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The egalitarian theory is considered as part of logic and not of set theory. Two months afterwards, Chevalley had a fifth version of the chapter on set theory Bourbaki b. For Chevalley, set theory is a theory without axioms; that is, a generic theory underlying all mathematical disciplines. Thus, the mentioning of these is completely nominal. The same axioms appear 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 book was written by Dixmier. Here, set theory loses the absolute role of the fifth version and is again considered a theory with axioms.
In the first chapter appear the four axiomatic schemes of Russell and Whitehead and the S5, S6 and S7 axioms of the final version Bourbaki b. The scheme of selection and reunion corresponds to the S8 scheme.
This version is very close to the final version. But, strictly speaking, it does not even correspond to the theory of Zermelo. In another article, Mathias shows that a model of this system exists, called Bou49 where the set axiom with two elements is false. In this history, academic spirit is highlighted along 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 all the work performed and questioned set theory as the only source of mathematics: the emergence of the theory of categories. Corry focuses his analysis on the opposition between the notion of structure and category theory. 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 them Samuel Eilenberg and Alexandre Grothendieck. We stop at a very specific issue of this controversy that is closely related to the central topic of this work.
The third chapter on Ordered Sets was published in and in the fourth and final chapter on Structures was published Fang , p. Examples of this ambivalence are that the group had decided to develop a fifth chapter of set theory on categories and functors; however, it was never published. Similarly, a previously agreed to volume on abelian categories was not published. At the same time, several members of the group, in their individual research-work, used and developed concepts of category theory.
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. Safety is related to the conviction that this theory bears no contradictions. In La Tribu No. The more a given branch has been developed, the less likely it becomes that contradictions may be met with in its further development. Bourbaki , p. Adhesion to this or that epistemo- logical principle or to a specific presentation of the theory comes from the empirical verification of its convenience in mathematical practice.
This practice philosophy is the determinant factor in the selection of research strategies more than the logical or philosophical considerations of such principle. This trust is based on the observation of its sound use in different fields.
In the Introduction of the book on set theory, published in , Bourbaki states: Nevertheless, during the half-century since the axioms of this theory were first precisely formulated, these axioms have been applied to draw conclusions in the most diverse branches of mathematics without leading to a contradiction, so that we 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 parts of this majestic edifice will never collapse as a result of the sudden appearance of a contradiction […]; but already for two thousand five hundred years math- ematicians have been correcting their errors to the consequent enrichment and not impoverishment of their science; and this gives them the right to face the future with serenity.
For Bourbaki was the axiomatic more convenient and productive at the time to generalize the theory of functions and functional analysis in abstract spaces Arboleda A standard problem to provide a foun- dation for category theory, from a set theoretical framework, is related to the size of the new objects. Starting with this conception, they presented some technical solutions related to the restriction of the domain.
A category is then any legitimate class in the sense of this axiomatic. Another device would be that of restricting the cardinal number, considering the category of all denumerable groups, of all groups of cardinal at most the cardinal of the continuum, and so on.
The subsequent developments may be suitable interpreted under any one of these viewpoints. Eilenberg and MacLane , p. This response, as expressed by Marquis , p. Particularly, Marquis examines the relationships between category theory and set theory and suggests the possibility to regard category theory itself as a foundational theory. Among them, the consideration is of a category of groups rather than the category of groups; but this solution has the difficulty of defining the composition of functors in general.
Similarly, it presents the possibility of adopting the theory of types as a foundation for the theory of classes, but it could complicate the study of natural isomorphisms because one would have to consider isomorphims between groups of different types.
Recall that in NBG primitive notion is that of class. Sets are special types of classes: a set is a class that is contained in another class.
The classes that are not sets are called proper classes. The notion de class is used to interpret the notion of size arising in category theory. Therefore, in this system, a category is any legitimate class of the system. Categories are classified as large and small. A large category is a category whose class of morphisms is a proper class of NBG. Otherwise, the category is said to be small. Another interesting aspect of NBG is the possibility of adopting a form of global choice.
These limitations will be raised in Sect. It was a 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 consistency NBG with global choice was only shown several years afterwards.
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. La Tribu No. Bourbaki a, p. But this system is rejected because it turns away from the viewpoint of the extension. Therefore, to be outside of set theory, it would mean to be outside of mathematics.
In this regard one 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, the group rejected this proposal and the system NBG remained an option to explore. In the 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 to formulating prohibitions which do no more than sanction current practice in the applications of the notion of set to various mathematical theories.
On the other hand, we cannot exclude the possibility that it may be easier to insert the basis of some mathematical theories into the framework of such systems than into the more rigid framework of the Zermelo—Fraenkel system. Bourbaki , pp. This option was emphasized by logician Daniel Lacombe, who had been consulted by Serre and Dixmier.
But, as we shall see below, this option was rejected by Grothendieck who proposed a different alternative. Hence we need to be able to consider classes of classes, and it would be naive to believe that one could stop at this second level. The usual operations of set theory i. By virtue of what I just said, such operations could only be carried out using a new axiom in set theory which will be formulated later. However, he understands that their main concerns lay in the applications and not in the categories themselves.
Similarly, Marquis , p. 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 , the functors and functor categories play a fundamental role.
Therefore, the mathematical practice of Grothendieck showed the insufficiencies of the NBG system. It was neces- sary to build a foundational framework from which these new mathematical concepts might have complete legitimacy.
To resolve this issue, without abandoning the set theoretic framework, Grothendieck proposes his notion of universe and the axiom of universes. The notion of universe is used to make a better distinction between small category and large category; and constitutes a natural way to accommodate the functor categories. These concepts are formally presented in the SGA4 — This work is considered a milestone in the history of category theory.
In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. It is now considered the cornerstone of modern algebraic geometry.
See particularly section 13 of the seminar. These definitions are in Grothendieck and Verdier , pp. Grothendieck , p. Thus, the method of Grothendieck universes constitutes a foundation for category theory starting 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.
It will be enough to include in the new Chapter 4 that replaces the previous unusable one anyways the complementary axioms of set theory and develops there the theory of categories as much as is desired. We owe this clarification to one of the anonymous reviewers of the manuscript.
In fact, he wrote a paper entitled Universes Bourbaki , pp. In this article, Bourbaki reviewed the relationship of Grothendieck universes with some types of structures with categories and with strongly inaccessible cardinals.
This is to say, that the existence of inaccessible cardinals or Grothendieck universes cannot be proved in ZFC. In his proof, Bour- baki proceeds by contradiction: he supposes the existence of the first non-numerable strongly inaccessible cardinal and presents a model of set theory where the axiom of universes fails.
Hence, the existence of strongly inaccessible cardinals68 is not be derived from the axioms of ZFC. This means that the existence of an inaccessible cardinal can only present itself through an axiom and shall only be accepted on the condition that such axiom does not pose contradiction with the others.
Unfortunately, this relative consistency cannot be proved. Kunen , p. Jech , p. However, we have reasons of a different nature which we believe help explain the conservative attitude of Bourbaki.
The first has to do with the exercise of a philosophy of mathematical practice. For Bourbaki, the acceptance of an additional axiom, as the axiom of universes, comes from the empirical verification of its convenience in mathematical practice.
We have seen that this practice philosophy is a determinant factor in the selection of research strategies more than the logical or philosophical considerations. The work with strongly inaccessible cardinals was very new and it was not easy for the group to leave the safe ground of set theory for a theory that was just beginning. This logical structure formed in the process of objectivation is imposed on our understanding and it is not possible to have access to new knowledge above it: Any attempt of transgressing this conservative order is viewed by conscience as an aggression from the outside.
Although there is no written evidence of a reflection by Bourbaki in this sense, this single logical consideration makes it unlikely that Bourbaki would be able to accept category theory at the time. Including these new ideas would have implied a profound change: moving from classical logic to a network of intermediate logics which are more close to an intuitionistic logic.
His formalist program always had the purpose of responding vigorously from classical logic to the criticisms of the intuitionists. Arboleda , p. See Grothendieck , pp. 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 developments of the time are treatable from the first-order classical logic and set theory, which was essential to his plan to write a textbook that would serve as a basis for the education of young mathematicians.
This project became a fundamental epistemological pro- posal for the development of mathematics. The influence of structuralism can not only be evidenced in the progress of modern mathematics but in the initial development of contemporary mathematics through several problems proposed by the group and widely discussed in his seminars.
Zalamea , p. Although con- temporary mathematics constitutes the expression of several structuralist proposals that 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 of mathematical education.
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