence of an inaccessible cardinal can only present itself throughan axiom and shall only be accepted on the condition that such axiom does not posecontradiction with the others.69

Consequently, Bourbaki needed to know that if by adding the axiom of universes tohis system, the new system was still relatively consistent. Unfortunately, this relativeconsistency cannot be proved. Kunen (1980, p. 145) shows that there exists no finite-type proof that the consistency of (ZFC) implies consistency (ZFC + axiom of existenceof strongly inaccessible cardinals), unless ZFC is inconsistent.70 Bourbaki had nocertainty of this result, although he intuited it: “It would be very interesting to prove thatthe axiom (A.6) of universes is harmless. This seems hard and even non-demonstrable,Paul Cohen said”.71 The impossibility of proving the relative consistency of the newsystem constituted a hard blow to the tranquility offered by his system.

According to Krömer (2007, p. 268) this fact could have influenced the rejection ofcategory theory by the group. Although Krömer himself has found no explicit evidencefor this conclusion in the sources consulted, we consider that it is very likely that thegroup, not having certainty of the relative consistency of the new system, felt unsure(the possibility that some inaccessible cardinals lead to contradictions was still latent)and decided not to consider this new theory.

Today, we know that these axioms have been studied for years without reaching con-tradictions and that there are “many discoveries that have proven with ample evidencethat there is a close relation between the large cardinals and several mathematical prob-lems”.72 These important mathematical developments offer a historical “guarantee”that the Bourbaki group did not have at the time. Grothendieck’s proposal was inter-

66 Tarski’s axiom says (Bourbaki 1972, p. 196): every cardinal is passed strictly for a strongly inaccessiblecardinal. That is, for each cardinal k, there is a strongly cardinal λ inaccessible, which is strictly greaterthan k. The notion of strongly inaccessible cardinal is located the first time in (Sierpinski 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 foracceptance 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’sincompleteness 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 etc’est même indémontrable, dit Paul Cohen. (Bourbaki 1972, p. 214).72 Numerosos descubrimientos que han probado una amplia evidencia de una estrecha relación entre losgrandes cardinales y varios problemas matemáticos. (Jech 2005, p. 374).

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esting but the risk of encountering an inaccessible cardinal that led to contradictionswas high.

4 Concluding remarks

We have seen that Bourbaki defines a logical system comparable to first-order predicatecalculus where he introduces the sign τ in the same sense as Hilbert’s ε. After severalyears of work, Bourbaki obtained with four explicit axioms and a logical scheme anaxiomatic system equivalent to ZFC−. But at the same time category theory began toemerge; and with it, doubts about the ZFC− system as the only foundational frame-work for mathematics. In particular, Grothendieck’s proposal was studied with thepurpose of making room for categories, but the group finally kept its original system.Knowing today’s breakthroughs in category theory and their enormous implications inthe development of contemporary mathematics, it might seem incredible that Bourbakihas not incorporated this theory into his project; and it may be even easier to develop acritical posture regarding this decision. However, we have reasons of a different naturewhich we believe help explain the conservative attitude of Bourbaki.

The first has to do with the exercise of a philosophy of mathematical practice. ForBourbaki, the acceptance of an additional axiom, as the axiom of universes, comesfrom the empirical verification of its convenience in mathematical practice. We haveseen that this practice philosophy is a determinant factor in the selection of researchstrategies more than the logical or philosophical considerations. For years Bourbakiconfirmed the convenience of his ZFC− system regarding non-contradiction. However,Bourbaki lacked this empirical affirmation in the late fifties and early sixties whenconsidering the axiomatic system formed by the ZFC− + axiom of Grothendieckuniverses. The Gödel-Cohen results had demonstrated that the continuum hypothesisis undecidable in ZFC, which opens a range of possibilities for the cardinality of realnumbers. Problems involving large cardinals, such as the measurement problem andWoodin’s cardinals were yet to come. The work with strongly inaccessible cardinalswas very new and it was not easy for the group to leave the safe ground of set theoryfor a theory that was just beginning.

On the other hand, many mathematicians, Schwartz (a member of Bourbaki’s group)included, share the conviction that his objects possess a “deep reality”. This convictionis manifested for Schwartz, as stated by Arboleda (2007), in that the knowledge,once acquired, is organized inside a field of consciousness provided with a “rigidstructure”. This logical structure formed in the process of objectivation is imposed onour understanding and it is not possible to have access to new knowledge above it:

Any attempt of transgressing this conservative order is viewed by conscienceas an aggression from the outside. If a new knowledge is proposed, it wouldtake time for the conscience to assimilate such need since it implies “reorder-ing a series of phenomena and integrating what was just learned to one’s ownschemes”.73

73 Cualquier intento de transgredir este orden conservador es visto por la conciencia como una agresión delexterior. Si se plantea por la necesidad de incorporar un conocimiento nuevo, tomará tiempo a la conciencia

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In particular, incorporating the theory of categories meant for Bourbaki the rethink-ing of his structuralist proposal for the organization of the field of mathematicalresearch. The objects and mathematical theories of this field are expressed in thelanguage of sets governed by the ZF axioms with classical first-order logic, whichconstituted a “rigid structure” that is not logically possible to violate.74

Moreover, in the decade of the 50s, Grothendieck’s work on category theory andtopos had just started and it was not easy to see the diverse associated logics.75 Aswe know today, there is one intrinsic correlation of a logic with each kind of categoryand vice versa, so “that there is no bias in advance for one particular logic” (Peruzzi2006, p. 449). Although there is no written evidence of a reflection by Bourbaki inthis sense, this single logical consideration makes it unlikely that Bourbaki wouldbe able to accept category theory at the time. Including these new ideas would haveimplied a profound change: moving from classical logic to a network of intermediatelogics which are more close to an intuitionistic logic.76 This leap was inconceivablefor Bourbaki. His formalist program always had the purpose of responding vigorouslyfrom classical logic to the criticisms of the intuitionists.77

Zalamea (2012) offers other epistemological reasons. He argues that the objects,methods and techniques that characterize contemporary mathematics78 (whose originscan be found in the theory of categories) are of a nature so complex and different thatmany of its practices are hard to “observe” from ZFC with first-order classical logic(Zalamea 2012, p. 46).79 The most relevant differences that support this thesis are: (i)the objects studied by contemporary mathematics “are not only collections of axiomsand their associated models, but also, from an inverse perspective, classes of structuresand their associated logics” (op. cit., p. 44). That is, mathematics and mathematicalpractice precede the logic; (ii) the way forward is different: instead of progressing froma known and safe interior toward a unknown exterior, “contemporary mathematics sets

Footnote 73 continuedasimilar tal necesidad, pues ello implica “reordenar toda una serie de fenómenos e imbricar lo que acabode aprender en mis propios esquemas”. (Arboleda 2007, p. 216).74 Grothendieck himself has produced beautiful images to represent a way of thinking inherent in certainmathematical 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 algebrasclasses 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 compu-tational theory in recent decades. The institutional developments of the logic in connection with computerscience 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, whosememory will undoubtedly survive only as a historical curiosity, has at least rendered the service of havingobliged its opponents, that is to say the vast majority of mathematicians, to clarify their own positions andto become more consciously aware of the reasons (whether logical or sentimental) for their confidence inmathematics (Bourbaki 1968, p. 336)”.78 Zalamea (2012), Zalamea (2011) identifies contemporary mathematics with all the accumulated knowl-edge from the middle of the twentieth century onwards.79 ZF is understood in the work of Zalamea as ZFC.

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itself within the determinate boundaries of the ‘non-’80 from the outset, and, setting outfrom those frontier, goes on to constructively explore new and astounding territories”(op. cit., p. 45); and (iii) the mixture of techniques has an essential role, not as usefultools in a deduction, but as proper objects, in which the construction of the disciplineis at stake. In this regard, “there is no space of contemporary mathematics that doesnot find itself mongrelized by the most diverse techniques; the mediatory (‘trans-’)condition, which in the first half of the twentieth century could be seen as one step inthe path of demonstration, is today becoming the very core of the discipline” (op. cit.,p. 45).

These differences make more understandable the conservative attitude of Bourbaki.From classical logic and set theory it is very hard to see all that the contemporary math-ematics involve. In contrast to this difficulty, the great majority of the developmentsof the time are treatable from the first-order classical logic and set theory, which wasessential to his plan to write a textbook that would serve as a basis for the educationof young mathematicians. This project became a fundamental epistemological pro-posal for the development of mathematics. The influence of structuralism can not onlybe evidenced in the progress of modern mathematics but in the initial developmentof contemporary mathematics through several problems proposed by the group andwidely discussed in his seminars. Zalamea (2011) talks about this kind of issues in thefollowing paragraph:81

Several inventions of the time come from the Bourbakist “spirit” incarnated in itssingular exponents: the “proper” coupling of local and global information on astructure, which originated the sheaves theory (Leray, Cartan); the natural transitbetween the structures of projective geometry and analytic geometry, that pro-moted the GAGA of sheaves on complex algebraic manifolds (Serre); generationof hierarchies and classifications of differential algebras, which gave place tothe theorems of representation ofgroups and to Lie algebras (Chevalley, Borel);the putting together of techniques of number theory and of abstract harmoniccalculus, which led to the adeles and ideals (Weil); the search of general back-grounds for functional analysis, which led to paracompactness (Dieudonné);the elimination of singular obstructions in favor of global convolutions, whichpropelled the theory of distributions (Schwartz), etc.82

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 singu-lares: el “buen” acoplamiento de la información local y global sobre una estructura, que originó la teoría dehaces (Leray, Cartan); el tránsito natural entre las estructuras de la geometría proyectiva y de la geometríaanalítica, que impulsó el GAGA de haces sobre variedades algebraicas complejas (Serre); la jerarquizacióny la clasificación de álgebras diferenciales, que dio lugar a los teoremas de representación de grupos y alge-bras de Lie (Chevalley, Borel); la conjugación de técnicas de teoría de números y de cálculo armónicoabstracto, que concluyó en los adeles e ideles (Weil); la búsqueda de entornos generales para el análisisfuncional, que llevó a la paracompacidad (Dieudonné); la eliminación de obstrucciones singulares a favorde convoluciones globales, que propulsó la teoría de distribuciones (Schwartz), etc. (Zalamea 2011, p. 115).

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For his great influence on the development of twentieth century mathematics, weconsider that a study on Bourbaki’s axiomatic system for set theory not only hashistorical significance but continues to be of great educational interest. Although con-temporary mathematics constitutes the expression of several structuralist proposalsthat have provided new alternatives for foundations of mathematics (lattice theory,universal algebra, category theory, topos theory, among others), it is modern math-ematics with its foundation in set theory that still prevails in university programs ofmathematical education.

Acknowledgments We want to express our thanks to Guillermo Ortiz (Universidad del Valle) for hisvaluable contributions and comments on the mathematical content of the document; also, Fernando Ramblaand 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 partof the project (code 1106-521-28616) funded by COLCIENCIAS and the Universidad del Valle. Specialthanks are extended to the anonymous reviewers for their valuable contributions and suggestions to improvethe quality of the manuscript.

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