A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory. NorthHolland, Amsterdam, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory ZF. Although it was clear that in ZF, the foundation axiom does not help in avoiding the paradoxes, it was added as a technical re nement. The separation axiom which replaced the unrestricted comprehension axioms is the one responsible for avoiding the paradoxes. This axiom goes as follows [23, 4]: Comprehension) For each open well formed formula , 9y8x[ x 2 y) x) where y is not free in (x) This unrestricted comprehension leads to a paradox by taking (x) to be : x 2 x) 9y8x[ x 2 y) x 2 x) 9y[ y 2 y) y 2 y) Such a comprehension axiom assumes that each open ....

Y. Bar-Hillel, A. Fraenkel and A. Levy. Foundations of set theory. NorthHolland, 1973.

....from the theorem on page 53 and the definition of infinite. Although we have seen that N is countable but R is not, we might still think that there is some smaller interval of the reals that can be paired to the naturals. ProofofPoincare (see [17] We show there is no bijection f : N [0, 1], in particular (#f : f : N [0, 1] f is not surjective) We do this by constructing for every function f : N [0, 1] a y [0, 1] such that f(n)# = y) We construct this y by means of a chain of segments (see paragraph 3.5.2) Let f : 0, 1] Let S n be an infinite chain of ....

....the definition of infinite. Although we have seen that N is countable but R is not, we might still think that there is some smaller interval of the reals that can be paired to the naturals. ProofofPoincare (see [17] We show there is no bijection f : N [0, 1] in particular (#f : f : N [0, 1]) f is not surjective) We do this by constructing for every function f : N [0, 1] a y [0, 1] such that f(n)# = y) We construct this y by means of a chain of segments (see paragraph 3.5.2) Let f : 0, 1] Let S n be an infinite chain of segments such that N : f(i) N : S ....

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Y. Bar-Hillel A.A. Fraenkel and A. Levy. Foundations of set theory. North-Holland Press, Amsterdam, 2 edition, 1973.

....provide classes of formulas which are assumed to be safe for this schema. Thus most of the axioms of ZF , the most famous axiomatic set theory, are just particular instances of this schema. The semantic guiding line behind the choice of these instances has been the limitation of size doctrine ([FBL73, Ha84]) according to which only collections which are not too big can be accepted as sets. However, this criterion is not constructive, so ZF uses some constructive substitutes to select formulas which seem to meet it. These principles are usually explained and justi ed ( Sh77] on semantic ground, ....

A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, North-Holland, Amsterdam, 1973.

....constructor for declarative programming can be traced back to [MW85] and has been advocated by other authors in the literature as well. The logical theory for such constructors is usually tacitly assumed to be some formal system of classical set theory, such as Zermelo Fraenkel (ZF ) set theory ([FBLD73], Sup72] 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 logic programs with finite sets, it is important to know exactly ....

....the Herbrand structure and show that Herbrand interpretations model SetAx. Finally, in x8, we give concluding remarks and brief comments on closely related work. Our usage of standard definitions of logic programming follows that in [Llo87] We also make use of a few results from [Sup72] and [FBLD73] in relating our set constructors to finite sets. 2 Set Constructors We consider the set constructors in a logical framework, i.e. a first order language L with equality, having an alphabet Sigma possessing a set of variables Sigma V , a set of constructor symbols Sigma C , a set of ....

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Fraenkel, A. A., Bar-Hillel, Y., Levy, A. and van Dalen, D.: Foundations of Set Theory, NorthHolland, 1973.

....variables if we want to avoid this slight complication. 2 accepted) among the axiomatic set theories. Most of the axioms of ZF are in fact just particular instances of the comprehension scheme 2 . The guiding line behind the choice of these instances has been the limitation of size doctrine ([FBL73], Ha84] according to which only collections which are not too big can be accepted as sets. Here not too big is an intuitive notion (which encompasses quite large infinite sets) With this intuitive notion in mind, a formula A of set theory may be called size limited w.r.t x, if the ....

A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, NorthHolland, Amsterdam, 1973.

....[3] for a detailed account. 5 cf. 5, ch. 3] 292 GABRIEL UZQUIANO It is evident that this axiom of infinity implies the existence of V# , which coincides with HF, the set of all hereditarily finite sets, as an immediate consequence, and even though it is mentioned in the second edition of [6] and figures as the o#cial axiom infinity in Azriel Levy s excellent text [7] it is seldom discussed in standard treatments of set theory. There are, to be sure, other variations on the axiom of infinity in the literature, but I am not now concerned to present an exhaustive review. My aim rather ....

....di#erences among common versions of the axiom of infinity, but I would not want to suggest that this oversight is too pervasive. For, as I should emphasize, the relative independence, modulo the axioms of Z , of alternative axioms of infinity is mentioned in [1] and in the second edition of [6]. And some of the drawbacks at which we shall look in the course of the discussion have been noticed before in the literature. See for example [4, pp. 110 111] and [9, Appendix B] 7 The construction appears, for example, in [9, p. 175] I borrow the term basic closure from Moschovakis. 8 A ....

Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, Foundations of set theory, North-Holland, 1973.

....and furthermore the comprehension principle is restricted. As language is the most important thing for them, interpretation is secondary. Thus it seems that the obvious semantics should be based on a substitutional interpretation. 3. Conceptualism Borrowing a sentence from Fraenkel (at the end of [Fraenkel 1973], page 336) Conceptualists are attracted neither by the luscious jungle flora of platonism nor by the ascetic desert landscape of neo nominalism . Concepts here are neither predicate expressions nor real properties. They are not objects but unsaturated entities, the saturation of which results ....

A. Fraenkel, Y. Bar-Hillel and A. Levy, Foundations of set theory, North-Holland, Amsterdam, 1973.

....state in Chapter 3 that the paradoxes before Russell were technical and affected only the most advanced parts of Cantor s theory . It would be more accurate to say that Cantor s methods were informal and intuitive, and that he just intuitively avoided what he called inkonsistenten Mengen (see [1] for further discussion) The descriptive set theory in Chapter 10 focuses on Baire space, N = N N , a countable product of countable discrete spaces. It will be a little difficult for most readers to see that the results also apply to more familiar spaces, such as R. Moschovakis does mention ....

A. A. Fraenkel, Y. Bar-Hillel, and A. L'evy, Foundations of Set Theory, Second Edition, North-Holland, 1984.

....worst case complexity. Our implementation of the algorithm is faster than our implementation of the Martelli Montanari algorithm on all examples that we tested. ffl We investigated different formulations of set theories (especially those with proper classes) Approaches using classes (e.g. Fra 73] were dropped as the modest gain in expressibility came at a continuing cost of proving that set objects were, in fact, sets. ffl We investigated various non standard logics, including temporal logics (specifically, the temporal logic called S 5 ) Temporal logic is not a part of EVES. There ....

A. A. Fraenkel, Y. Bar-Hillel and A. Levy. Foundations of Set Theory. North-Holland.

....we should have less confidence in this kind of perception, i.e. in mathematical intuition, than in sense perception. In this survey, we first give a brief review of classical set theories, trying to avoid the technical details which the reader can find in classical texts like (Halmos 1974) or (Fraenkel et al. 1973) and instead focusing on the underlying concepts. We then consider the alternative set theories which have been proposed throughout the century to overcome the limitations of classical theories. Later, we investigate the properties of a possible commonsense set theory, treating different aspects ....

Fraenkel, A. A., Bar-Hillel, Y. and Levy, A. (1973). Foundations of Set Theory . NorthHolland, Amsterdam.

....in AI because there is considerable beauty, economy, and naturalness in using sets for information modeling and knowledge representation. In this paper, we first give a brief review of classical set theory. We avoid the technical details which the reader can find in texts like (Halmos, 1974) (Fraenkel et al. 1973), and (Suppes, 1972) and instead focus on the underlying concepts. While we assume little or no technical background in set theory per se, we hope that the reader is interested in the applications of this formal theory to the problems of intelligent information management. We then consider the ....

Fraenkel, A. A., Bar-Hillel, Y., and Levy, A. (1973). Foundations of Set Theory , North-Holland, Amsterdam.

....constructor for declarative programming was first introduced in [JP87] and has been advocated by other authors in the literature as well. The logical theory for such constructors is usually tacitly assumed to be some formal system of classical set theory, such as ZermeloFraenkel (ZF ) set theory ([FBLD73], Sup72] 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 logic programs with finite sets, it is important to know exactly ....

....the Herbrand structure and show that Herbrand interpretations model SetAx. Finally, in x8, we present our conclusions and further comparisons with related work. Our usage of standard definitions of logic programming follows that in [Llo87] We also make use of a few results from [Sup72] and [FBLD73] in relating our set constructors to finite sets. 2 Set Constructors We consider the set constructors in a logical framework, i.e. a first order language L with equality, having an alphabet Sigma possessing a set of variables Sigma V , a finite set of constructor symbols Sigma C , a finite ....

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Fraenkel, A. A., Bar-Hillel, Y., Levy, A. and van Dalen, D.: Foundations of Set Theory, North-Holland, 1973.

....about the form and function of products. A brief overview of the commitments found so far by the author is given in this section. The author is formalizing ontological commitments as they are discovered, in the Axiomatic Information Model for Design (AIM D) In AIM D, axiomatic set theory (Fraenkel, Bar Hillel, Levy 1973) is interpretted from the point of view of product modeling. This loads the axioms with a semantics specific to design; that is, AIM D is an ontological commitment. The theory itself is beyond the scope of this article (refer to (Salustri 1996) for details) however, a brief overview of the theory ....

Fraenkel, A. A.; Bar-Hillel, Y.; and Levy, A. 1973. Foundations of Set Theory. North-Holland.

....we translate type theory into an untyped setting we shift from one restriction to the other, indeed the formation of proposition is not restricted anymore, but the relativization of quantifier restricts the comprehension schemes. Relativization of quantifiers shows that, as already remarked in [42, 18], the two ways to avoid Russell s paradox: restricting the formation of propositions from objects (as in type theory) and restricting the formation of objects from propositions (as in set theory) lead to a somehow similar result. 3.2.2 A single symbol for application From the theory above, we ....

A. A. Fraenkel, Y. Bar-Hillel, A. Levy, Foundations of set theory, North-Holland (1973).

....evolution, both a formal theory and modeling language for artifact systems have been synthesized. The formal theory represents the logical foundations upon which the modeling language is built. The formal theory, called the Hybrid Model (HM) is an interpretation of classical, axiomatic set theory [2] from the point of view of design. It is founded on an isomorphism that relates the primitive notions of set theory (individuals and sets) to primitive information items in design (quantities and design entities) From this isomorphism, the axioms of set theory are re interpreted as axioms of ....

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. North-Holland, 1973.

....possible primitive part of relations exhaustively. The goal of the current authors work is to develop a universal formal theoretical framework for the description of designed products. The theory, the Axiomatic Information Model of Design (AIM D) presents an interpretation of axiomatic set theory [4] as such a framework. Currently, the theory supports only a na ve notion of a single, universal part of relation. The work reported herein extends AIM D to support various different kinds of part of relations in a consistent manner. AIM D is discussed in detail elsewhere [8] 3 A New Notion of ....

Fraenkel, A. A., Bar-Hillel, Y., and Levy, A., Foundations of set theory, North-Holland, 1973.

....knowledge representation and reasoning, two basic mental (cognitive) processes are abstraction and categorization. In fact, a set might be considered as the image (in mind) of a collection of things under some abstraction mechanism. This mechanism should be similar to the one Cantor thought [13, 11, 32, 36, 38] and can use the color, size, use, location, and other things that might be useful in this categorization process. Here we did not say property instead of thing because the differentiation of the objects might not depend on the object, viz. location. According to Piaget [27] human ....

....objects, and (iii) to give the rules of the game to be played with newly introduced symbols [36] In general, ZF is the basic axiomatization used heavily in mathematics. The origin and the underlying mathematical properties and results of its axioms were extensively discussed in the literature [9, 13, 15, 16, 17, 18, 19, 20, 30, 31, 33, 35, 36, 37]. The axioms are defined in first order logic and only the membership relation (2) is considered to be a basic relation. The axioms of ZF are Extensionality, Null Set , Pair Set , Union, Infinity, Power Set , Separation (Subset) Replacement , and Foundation (Regularity) Choice is not an ....

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A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory . North-Holland, Amsterdam, 1973.

....is sufficiently strong to serve as a foundation for mathematics, but is (so everybody believes) consistent. Most of the axioms of ZF are certain particular instances of the comprehension scheme 2 . The guiding line behind the choice of these instances has been the limitation of size doctrine ([FBL73], Ha84] according to which only collections which are not too big can be accepted as sets. Here not too big is an intuitive notion (which encompasses quite large infinite sets) With this intuitive notion in mind, a formula A of set theory may be called size limited w.r.t x, if the ....

A. Fraenkel, Y. Bar-Hillel, and A. Levy, Foundations of Set Theory, NorthHolland, Amsterdam, 1973.

....new model for engineering computing. We will then introduce the language we have devised, called Designer, including some examples of its use. Finally, we discuss our results and directions for future work. 3 2 THEORETICAL FOUNDATIONS The Hybrid Model is an extended form of axiomatic set theory [5] specific to design information. The details of HM are given in [4] only a brief summary is presented here. The domain of HM is that of design entities, defined to be information units of relevance to designers. It describes design information independently of the design processes that gave rise ....

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. North-Holland, 1973.

....The present paper is essentially a revised version of the technical report [Bal94] In this report, some proofs that are only sketched here can be found in full detail. 2. Preliminaries 2.1. Foundations. We claim to work informally in Aczel s variant of Zermelo Fraenkel axiomatic set theory ([FBL73], Acz88] Thus our system of set theoretic axioms is ZFC Gamma AFA, ZFC Gamma being the usual ZFC system without the Axiom of Foundation (but with the Axiom of Choice) Now let us recall AFA ( Acz88] 1 Anti Foundation Axiom. Let G be an accessible directed graph, i.e. G = V ; E V ....

A.A. Fraenkel, Y. Bar--Hillel, and A. Levy. Foundations of Set Theory. North-- Holland, 1973.

....its underlying philosophical principles. Collecting entities into an abstraction for further thought (i.e. set construction) is an important process in mathematics, and this brings in assorted problems [5] The theory had many ground shaking crises (like the discovery of the Russell s Paradox [6]) throughout its history, which were nevertheless overcome by new axiomatizations. The most popular of these is the Zermelo Fraenkel axiomatization with Choice (ZFC) ZFC is an elegant theory which inhabits a stable place among other axiomatizations as the mainstream set theory. It provides a ....

.... by the Axiom of Foundation (FA) which forbids infinite descending sequences of sets under the membership relation 2, such as : 2 a 2 2 a 1 2 a 0 2 a (thereby not allowing sets which can be constituents of themselves) and which has sometimes been regarded as a somewhat superficial limitation [6]. Sets which obey the FA are called well founded sets. The cumulative hierarchy has provided a precise framework for the formalization of many mathematical concepts [7] However, it may be asked whether the hierarchy is limiting, in the sense that it might be omitting some sets one would like to ....

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A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory. NorthHolland, Amsterdam, 1973.

.... the numeric form of choice from numbers to sets: 102 8x9Q [x; Q] 9Z8x [x; Z x ] When referring to choice over formulas that mention set quantifiers, these setexistence principles are circular, since the existence of the set Z is based 99 See e.g. the introduction of [ Mendelson, 1964] or [Fraenkel et al. 1973, Ch. 1 ] for concise and informative surveys of that crisis. Fraenkel and Bar Hillel, 1958, Section I.6] contains a detailed bibliography through 1956. 100 [Whitehead and Russell, 1929; Zermelo, 1908; Zermelo, 1930; Fraenkel, 1922 ] 101 The notion of existence is understood here to ....

....Then, the l.u.b. b of a bounded set A of reals is defined as fr 2 Q j (9a 2 A) r 2 a) g, i.e. one uses Comprehension: 9b 8r 2 Q(r 2 b (9a 2 A) r 2 a) 106 Depending on the exact formulation of the sequential calculus, Pi 0 may need to be slightly repaired to yield a correct proof 107 [Fraenkel et al. 1973] (p. 132 fn. 2) attributes this use of cut free proofs to Paul Cohen. A proof theoretic proof that does not depend on cut elimination runs as follows. See [ Troelstra, 1973] for details, for the case of first order arithmetic and the schema of induction. Suppose Delta is a proof in T 2 of a ....

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A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory. North-Holland, Amsterdam, second edition, 1973.

....referring to choice over formulas that mention set quantifiers, these setexistence principles are circular, since the existence of the set Z is based 99 See e.g. the introduction of [ Mendelson, 1964] or [Fraenkel et al. 1973, Ch. 1 ] for concise and informative surveys of that crisis. Fraenkel and Bar Hillel, 1958, Section I.6] contains a detailed bibliography through 1956. 100 [Whitehead and Russell, 1929; Zermelo, 1908; Zermelo, 1930; Fraenkel, 1922 ] 101 The notion of existence is understood here to encompasses disjunction, since, for example, p q is equivalent to 9x (x = 0 p) x 6= 0 q) ....

A. A. Fraenkel and Y. Bar-Hillel. Foundations of Set Theory, Second edition. North-Holland, Amsterdam, 1958.

....which may arise with implementation. 2 A Real Time Logic The real time logic we are going to use for the semantics of the programming language is constructed from conservative extensions to first order predicate logic: this allows the use of the standard first order proof system (as used in [7] for example) The logic formalises the notion of a timed communication channel. Time is represented by the set of positive integers (denoted N ) and a timing function is used to represent the values found in channels at specific times. Formulae are therefore constraints on the relationship ....

A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory. North-Holland, 1973.

....and workshops. 1 Motivation In the foundations of mathematics, the most popular approach is set theory. All of the mathematical objects can be constructed out of sets. In this view, mathematics deals only with the properties of sets, all of which can be deduced from a suitable list of axioms [5, 6]. In general, Zermelo Fraenkel (ZF) is the basic axiomatization used heavily in mathematics [3] Its origin and the underlying mathematical ideas for its axioms were extensively discussed in the literature [5, 10, 11] The axioms are defined in first order logic and only the membership relation ....

.... the properties of sets, all of which can be deduced from a suitable list of axioms [5, 6] In general, Zermelo Fraenkel (ZF) is the basic axiomatization used heavily in mathematics [3] Its origin and the underlying mathematical ideas for its axioms were extensively discussed in the literature [5, 10, 11]. The axioms are defined in first order logic and only the membership relation (2) is considered to be a basic relation [9] A fundamental axiom of ZF is Extensionality : 8x8y8z[ z 2 x z 2 y) x = y] Basically, this axiom formalizes the notion of being a set: a set is a collection of ....

A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory . NorthHolland, Amsterdam, 1973.

....the status of hypertext is even more tenuous [17] It became necessary to return to more basic first principles, and it was during our study of symbolic logic that axiomatic set theory presented itself as a very viable system upon which to base HM. Axiomatic set theory has taken on various forms [18, 19, 20] but every form is based on the classical theory developed by Zermelo and Fraenkel [18] and which is generally referred to as ZF set theory, or just ZF. We will, for brevity s sake, adopt this convention also. ZF deals with groups of completely general entities; a group of entities is called a ....

....ZF. We will, for brevity s sake, adopt this convention also. ZF deals with groups of completely general entities; a group of entities is called a set. The theory formalizes the nature of sets to such a degree as to permit the derivation of almost all the classical branches of mathematics and logic [19]. The most interesting implication of set theory as far as the authors are concerned regards consistency of theories that are supersets of classical axiomatic set theory. In [18] it is proved that any axiom system that can be rewritten in terms Department of Mechanical Engineering University of ....

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Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. North-Holland, 1973.

....up with several examples indicating how Designer can capture design information effectively and concisely. We then present a discussion of our results and conclusions that will guide our future efforts. 2 Theoretical Foundations The Hybrid Model (HM) is an extended form of axiomatic set theory [9] specifically oriented to describe design information. The details of HM are given in [8] only a brief summary is presented here. HM is the formal domain model upon which the Designer language is based. The control of impedance mismatches between HM and the cognitive models of designers is dealt ....

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. North-Holland, 1973.

....a brief overview and status report of the authors efforts. Each of the three design theoretic tools in figure 1 will be covered in turn. THEORETICAL FOUNDATIONS The authors first step was the derivation of the Hybrid Model (HM) of design information [6] a variant of axiomatic set theory [7], extended to address the specific requirements of design. Since HM adheres to the rules of logic governing the extension of classical set theory, it can be proved that HM is a logically valid axiomatic system. The domain of HM is that of design entities, defined to be (possibly complex) ....

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy. Foundations of Set Theory. North-Holland, 1973.

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A. A. Fraenkel, Y. Bar-Hillel, and A. Levy. Foundations of Set Theory. NorthHolland, Amsterdam, 1973.

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Y. Bar-Hillel, A. Fraenkel, and A. Levy, Foundations of set theory, North-Holland, 1973.

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A. Fraenkel, Y. Bar-Hillel and A. Levy, Foundations of set theory, North-Holland, Amsterdam, 1973.

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Fraenkel, A., A., Bar-Hillel, Y., and Levy, A. (1973) Foundations of set theory. North-Holland, Amsterdam.

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A.A. Fraenkel, Y. Bar-Hillel and A. Levy (1973). Foundations of Set Theory. Second revised edition. Studies in Logic and the Foundations of Mathematics 67. North-Holland Publishing Company, Amsterdam.

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