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A.Lewis, Infinite time turing machines
 Journal of Symbolic Logic
"... Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ..."
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Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ∆1 2 sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators. In these days of superfast computers whose speed seems to be increasing without bound, the more philosophical among us are perhaps pushed to wonder: what could we compute with an infinitely fast computer? By proposing a natural model for supertasks—computations with infinitely many steps—we provide in this paper a theoretical foundation on which to answer this question. Our model is simple: we simply extend the Turing machine concept into transfinite ordinal time. The resulting machines can perform infinitely many steps of computation, and go on to more computation after that. But mechanically they work just like Turing machines. In particular, they have the usual Turing machine hardware; there is still the same smooth infinite paper tape and the same mechanical head moving back and forth according to a finite algorithm, with finitely many states. What is new is the definition of the behavior of the machine at limit ordinal times. The resulting computability theory leads to a notion of computation on the reals, concepts of decidability and semidecidability for sets of reals as well as individual reals, two kinds of jumpoperator, and a notion of relative computability using oracles which gives a rich degree structure on both the collection of reals and the collection of sets of reals. But much remains unknown; we hope to stir interest in these ideas, which have been a joy for us to think about.
Nonstandard Set Theories and Information Management
"... . The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of thes ..."
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. The merits of set theory as a foundational tool in mathematics stimulate its use in various areas of artificial intelligence, in particular intelligent information systems. In this paper, a study of various nonstandard treatments of set theory from this perspective is offered. Applications of these alternative set theories to information or knowledge management are surveyed. Keywords: set theory, knowledge representation, information management, commonsense reasoning, nonwellfounded sets (hypersets) 1. Introduction Set theory is a branch of modern mathematics with a unique place because various other branches can be formally defined within it. For example, Book 1 of the influential works of N. Bourbaki is devoted to the theory of sets and provides the framework for the remaining volumes. Bourbaki said in 1949 (Goldblatt, 1984) 1 : "[A]ll mathematical theories may be regarded as extensions of the general theory of sets : : : [O]n these foundations I can state that I can build up t...
Axiomatic Set Theories And Common Sense
, 1994
"... Various axiomatic set theories (ZF, NBG, NF, and KPU) are studied with a critical eye. The basic mathematical and philosophical reasons behind their axioms are given, as well as their review from the commonsense point of view. An introduction to a "commonsense set theory" is attempted at t ..."
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Various axiomatic set theories (ZF, NBG, NF, and KPU) are studied with a critical eye. The basic mathematical and philosophical reasons behind their axioms are given, as well as their review from the commonsense point of view. An introduction to a "commonsense set theory" is attempted at the end. 1. Introduction In the foundations of mathematics, one of the most popular approaches is set theory. Almost all of the mathematical objects can be constructed out of sets. In this view, mathematics deals only with the properties of sets and all of these properties can be deduced from a suitable list of axioms [14, 21]. In an application of set theory to commonsense 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 col...