Luc, Joanna (2026) Formalised axiomatic theories and Bourbaki's concept of species of set-theoretical structure. [Preprint]

Text Joanna Luc, Formalised axiomatic theories and Bourbaki’s concept of species of set-theoretical structure.pdf Download (520kB)

Abstract

This paper investigates Bourbaki's concepts of set-theoretical structure and isomorphism. First, I introduce Bourbaki's formalism in a modern guise. Second, I show that there is a one-to-one correspondence (up to logical equivalence) between theories consisting of finitely many axioms expressed in some formal (first-order or higher-order) language and t-species of set-theoretical structure---that is, species of structure S which are transportable in the sense that the extension of the predicate ``being a structure of species S'' is closed under isomorphisms. Third, I examine the significance of the formal definitions and results presented in this paper for the philosophy of mathematics and the philosophy of the empirical sciences. In particular, I argue that: (1) since definitions of predicates ``being a structure of species S'' are conservative and eliminable relative to ZFC, such predicates can be used to achieve a reduction of (large parts of) mathematics to ZFC; (2) since t-species of structure are transportable, they can provide a basis for a set-theoretical form of structuralism by enabling us to ``forget'' those aspects of structures that are not isomorphism-invariant; and (3) the correspondence between t-species of structure and formal axiomatic theories shows that there is no deep divide between the so-called semantic and syntactic approaches in the philosophy of science.

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Item Type:	Preprint
Creators:	Creators Email ORCID Luc, Joanna joanna.luc.poczta@gmail.com 0000-0003-2758-8230
Keywords:	Structure; Isomorphism; Higher-order languages; Bourbaki
Subjects:	Specific Sciences > Mathematics > Foundations General Issues > Structure of Theories
Depositing User:	Joanna Luc
Date Deposited:	11 Jun 2026 15:21
Last Modified:	11 Jun 2026 15:21
Item ID:	30020
DOI or Unique Handle:	10.1007/s10992-026-09844-8
Subjects:	Specific Sciences > Mathematics > Foundations General Issues > Structure of Theories
Date:	2026
URI:	https://philsci-archive.pitt.edu/id/eprint/30020

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