First-order Set Theory Research Articles

A Reassessment of Cantorian Abstraction based on the varepsilon -operator

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s varepsilon -operator. Moreover, it is argued that the BK system and the varepsilon -operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the varepsilon -operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the varepsilon -operator in the BK definition of cardinal numbers.

Relating first-order set theories, toposes and categories of classes

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 (dssi) 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 (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). 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 (up to equivalence) 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.

Satisfaction relations for proper classes: Applications in logic and set theory

AbstractWe develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate (⊨*) appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension Θ of ZF there is a finitely axiomatizable extension *Θ′ of GB that is a conservative extension of Θ. We also prove a conservative extension result that justifies the use of ⊨* to characterize ground models for forcing constructions.

A Role-Based Authorization Model for Service-Oriented Architecture

Service-oriented architecture (SOA) is widely recognized as an especially effective solution for integrating loosely coupled and distributed resources. One of the major challenges in developing SOAbased applications is the management of authorization requirements in distributed environments. This paper proposes a formal authorization model based on a role-based access control model to demonstrate the approach for authorizing service requesters, to access a particular service and information under specific permissions in an SOA. The proposed model defines the authorization relationships and constraints among users (service requesters), roles, permissions, and services according to first-order logic and set theory. Furthermore, role hierarchy and service hierarchy are discussed to fulfill the requirements for authorization hierarchy. Finally, we use a health care system developed in a service-based architecture to demonstrate the utility of the concepts of the proposed authorization model.

Relating First-Order Set Theories and Elementary Toposes

AbstractWe show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo–Fraenkel set theory (IZF).

Schemata: The Concept of Schema in the History of Logic

AbstractSchemata have played important roles in logic since Aristotle'sPrior Analytics. The syllogistic figures and moods can be taken to be argument schemata as can the rules of the Stoic propositional logic. Sentence schemata have been used in axiomatizations of logic only since the landmark 1927 von Neumann paper [31]. Modern philosophers know the role of schemata in explications of the semantic conception of truth through Tarski's 1933 Convention T [42]. Mathematical logicians recognize the role of schemata in first-order number theory where Peano's second-order Induction Axiom is approximated by Herbrand's Induction-Axiom Schema [23]. Similarly, in first-order set theory, Zermelo's second-order Separation Axiom is approximated by Fraenkel's first-order Separation Schema [17]. In some of several closely related senses, a schema is a complex system having multiple components one of which is atemplate-textorscheme-template, a syntactic string composed of one or more “blanks” and also possibly significant words and/or symbols. In accordance with aside conditionthe template-text of a schema is used as a “template” to specify a multitude, often infinite, of linguistic expressions such as phrases, sentences, or argument-texts, calledinstancesof the schema. The side condition is a second component. The collection of instances may but need not be regarded as a third component. The instances are almost always considered to come from a previously identified language (whether formal or natural), which is often considered to be another component. This article reviews the often-conflicting uses of the expressions ‘schema’ and ‘scheme’ in the literature of logic. It discusses the different definitions presupposed by those uses. And it examines the ontological and epistemic presuppositions circumvented or mooted by the use of schemata, as well as the ontological and epistemic presuppositions engendered by their use. In short, this paper is an introduction to the history and philosophy of schemata.

Map Theory: From Well-Foundation to Antifoundation

Map Theory is a powerful extension of type-free lamba-calculus (with only a few term constants added). Due to Klaus Grue, it was designed to be a common foundation for Computer Sciences and for Mathematics. All the primitive notions of first-order logic and set theory, including truth values, connectives and quantifiers, set-membership and set-equality, are interpreted as terms. All the usual set-theoretic constructs, including inductive data-types, get computational interpretations. Now, Grue’s version of Map Theory is founded, in the sense that it only considers mathematical sets or classes which are well-founded with respect to the membership relation. In [19], we have shown that it was possible to design an alternative version which takes non-well-founded sets into account, and allows for co-inductive reasoning over them. This new version opens the way to a direct representation of co-inductive data-types and of circular processes and phenomena in Map Theory. In this article, we give parallel presentations of the two versions of Map Theory and of their relations with ZFC. We also give a flavor of the proofs of their relative consistency with respect to the existence of a strongly inaccessible cardinal. These proofs take place inside the κ-continuous semantics, which is an extension of Scott’s continuous semantics (where κ = ω) to any regular cardinal κ.

Second-Order Logic and Foundations of Mathematics

AbstractWe discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

Relating Z and First-Order Logic

Abstract. Despite being widely regarded as a gloss on first-order logic and set theory, Z has not been found to be very supportive of proof. This paper attempts to distinguish between the different philosophies of proof in Z. It discusses some of the issues which must be addressed in creating a proof technology for Z, namely schemas, undefinedness, and what kind of logic to use.

Reasoning about partial functions with the aid of a computer

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Godel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems.

A set-theoretic translation method for polymodal logics

The paper presents aset-theoretic translation method for polymodal logics that reduces derivability in a large class of propositional polymodal logics to derivability in a very weak first-order set theory Ω. Unlike most existing translation methods, the one we propose applies to any normal complete finitely axiomatizable polymodal logic, regardless of whether it is first-order complete or an explicit semantics is available. The finite axiomatizability of Ω allows one to implement mechanical proof-search procedures via the deduction theorem. Alternatively, more specialized and efficient techniques can be employed. In the last part of the paper, we briefly discuss the application ofset T-resolution to support automated derivability in (a suitable extension of) Ω.

Logical foundations for programming semantics

The paper provides an introduction to a natural-deduction-based set theory, NaDSet, and illustrates its use in programming semantics. The need for such a set theory for the development of programming semantics is motivated by contrasting the presentation of recursive definitions within first-order logic with their presentation within NaDSet. Within first-order logic such definitions are always incomplete in a very simple sense: Induction axioms must be added to the given definitions and extended with every new recursive definition. Within a set theory such as NaDSet, recursive definitions of sets are represented as terms in the theory and are complete, in the sense that all properties of the set can be derived from its definition. Such definitions not only have this advantage of completeness, but they also permit recursively defined sets to be members of the universe of discourse of the logic and thereby be shown to be members of other defined sets. The presentation of the semantics within NaDSet is not only fully formal, in contrast to the simply mathematical presentation of denotational semantics, but because NaDSet is formalized as a natural-deduction logic, computer-assisted proof constructions are plausible. A consistency proof for NaDSet is provided elsewhere. The resolution of the paradoxes provided by NaDSet is dependent upon replacing the naive comprehension axiom scheme of an inconsistent first-order set theory with natural-deduction rules for the introduction of abstraction terms into arguments. The abstraction terms admitted are a generalization of the abstraction terms usually admitted into set theory. In order to avoid a confusion of use and mention, the nominalist interpretation of the atomic formulas of the logic forces NaDSet to be second-order, although only a single kind of quantifier and variable is required.

Forcing and reducibilities. III. Forcing in fragments of set theory

We conclude here the treatment of forcing in recursion theory begun in Part I and continued in Part II of [31]. The numbering of sections is the continuation of the numbering of the first two parts. The bibliography is independent.In Part I our language was a first-order language: the only set we considered was the (set constant for the) generic set. In Part II a second-order language was introduced, and we had to interpret the second-order variables in some way. What we did was to consider the ramified analytic hierarchy, defined by induction as:A0 = {X ⊆ ω: X is arithmetic},Aα+1 = {X ⊆ ω: X is definable (in 2nd order arithmetic) over Aα},Aλ = ⋃α<λAα (λ limit),RA = ⋃αAα.We then used (a relativized version of) the fact that (Kleene [27]). The definition of RA is obviously modeled on the definition of the constructible hierarchy introduced by Gödel [14]. For this we no longer work in a language for second-order arithmetic, but in a language for (first-order) set theory with membership as the only nonlogical relation:L0 = ⊘,Lα+1 = {X: X is (first-order) definable over Lα},Lλ = ⋃α<λLα (λ limit),L = ⋃αLα.