Nonstandard Set Theories and Information Management VAROL AKMAN akmantroycsbilkentedutr Department of Computer Engineering and Information Science Bilkent University Bilkent Ankara Turkey MUJDATPAKKAN pakkantrb ounbitnet Department of Computer Engineering Bosphorus University Bebek Istanbul Turkey Abstract The merits of set theory as a foundational to ol in mathematics stimulate its use in various areas of articial intelligence in particular intelligent information systems In this pap er a study of various nonstandard treatments of set theory from this p ersp ective is oered Applications of these alternative set theories to information or knowledge management are surveyed Keywords set theory knowledge representation information management commonsense rea soning nonwellfounded sets hyp ersets Intro duction Set theory is a branch of mo dern mathematics with a unique place b ecause various other branches can b e formally dened within it For example Book of the inuential works of N Bourbaki is devoted to the theory of sets and provides the framework for the remaining 1 volumes Bourbaki said in Goldblatt All mathematical theories may b e regarded as extensions of the general theory of sets On these foundations I can state that I can build up the whole of the mathematics of the present day This brings up the p ossibility of using set theory in foundational studies in AI Mc Carthy has emphasized the need for fundamental research in AI and claimed that AI needs mathematical and logical theory involving conceptual innovations In an op ening address McCarthy he stressed the feasibility of using set theory in AI b e cause there is considerable b eauty economy and naturalness in using sets for information mo deling and knowledge representation In this pap er we rst give a brief review of classical set theory We avoid the technical detailswhich the reader can nd in texts like Halmos Fraenkel et al and Supp es and instead fo cus on the underlying concepts While we assume little or no technical background in set theory p er se we hop e that the reader is interested in the applications of this formal theory to the problems of intelligent information management We then consider the alternative set theories which have b een prop osed to overcome the limitations of the standard theory Finally we survey various nonstandard treatments of set theory each innovating dierent asp ects such as urelements cumulative hierarchy selfreference cardinality wellorderings and so on It is shown that such treatments which are all very recent and sometimes esotericare quite useful to the I IS community for there are assorted technical problems in information management eg commonsense reasoning terminological logics etc that may prot from such nonstandard approaches Early developments in set theory G Cantors work on the theory of innite series should b e considered as the foundation of the research in set theory In Cantors conception a set is a collection into a whole of denite distinct ob jects of our p erception or our thought the elements of the set This prop erty of deniteness implies that given a set and an ob ject it is p ossible to determine if the ob ject is a memb er of that set In other words a set is completely determined by its memb ers In the initial stages of his research Cantor did not work from axioms However all of his theorems can b e derived from three axioms Extensionality which states that two sets are identical if they have the same memb ers Abstraction which states that for any given prop erty there is a set whose memb ers are just those entities having that prop erty and Choice which states that if b is a set all of whose elements are nonempty sets no two of which have any elements in common then there is a set c which has precisely one element in common with each element of b The theory was so on threatened by the intro duction of some paradoxes which led to its evolution In B Russell found a contradiction in G Freges foundational system which was develop ed on Cantors naive conception van Heijenho ort This contra diction could b e derived from the Axiom of AbstractionAxiom V in Freges systemby considering the set of all things which have the prop erty of not b eing memb ers of them selves This prop erty can b e denoted as x x or simply x x in the language of rst order logic The Axiom of Abstraction itself can b e formulated as xy x y x where x is a formula in which y is free In the case of Russells Paradox x x x and we have xy x y x x Substituting y for x we reach the contradiction y y y y Another antinomy o ccurred with the set of all sets V fx x xg The well known Cantors Theorem states that the p ower set of V has a greater cardinality than V itself This is paradoxical since V by denition is the most inclusive set This is the so called Cantors Paradox and led to discussions on the size of comprehensible sets Strictly sp eaking it was Freges foundational system that was overthrown by Russells Paradox not Cantors naive set theory The latter came to grief precisely b ecause of the preceding limitation of size constraint Later J von Neumann would clarify this problem as follows Goldblatt Some predicates have extensions that are to o large to b e successfully encompassed as a whole and treated as a mathematical ob ject Paradoxes of the preceding sort were instrumental in new axiomatizations of the set theory and in alternate approaches However it is b elieved that axiomatic set theory would still have evolved in the absence of paradoxes b ecause of the continuous search for foundational principles Alternate axiomatizations The new axiomatizations to ok a common step for overcoming the deciencies of the naive approach by intro ducing classes NBG which was prop osed by von Neumann and later revised and simplied by P Bernays and Godel was the most p opular of these In NBG there are three primitive notions set class and memb ership Classes are considered as totalities corresp onding to some but not necessarily all prop erties The classical paradoxes are avoided by recognizing two typ es of classes sets and prop er classes A class is a set if it is a memb er of some class Otherwise it is a prop er class Russells Paradox is avoided by showing that the class Y fx x xg is a prop er class not a set V is also considered as a prop er class The axioms of NBG are simply chosen with resp ect to the limitation of size constraint Other approaches against the deciencies of the naive approach alternatively played with its language and are generally dubb ed typetheoretical Russell and Whiteheads Theory of Types is the earliest and most p opular of these Whitehead Russell In this theory a hierarchy of typ es is established to forbid circularity and hence avoid paradoxes For this purp ose the universe is divided into typ es starting with a collection M of individuals The elements of M are of typ e Sets whose memb ers are of typ e are said to b e of typ e sets whose elements are of typ e are said to b e of typ e and so n n+1 on The memb ership relation is dened b etween sets of dierent typ es eg x y Therefore x x is not even a valid formula in this theory and Russells Paradox is trivially avoided Similar to the Theory of Typ es is Quines New Foundations NF which he invented to overcome some unpleasant asp ects of the former Quine NF uses only one kind of variable and a binary predicate letter for memb ership A notion called stratication is 2 intro duced to maintain the hierarchy of typ es In NF Russells Paradox is avoided as in the Theory of Typ es since the problematic w is not stratied ZF set theory ZermeloFraenkel ZF is the earliest axiomatic system in set theory The rst axiomati zation was by E Zermelo A A Fraenkel observed a weakness of Zermelos system and prop osed a way to overcome it His prop osal was reformulated by T Skolem by intro ducing a new axiom ZF is carried out in a language which includes sets as ob jects and for memb ership Equality is dened externally by the Axiom of Ex tensionality which states that two sets are equal if and only if i they have the same elements ZFs essential feature is the cumulative hierarchy it prop oses Parsons The intention is to build up mathematics by starting with and then construct further sets in a stepwise manner by various dened op erators Hence there are no individuals urelements in the universe of this theory The cumulative hierarchy works as follows Tiles The Nul l Set Axiom guarantees that there is a set with no elements ie This is the only set whose existence is explicitly stated The Pair Set Axiom states the existence of a set which has a memb er when the only existing set is So the set fg can now b e formed now and we have two ob jects and fg The application of the axiom rep etitively yields any nite numb er of sets each with only one or two elements It is the Sum Set Axiom which states the existence of sets containing any nite numb er of elements by dening the S union of already existing sets Thus ff fgg ff fgggg f fg f fggg However it should b e noted that all these sets will b e nite b ecause only nitely many sets can b e formed by applying Pair Set and Sum Set nitely many times It is the Axiom of Innity which states the existence of at least one innite set from which other innite sets can b e formed The set which the axiom asserts to exist is f fg f fgg f fg f fggg g Figure The cumulative hierarchy The cumulative hierarchy is depicted in Figure Thus the ZF universe simply starts with the and extends to innity It can