Franco Parlamento

Decision procedures for elementary sublanguages of set theory: XIII. Model graphs, reflection and decidability

Positive solutions to the decision problem for a class of quantified formulae of the first order set theoretic language based on ϕ, ε, =, involving particular occurrences of restricted universal quantifiers and for the unquantified formulae of ϕ, ε, =, {...}, η, where {...} is the tuple operator and η is a general choice operator, are obtained. To that end a method is developed which also provides strong reflection principles over the hereditarily finite sets. As far as finite satisfiability is concerned such results apply also to the unquantified extention of ϕ, ε, =, {...}, η, obtained by adding the operators of binary union, intersection and difference and the relation of inclusion, provided no nested term involving η is allowed.

Expressing infinity without foundation

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed. The problem of expressing the existence of infinite sets in the first order settheoretic language by means of formulae of low logical complexity has been addressed in [PP88] and [PP9Ob]. While the usual formulations of the infinity axiom (Inf) make use of formulae involving (at least) alternations of universal and existential restricted quantifiers, [PP88] provided the first example of a formula involving only restricted universal quantifiers, whose satisfiability entails the existence of infinite sets, provided the foundation axiom (FA) is assumed together with the usual axioms of Zermelo-Fraenkel except, of course, the infinity axiom. It was then observed in [PP9Ob] that an even shorter formula had the same property. As explained in [PP88], the above problem is related to the so-called decision problem for fragments of set theory (see [CFO90]). Set theories not assuming FA but rather contradicting it in various forms have come to attract considerable interest (see [Acz88]), and the corresponding decision problem has begun to be investigated (see [PP9Oa]). It is therefore of particular interest to ask whether there are restricted purely universal formulae which are satisfiable but not finitely satisfiable, even when FA is dropped. In this note we show that a positive answer can be obtained through an appropriate merging of the two formulae in [PP88] and [PP9Ob], although neither of them suffices alone. Let SE be the first order set-theoretic language with identity, based on the membership relation e. A formula of Ad is restricted if it does not contain quantifiers except for the restricted quantifiers (Vx e y) and (3x e y). Let ZFdenote ZF Inf and ZF-denote ZF FA. In ZF-one can define the ordinals as transitive sets well-ordered by e and the nonzero natural numbers as successor ordinals with zero and successor ordinals only, as elements. Finiteness is taken to stand for equinumerousity with a natural number, and Inf can be stated as the existence of a set containing all the natural numbers. Received May 30, 1990; revised November 14, 1990. This work was supported by funds from the MPI, and by the AXL project of ENI and ENIDATA. ? 1991, Association for Symbolic Logic 0022-4812/91/5604-0005/

Decidability of ∀*∀‐Sentences in Membership Theories

01 .60

THE DECISION PROBLEM FOR RESTRICTED UNIVERSAL QUANTIFICATION IN SET THEORY AND THE AXIOM OF FOUNDATION

The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented. Mathematics Subject Classification: 03B25, 03E30.

Truth-Value Semantics and Functional Extensions for Classical Logic of Partial Terms Based on Equality

The still unsettled decision problem for the restricted purely universal formulae ((∀)0-formulae) of the first order set-theoretic language based over =, ∈ 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 (∀)0-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 Boffas antifoundation axiom, the complement of the set-satisfiability problem for the full class of Δ0-formulae is shown to be semidecidable; a result that is known not to hold, for the set-satisfiability problem itself, even for a very restricted subclass of the Δ0-formulae.

Cut Elimination for Gentzen’s Sequent Calculus with Equality and Logic of Partial Terms

We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality, and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption. Mathematics Subject Classification: 03B20

Henkin’s Completeness Proof and Glivenko’s Theorem

We provide a natural formulation of the sequent calculus with equality and establish the cut elimination theorem. We also briefly outline and comment on its application to the logic of partial terms, when “existence” is formulated as equality with a (bound) variable.

FINITE FAMILIES WITH FEW SYMMETRIC DIFFERENCES

We observe that Henkin’s argument for the completeness theorem yields also a classical semantic proof of Glivenko’s theorem and leads in a straightforward way to the weakest intermediate logic for which that theorem still holds. Some refinements of the completeness theorem can also be obtained.

Truth in V for ∃*∀∀-sentences is decidable

We show that 2[log2(m)1 is the least number of symmetric differences that a family of m sets can produce. Furthermore we give two characterizations of the set-theoretic structure of the families for which that lower bound is actually attained.