Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a first-order set theory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions on the topos, a forcing-style interpretation of the language of first-order set theory in the topos is given, which conservatively extends the internal logic of the topos. This forcing interpretation applies to an arbitrary elementary topos, since any such is equivalent to (...) one carrying a dssi. We prove that the set theory BIST+Coll is sound and complete relative to forcing interpretations in toposes with natural numbers object. Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that all cocomplete and realizability toposes can be endowed with such superdirected systems of inclusions.A large part of the paper is devoted to an alternative notion of category-theoretic model for BIST, which, following the general approach of Joyal and Moerdijkʼs Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with BIST. We prove soundness and completeness results for BIST relative to the class-category semantics. Furthermore, BIST+Coll is complete relative to the restricted collection of categories of classes given by categories of ideals over elementary toposes with nno and dssi. It is via this result that the completeness of the original forcing interpretation is obtained, since the internal logic of categories of ideals coincides with the forcing interpretation. (shrink)

Category theory is often treated as an algebraic foundation for mathematics, and the widely known algebraization of ZF set theory in terms of this discipline is referenced as “categorical set theory” or “set theory for category theory”. The method of algebraization used in this theory has not been formulated in terms of universal algebra so far. In current paper, a _universal algebraic_ method, i.e. one formulated in terms of universal algebra, is presented and used for algebraization of a ground mereological (...) system described as “quantitative” (QMS) and of the standard set theories built upon it. The QMS is algebraized by using abelian generalized groups (AGGs) generalizing both abelian groups and Boolean algebras. To algebraize set theory, the AGG are enriched with operators to produce _extensional algebras_ (EA) of Tarskian “Boolean algebras with operators” type. EAs explicate the intuition of set theory’ universes (of discourse). The axiomatic set theory presented in this paper, named _extension theory_, previously went through an intuitive phase of development during its use by the author in computer science projects, where it served as the basis for development of a foundational data science framework and an extensional model of natural languages also outlined here. (shrink)

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such (...) axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework, despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry in the relationship between algebra and coalgebra. (shrink)

Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic. They were introduced in the 1970's and they represent a formal context within which to codify mathematics based on intuitionistic logic. They are formulated on the basis of the standard first order language of Zermelo-Fraenkel set theory and make no direct use of inherently constructive ideas. In working in constructive and intuitionistic ZF we can thus (...) to some extent rely on our familiarity with ZF and its heuristics. -/- Notwithstanding the similarities with classical set theory, the concepts of set defined by constructive and intuitionistic set theories differ considerably from that of the classical tradition; they also differ from each other. The techniques utilised to work within them, as well as to obtain metamathematical results about them, also diverge in some respects from the classical tradition because of their commitment to intuitionistic logic. In fact, as is common in intuitionistic settings, a plethora of semantic and proof-theoretic methods are available for the study of constructive and intuitionistic set theories. -/- The entry introduces the main features of Constructive and Intuitionistic ZF and offers links to the relevant bibliography. (shrink)

Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby. In order to stay focused on the “big picture”, we merely sketch the overall form of each construction, referring to the (...) literature for details. Each of the three steps considered below is based on more recent logical research than the preceding one. The first step from sets to types is essentially the familiar idea of set theoretic semantics for a syntactic system, i.e. giving a model; we take a brief glance at this step from the current point of view, mainly just to fix ideas and notation. The second step from types to categories is known to categorical logicians as the construction of a “syntactic category”; we give some specifics for the benefit of the reader who is not familiar with it. The third step from categories to sets is based on quite recent work, but captures in a precise way an intuition from the early days of foundational studies. With these pieces in place, we can then draw some conclusions regarding the differences between the three schemes, and their relative merits. In particular, it is possible to state more precisely why the methods of category theory are more appropriate to philosophical structuralism. (shrink)