Set-theoretic Constructions Research Articles

“Map Theory et Antifondation”

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. Indeed, it was shown to be at least as powerful as ZFC+FA, where FA is the usual well-foundation axiom of set theory.In this thesis, we show that it is 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 the first part of the thesis we present the axiomatization of this new system, called MTA, and we show that it is powerful enough to interpret ZFC+AFA, where AFA is the Aczel-Forti-Honsell Antifoundation axiom. In particular, this interpretation implies the embedding of the first order reasoning on formulas into equational reasoning on terms which translate these formulas. In the second part, we show the consistency of MTA inside the framework of the k-continuous semantics for k > σ, where σ is any strongly inaccessible cardinal. The proof uses the k-premodels of Berline-Grue (k-cpos satisfying few simple additional properties). The main issue in modelling MTA is the construction, inside any k-premodel M, of an adequate k-open subset Φ of M which, once enriched with adequate equality and membership relations, will be a model of ZFC+AFA.

WELL-FORMED SET REPRESENTATIONS OF SOLIDS

All point Membership Classification (PMC) algorithms on solids constructed using regularized set operations require representing and computing neighborhoods of a point with respect to the represented solid. Such computations involve some of the more sensitive and difficult algorithms in solid modeling. We establish the necessary and sufficient conditions under which set-theoretic constructions are well-formed so that PMC amounts to evaluating a ternary logic expression and does not require any neighborhood computations. We demonstrate well-formedness of any monotone set expression whose primitives form a simple arrangement in Ed, and of common representations for polyhedral objects constructed using 'decreasing convex hull' algorithms. Well-formed representations exist for every solid but not for every fixed collection of primitives. We also give a necessary and sufficient test for existence of such representations. Identifying and eliminating neighborhood computations whenever possible should lead to simpler, more efficient, and robust geometric algorithms that are also more suitable for hardware implementation. Finally, well-formed set representations can be trivially translated into representations of solids by real-valued implicit functions.

Making Beliefs Coherentl. The Subtraction and Addition Strategies.

The notion of epistemic coherence is interpreted as involving not only consistency but also stability. The problem how to consolidate a belief system, i.e., revise it so that it becomes coherent, is studied axiomatically as well as in terms of set-theoretical constructions. Representation theorems are given for subtractive consolidation (where coherence is obtained by deleting beliefs) and additive consolidation (where coherence is obtained by adding beliefs).

The consistency theorem for free type definitions in Z

Abstract Successive editions of the author's reference manual for the Z specification language [ZRM89, ZRM92] gave different sufficient conditions for the consistency of free type definitions. Both were expressed in terms of the settheoretic constructions present in the type definition. Whilst the first edition specified that these constructions should be continuous , in the sense that they preserve limits of ascending chains of sets, the second edition specified that they should be finitary , i.e., that the result of each construction should be the union of a set of finite approximations. This short note clarifies the relationship between these two conditions.

Noemata and their formalization

The presentation of the formal conception of noemata is the main aim of the article. In the first section, three informal approaches to noemata are discussed. The goal of this chapter is specifying main controversies and their sources concerned with different ways of the understanding of noemata. In the second section, basic assumptions determining the proposed way of understanding noemata are presented. The third section is devoted to the formal set-theoretic construction needed for the formal comprehension of noemata. In the fourth section, definitions of noemata and their various kinds, as well as definitions of other phenomenological notions are formulated. In the last section, possibilities of further developing the proposed formal conception are indicated.

Algebra of constructions II: an algebraic approach to Martin-Löf type theory and the calculus of constructions

We present an algebraic approach to the syntax and semantics of Martin-Löf type theory and the calculus of constructions developed by T. Coquand and G. Huet. In our approach, models of this theory and this calculus are treated as three-sorted partial algebras, called ITSΠ-structures, described by essentially algebraic theories. We give a proof that derived statements of Martin-Löf type theory hold in appropriate ITSΠ-structures. In this proof, a formal translation from the language of terms and types into the language of terms of an appropriate essentially algebraic theory of ITSΠ-structures is used. A straightforward set-theoretic construction of ITSΠ-structures that are models of Martin-Löf type theory is demonstrated. We present a construction of a recursive ITSΠ-structure(i.e. its partial and total operations are recursive functions over some alphabet) that is a model of the calculus of constructions and demonstrates the decidability of this calculus.

Set-theoretic constructions of nonshrinking open covers

A family { M α | αϵA} is a shrinking of a cover { O α | αϵA} of a topological space if { M α | αϵA} also covers and M α ⊂ O α for all αϵA. ◊ ++ implies that there is a normal space such that every increasing open cover of it has a clopen shrinking but there is an open cover having no closed shrinking. ◊ implies that there is a P-space (i.e. a space having a normal product with every metric space), which has an increasing open cover having no closed shrinking. This space is used in [17] to show that any space which has a normal product with every P-space is metrizable.

Some set-theoretic constructions in topology

Some set-theoretic constructions in topology

A Sketch of the Use of Formalized Languages for the Assertion of Music

After a brief explanation formalized languages, a formalized language adequate for the assertion a few twelve-note-class compositions presented. Elementary set-theoretic concepts are introduced to specify properties the formalized language and particular extensions it. The paper concludes with discussion applications other formalized languages to various musical situations. O. Preliminary remarks. The concept set central to much mathematics. Let us begin our discussion informally by presuming knowledge this concept, that is, understanding the meaning statements, like an element the set natural numbers, when the immediate constituents (in our example, seven and all natural numbers) the phrase is an element the set of have understood meanings. (We presume also some knowledge numbers. The set natural numbers such that zero and every positive integer are the only elements the set.) Let us proceed to introduce certain usual set-theoretic constructs, in the informal style common to much mathematical writing (for example, Halmos [1]).1 One set a subset another set and only every element the one set an element the other set.2 One set equal to another set and only the one set a subset the other set and the other set a subset the one set. One set a proper subset another set and only the one set a subset the other set and the one set not equal to the other set. Any set a subset itself; any set equal to itself; no set a proper subset itself. The reader should be made aware, even after reading only the two above paragraphs informal writing, several deficiencies characteristic such writing. For the avoidance these deficiencies in formal asserting an important motivation for the construction 1 Numerals enclosed by square brackets refer to entries so numbered in the list references given at the end this paper. 2 Notice that notions (e.g., and only if and every), which may be explicated by a formal treatment propositional calculus or quantification theory (for which explication see Church [2]), are involved in our informal discussion.

Free monadic algebras

A monadic (Boolean) algebra is a Boolean algebra A together with an operator 3 on A (called an existential quantifier, or, simply, a quantifier) such that 30=0, pfk 3p, and 3(^A 3q) = 3p* 3g whenever p and q are in A. Most of this note uses nothing more profound about monadic algebras than the definition. The reader interested in the motivation for and the basic facts in the theory of monadic algebras may, however, wish to consult [2]. Every Boolean algebra can be converted into a monadic algebra, usually in several ways. (One way is to write 3p = p for all p; another is to write Bp =0 or 1 according p =0 or py^O. These special operators are known the discrete and the simple quantifier, respectively.) It follows, a fortiori, that every Boolean algebra can be embedded into a monadic algebra, and it is clear, on grounds of universal algebra, that among the monadic extensions of a Boolean algebra there is one that is as free possible. To be more precise, let us say that a monadic algebra A is a free monadic extension of a Boolean algebra B if (i) B is a Boolean subalgebra of A, (ii) A is (monadically) generated by B, (iii) every Boolean homomorphism g that maps B into an arbitrary monadic algebra C has a (necessarily unique) extension to a monadic homomorphism f that maps A into C. The statement that a monadic extension as free possible always exists means that every Boolean algebra B has a free monadic extension A; the algebra A is uniquely determined to within a monadic isomorphism that is equal to the identity on B. The purpose of what follows is to give a constructive proof of this fact, i.e., a proof that exhibits A by means of certain set-theoretic constructions based on B (instead of a structurally not very informative proof via equivalence classes of strings of symbols). A by-product of the proof is a rather satisfying insight into the structure of the (Stone) dual space of the free monadic extension. The theorem can (and will) be formulated in such a way to subsume the results of Hyman Bass [l ] on the cardinal number of finitely generated free monadic algebras. The idea of the construction is this. Step 1: form a free Boolean algebra generated by a copy of B. Step 2: adjoin the elements of that