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In Bourbaki's "Elements of Mathematics: Theory of sets", after an account of a set theory, a "theory of structures" is introduced. This book is difficult to read, but it seems, Bourbaki treat a structure as an object "defined" in set theory. Are there different approaches in defining what are structures?

Ioachim Drugus
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  • It's more of a question of philosophy. If you're Constructivist then you can think of numbers simply in terms of finitary sequences of formal symbols. "Playing with lego" is a groupoid, even though we don't have sets. – Luiz Cordeiro May 23 '18 at 20:22
  • The answer is obviously yes. First order logic, many-sorted first order logic, universal algebra, etc. are different ways to define mathematical structures. They differ in their level of generality and, thus, in what theorems can be proven about the languages and the structures that they define. The closest to Bourbaki would be many-sorted logic. Note that Bourbaki theory of structures is not used in practice, not even in Bourbaki itself, too general. – Dominic108 Jun 11 '18 at 16:09
  • I might have misinterpreted your question. Formal languages are usually interpreted with structures defined in set theory, but I see no reason why they could not be interpreted with a different language. I see an analogy with programming languages. They are normally executed in a machine language, but they can be interpreted in other languages. Set theory is analogous to the machine language. When I wrote that Bourbaki is too general, I meant that it is too close to the machine. We almost do not see a duality formal language vs interpretation language. – Dominic108 Jun 11 '18 at 16:43
  • @Dominic108: Bourbaki's theory of structures is in fact used throughout Bourbaki's treatise. – Fred Rohrer Jun 11 '18 at 18:30
  • @FredRohrer, you might be right. I only read and understood the definition of structures that they provide, nothing else from Bpurbaki. However, I read a few comments saying that it's not used within Bourbaki. Of course, I am sure that all the structures are defined using set theories, as it is the case every where, even with universal algebra. The question is do they use the Bourbaki definition of structures in the same way as we use universal algebra to study algebraic structures. – Dominic108 Jun 11 '18 at 18:46
  • @FredRohrer, in other words, I could consider that when I interpret a language of Universal algebra, I interpret in terms of Bourbaki structures, but if I don't make a concrete use of this fact, I might as well say that I don't use Bourbaki structures. I would be happy to know how exactly they use the specific of Bourbaki's definition, say within algebra and every where, as you say. Again, you might be right. I am curious, in fact, interested to know. – Dominic108 Jun 11 '18 at 19:15
  • @Dominic108: I do not understand your question. I am just saying that (in spite of what one hears time and again) Bourbaki refers to his theory of structures and uses its language, also in later books. – Fred Rohrer Jun 11 '18 at 20:02
  • Can you name a theorem that refers to it. What you say is a bit surprising, in a way. Everyone else develops universal algebra, etc. with models that are created within set theory, without any need for an extra language. I am saying if the use of Bourbaki's structures is a bit artificial in the sense that they could do very well without it, then I would not call this an application of Bourbaki's definition of structures. I mean it does not convey its usefulness in the same way as the three isomorphism theorems of universal algebra convey the usefulness of universal algebra, for example. – Dominic108 Jun 11 '18 at 20:35
  • I meant "... without the need for an extra language beside the language of universal algebra". – Dominic108 Jun 11 '18 at 20:40

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General idea of a structure - the core aspect of a structure (as opposed to a theory) is to imply a concept of truth (e.g. using meta language or some internal concept) e.g. to assess whether or not the structure is a model of some theory.

Toposes as structures - this said, topos theory can be regarded as an alternative approach to structures: A topos is a certain type of nice (in a way set-like) category with a(n internal) truth concept (by something called a subobject classifier). Depending on the regarded topos, the truth concept given by its subobject classifier might look very different to the classical true/false and in general induces an intuitionistic logic to the category.

Toposes as alternative - category theory can be developed without reference to set theory, i.e. without a concept of membership relationship, see e.g. Category theory without sets (incl. references for further reading).

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