A New Approach to Predicative Set Theory

Notre Dame Journal of Formal Logic, 2014

A "new" criterion for set existence is presented, namely, that a set ¹x j 'º should exist if the multigraph whose nodes are variables in ' and whose edges are occurrences of atomic formulas in ' is acyclic. Formulas with acyclic graphs are stratified in the sense of New Foundations, so consistency of the set theory with weak extensionality and acyclic comprehension follows from the consistency of Jensen's system NFU. It is much less obvious, but turns out to be the case, that this theory is equivalent to NFU: it appears at first blush that it ought to be weaker. This paper verifies that acyclic comprehension and stratified comprehension are equivalent by verifying that each axiom in a finite axiomatization of stratified comprehension follows from acyclic comprehension.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1992

The still unsettled decision problem for the restricted purely universal formulae ((V),-formulae) of the first order set-theontic language based over =, E is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the (V),-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite sets, and various extensions of this result are obtained. When variables are restricted to range only over sets, in universes with infinitely many urelements the set-satisfiability problem is shown to be solvable provided the axiom of foundation is assumed; if it is not, then the decidability of a related derivability problem still holds. That, in turn, suggests the alternative adoption of an antifoundation axiom under which the set-satisfiability problem is also solvable (of course with different answers). Turning to set theory without urelements, assuming a form of Boffa's antifoundation axiom, the complement of the setsatisfiability problem for the full class of do-formulae is shown to be sernidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the do-formulae. MSC: 03B25, 03E30.

Communications on Pure and Applied Mathematics, 1980

The Journal of Logic Programming, 1999

The use of sets in declarative programming has been advocated by several authors in the literature. A representation often chosen for finite sets is that of scons, parallel to the list constructor cons. The logical theory for such constructors is usually tacitly assumed to be some formal system of classical set theory. However, classical set theory is formulated for a general setting, dealing with both finite and infinite sets, and not making any assumptions about particular set constructors. In giving logical-consequence semantics for programs with finite sets, it is important to know exactly what connection exists between sets and set constructors. The main contribution of this paper lies in establishing these connections rigorously. We give a formal system, called SetAx, designed around the scons constructor. We distinguish between two kinds of set constructors, scons(x,y) and dscons(x,y), where both represent {x} t.J y, but x ~ y is possible in the former, while x g y holds in the latter. Both constructors find natural uses in specifying sets in logic programs. The design of SetAx is guided by our choice of scons as a primitive symbol of our theory rather than as a defined one, and by the need to deduce nonmembership relations between terms, to enable the use of dscons. Alter giving the axioms Se-tAx, we justify it as a suitable theory for finite sets in logic programming by (i) showing that the set constructors indeed behave like finite sets; (ii) providing a framework for establishing the correctness of set unification; and (iii) defining a Herbrand structure and providing a basis for discussing logical consequence semantics tbr logic programs with finite sets. Together, these results provide a rigorous tbundation for the set constructors in the context of logical semantics.

Information and Computation, 1993

We describe an axiomatic theory for the concept of one-place, partial function, where function is taken in its extensional sense. The theory is rather general, i.e., concepts like natural numbers and sets are definable, and topics as non-strictness and self application can be handled. It contains a model of the (extensional) lambda calculus, and commonly applied mechanisms (like currying, inductive definability) are possible. Furthermore, the theory is equi-consistent with and equally powerful as ZF-Set Theory. The theory (called Axiomatic Function Theory, AFT) is described in the language of classical first order predicate logic with equality and one non-logical predicate symbol for function application. By means of some notational conventions, we describe a method within this logic to handle undefinedness in a natural way.

1985

model theory is the attempt to systematize the study of logics by studying the relationships between them and between various of their properties. The perspective taken in abstract model theory is discussed in Section 2 of Chapter I. The basic definitions and results of the subject were presented in Part A. Other results are scattered throughout the book. This final part of the book is devoted to more advanced topics in abstract model theory.

Journal of Philosophical Logic

2000

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so classvalued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the imps Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.

arXiv (Cornell University), 2013

We construct a model of the comprehension schema in the logic LP⇒. We are concerned with the following problem: can we formulate a paraconsistent set theory, with a full complement of self-containing sets, which nonetheless recaptures classical set theory, e.g. in the sense of proving all theorems of ZFC, and nothing false in ZFC, under an appropriate translation? We continue with the approach to this problem begun in Thomas [2013]. That paper defined a logic called LP ⇒ , an extension of LP adding a strengthened conditional operator, ⇒. In this paper we prove that the comprehension schema in LP ⇒ is nontrivial. We assume familiarity with Thomas [2013]. First some definitions. "The language of set theory" is the signature consisting of the sole binary relation ∈. A "⇒-free formula" is a formula which does not use the ⇒ connective. We write x to denote a sequence of variables x 1 , ..., x n. Definition 1. The comprehension schema (of LP ⇒) is the LP ⇒ theory consisting of all formulas of the form ∀ x ∃Y ∀z(z ∈ Y ⇔ φ(x, z)), where φ(x, z) is a ⇒-free formula in the language of set theory with its free variables among x, z (so in particular Y is not free in φ). This differs from, e.g., the comprehension schema of Restall [1992], in using the ⇔ connective instead of ↔. Because of the behavior of ⇔, it is a stronger axiom, more similar to the "Comprehension Rule" of that paper. The requirement for formulas to be ⇒-free is necessary to avoid trivializing paradoxes. Otherwise we can form sets such as R = {x : (x ∈ x) ⊥ }. (Recall that ⊥ is a logical operator defined in terms of ⇒.) Consideration of the self-membership of R leads to triviality, as the reader may verify. The restriction to ⇒-free formulas makes LP ⇒ comprehension not truly a "naïve" comprehension, in the sense that not every formula may be used to 1

1992

Abstract Zermelo-Fraenkel (ZF) set theory is widely regarded as unsuitable for automated reasoning. But a computational logic has been formally derived from the ZF axioms using Isabelle. The library of theorems and derived rules, with Isabelle's proof tools, support a natural style of proof. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor's Theorem, the Composition of Homomorphisms challenge [3], and Ramsey's Theorem [2].