B Local theory of sets as a foundation for category theory and its connection with the Zermelo – Fraenkel set theory

2018 ◽  
pp. 281-360
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Local Theory

Local theory of sets as a foundation for category theory and its connection with the Zermelo-Fraenkel set theory

2006 ◽  
Vol 138 (4) ◽  
pp. 5763-5829 ◽  
Author(s):  
V. K. Zakharov ◽  
E. I. Bunina ◽  
A. V. Mikhalev ◽  
P. V. Andreev
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Local Theory

Categories: From Zero to Infinity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Pierre Schapira
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
And Topology

Set theory, category theory, and topology. Pierre Schapira explores the concept of identity within category theory, and what it means for the properties to be satisfied only up to homotopy.

Category theory, introduction to

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Category Theory ◽  
Standard Set ◽  
Universe Of Sets ◽  
The Universe

A ‘category’, in the mathematical sense, is a universe of structures and transformations. Category theory treats such a universe simply in terms of the network of transformations. For example, categorical set theory deals with the universe of sets and functions without saying what is in any set, or what any function ‘does to’ anything in its domain; it only talks about the patterns of functions that occur between sets. This stress on patterns of functions originally served to clarify certain working techniques in topology. Grothendieck extended those techniques to number theory, in part by defining a kind of category which could itself represent a space. He called such a category a ‘topos’. It turned out that a topos could also be seen as a category rich enough to do all the usual constructions of set-theoretic mathematics, but that may get very different results from standard set theory.

Global quantification in Zermelo-Fraenkel set theory

1985 ◽  
Vol 50 (2) ◽  
pp. 289-301
Author(s):  
John Mayberry
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Prima Facie ◽  
Local Theory ◽  
Axiom Schema ◽  
First Order ◽  
First Order Theory ◽  
Naturally Occurring ◽  
Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

Category theory, applications to the foundations of mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Classical Logic ◽  
Elementary Theory ◽  
Mathematical Structure ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Logical Foundations ◽  
Classical Mathematics ◽  
The 1960S

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.

NEITHER CATEGORICAL NOR SET-THEORETIC FOUNDATIONS

2012 ◽  
Vol 6 (1) ◽  
pp. 16-23
Author(s):  
GEOFFREY HELLMAN
Keyword(s):  
Modal Logic ◽  
Set Theory ◽  
Category Theory ◽  
Top Down ◽  
Ongoing Debate ◽  
Modest Proposal ◽  
Categories And Functors ◽  
Theoretic Foundations

AbstractFirst we review highlights of the ongoing debate about foundations of category theory, beginning with Feferman’s important article of 1977, then moving to our own paper of 2003, contrasting replies by McLarty and Awodey, and our own rejoinders to them. Then we offer a modest proposal for reformulating a theory of category of categories that would actually meet the objections of Feferman and Hellman and provide a genuine alternative to set theory as a foundation for mathematics. This proposal is more modest than that of our (2003) in omitting modal logic and in permitting a more “top-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

Building domains from graph models

1992 ◽  
Vol 2 (3) ◽  
pp. 277-299 ◽  
Author(s):  
Wesley Phoa
Keyword(s):  
Internal Structure ◽  
Set Theory ◽  
Category Theory ◽  
Domain Theory ◽  
Equivalence Relations ◽  
Graph Models ◽  
Constructive Set Theory ◽  
Partial Equivalence Relations

In this paper we study partial equivalence relations (PERs) over graph models of the λcalculus. We define categories of PERs that behave like predomains, and like domains. These categories are small and complete; so we can solve domain equations and construct polymorphic types inside them. Upper, lower and convex powerdomain constructions are also available, as well as interpretations of subtyping and bounded quantification. Rather than performing explicit calculations with PERs, we work inside the appropriate realizability topos: this is a model of constructive set theory in which PERs, can be regarded simply as special kinds of sets. In this framework, most of the definitions and proofs become quite smple and attractives. They illustrative some general technicques in ‘synthetic domain theory’ that rely heavily on category theory; using these methods, we can obtain quite powerful results about classes of PERs, even when we know very little about their internal structure.

scholarly journals Towards a dual ontology: duality, a case study

Sofia ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 62-79
Author(s):  
Francesco Maria Ferrari
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Formal System ◽  
Basic Mechanism ◽  
Spontaneous Breaking ◽  
Theoretic Approach ◽  
Non Linear ◽  
Linear Thermodynamics ◽  
Far From Equilibrium

The main aim of this work is to depict the interconnection of the most relvantformal concepts of modal logic and category theory, i.e., bisimulation andduality, arising from the mathematical analysis of physical processes and toshow their relevance with respect to some foundational issues related to the actual ontological debates. Current foundamental physics concerns the non-linear thermodynamics of the quantum eld, whose range is made of far from equilibrium systems and whose basic mechanism of symmetries (patterns) formation supposes the spontaneous breaking of symmetries (SBS). SBS implies that such systems reach unpredictable states. Thus, evolutive and/or far from equilibrium systems are to be conceived primarily as processes and just in a secondary way as objects, for the information they display is always incomplete with respect to their evolution. Formally, this is due to their non-linear mathematical behaviour.This make a question about the ontology of such systems, given thatthe actual most widespread ontologies conceive existent entities just as objects(actualist ontologies). It is claimed that the fundamental dierence and advantage of category theoretic approach to foundation is that, instead of considering objects and operations for what they 'are', as it is in set theory, in and through category theory we are considering them for what they 'do'. This, of course, would constitute a signicative shifting in mathematical philosophy and in foundationof mathematical physics: from a Platonic to an Aristotelian ontology ofmathematics (and, then, of physics). Actually, providing a contribution to thisvery shift is what this paper want to be focused on. In fact, the implicit pointthe present investigation is concerned with is how to treat the potential innite:the modalization of the existence of each object of the domain of quanticationmeans a potentially innite variation of the domain of quantication. The Aristotelian notion of potentiality diers with the usual one (employed by Platonism and/or formalism and/or conceptualism) inasmuch it does not presupposes any actuality. For instance, it is well known that the Platonic presupposition of set theory consists in the fact "that each potential innite, if it is rigorously applicable mathematically, presupposes an actual innite" [Hallett (1984, p. 25)]. In turn, the formalist notion of (absolute) completeness derives directly from that, if only for the actuality of the information a formal system was intended to dispaly.

scholarly journals Beyond Recognition: Badiou’s Mathematics of Bodily Incorporation

2020 ◽  
Vol 41 (2) ◽  
Author(s):  
Norman Madarasz
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Alain Badiou ◽  
General Logic ◽  
Phenomenological Methodology ◽  
The Absolute ◽  
The One

In Being and Event, Alain Badiou disconnects the infinite from the One and the Absolute, thus recasting the basis from which to craft a new theory of generic subject, the existence of which is demonstrated through set theory. In Logics of Worlds, Badiou turns his attention to the modes by which this subject appears in a world. It does so by being incorporated as a subjectivizable body, a body of truth. As opposed to Being and Event, the demonstration of this argument takes shape according to two distinct levels, that of a “calculated phenomenology” and that of a formalism in which category theory provides a general logic, the combination of which delineates an “onto-logic”. In this essay, we trace Badiou’s derivation of the notion of body of truth and evaluate the innovative phenomenological methodology applied to explain its association with a world.

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