He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and nondenumerable sets (uncountably infinite sets).A countable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
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Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.
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His address was published in the Conference Proceedings (The Theory of Models, North-Holland Publishing Co., 1965) as "On the denumerable models of theories with extra predicates", pp 376–389. In this paper he characterizes the countable ("denumerable") structures which can be made into models of a theory by adding interpretations of the extra predicates used in defining the theory. His characterization involves (infinite) expressions beginning with an infinite sequence of alternating quantifiers. Such expressions are now interpreted using infinite two-person games.
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Aristotle's solution to Zeno's paradoxes involves the idea that time is not made out of durationless instants, but ever smaller temporal intervals. Every interval of time can be divided into smaller and smaller intervals, without ever terminating in some privileged set of durationless instants. In other words, motion is possible because time is gunky. Despite having been a relatively common position in metaphysics, after Cantor's discovery of the distinction between denumerable and non-denumerable infinite cardinalities, and mathematical work by Adolf Grünbaum, gunk theory was no longer seen as a necessary alternative to a topology of space made out of points.
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Within the formalism of QFT such a picture generally does not exist, because these two representations are unitarily inequivalent. Thus the practitioner of QFT is confronted with the so-called choice problem, namely the problem of choosing the 'right' representation among a non-denumerable set of inequivalent representations.
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W. Marek, On the metamathematics of impredicative set theory. Dissertationes Mathematicae 98, 45 pages, 1973 He proved that the so-called Fraïssé conjecture (second-order theories of denumerable ordinals are all different) is entailed by Gödel's axiom of constructibility. Together with Marian Srebrny, he investigated properties of gaps in a constructible universe.
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To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.
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The original formulation by Vaught was not stated as a conjecture, but as a problem: Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly ℵ1 non-isomorphic denumerable models? By the result by Morley mentioned at the beginning, a positive solution to the conjecture essentially corresponds to a negative answer to Vaught's problem as originally stated.
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The paradox can be interpreted as an application of Cantor's diagonal argument. It inspired Kurt Gödel and Alan Turing to their famous works. Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows: Let E be the set of real numbers that can be defined by a finite number of words. This set is denumerable.
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The Dartmouth mathematics department professors also wrote Finite Mathematical Structures (1959) and Finite Mathematics with Business Applications (1962). Other colleges and universities followed this lead and several more textbooks in Finite Mathematics were composed elsewhere. The topic of Markov chains was particularly popular so Kemeny teamed with J. Laurie Snell to publish Finite Markov Chains (1960) to provide an introductory college textbook. Considering the advances using potential theory obtained by G. A. Hunt, they wrote Denumerable Markov Chains in 1966.
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This sequence has order type \omega, so \omega+\omega is the limit of a sequence of type less than \omega+\omega whose elements are ordinals less than \omega+\omega; therefore it is singular. \aleph_1 is the next cardinal number greater than \aleph_0, so the cardinals less than \aleph_1 are countable (finite or denumerable). Assuming the axiom of choice, the union of a countable set of countable sets is itself countable. So \aleph_1 cannot be written as the sum of a countable set of countable cardinal numbers, and is regular.
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An example is the set of natural numbers, In Cantor's formulation of set theory, there are many different infinite sets, some of which are larger than others. For example, the set of all real numbers is larger than , because any procedure that you attempt to use to put the natural numbers into one-to-one correspondence with the real numbers will always fail: there will always be an infinite number of real numbers "left over". Any infinite set that can be placed in one-to-one correspondence with the natural numbers is said to be "countable" or "denumerable". Infinite sets larger than this are said to be "uncountable".
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The proof of the "razor" is based on the known mathematical properties of a probability distribution over a countable set. These properties are relevant because the infinite set of all programs is a denumerable set. The sum S of the probabilities of all programs must be exactly equal to one (as per the definition of probability) thus the probabilities must roughly decrease as we enumerate the infinite set of all programs, otherwise S will be strictly greater than one. To be more precise, for every \epsilon > 0, there is some length l such that the probability of all programs longer than l is at most \epsilon.
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J. L. Doob (1966) Review: Denumerable Markov Chains, This textbook, suitable for advanced seminars,Preface, page vi was followed by a second edition in 1976 when an additional chapter on random fields by David Griffeath was included. Kemeny and Kurtz were pioneers in the use of computers for ordinary people. After early experiments with ALGOL 30 and DOPE on the LGP-30, they invented the BASIC programming language in 1964, as well as one of the world's first time-sharing systems, the Dartmouth Time- Sharing System (DTSS). In 1974, the American Federation of Information Processing Societies gave an award to Kemeny and Kurtz at the National Computer Conference for their work on BASIC and time-sharing.
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The four Merton ‘Calculators’ were not only well versed in the current issues of philosophy during the fourteenth century; they actually initiated new groundbreaking scientific postulations. John of Dumbleton was no exception. Because he concurred with many of the positions held by William Ockham (1288–1348)—especially the idea that is commonly referred to as Ockham’s razor, which states that the most simplistic explanations are ideal—he may have learned how to succinctly formulate his scientific conjectures. Of Dumbleton’s many scientific theories there is one in particular that is worth mentioning here. By making the assumption that bodies are finite, Dumbleton was able to conjecture that contraction or expansion, as in cases of condensation or rarefaction, does not eliminate any parts of a body; rather, a “denumerable number of parts” always exists.
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Each set in the countable sequence of sets (Si) = S1, S2, S3, ... contains a nonzero, and possibly infinite (or even uncountably infinite), number of elements. The axiom of countable choice allows us to arbitrarily select a single element from each set, forming a corresponding sequence of elements (xi) = x1, x2, x3, ... The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. I.e., given a function A with domain N (where N denotes the set of natural numbers) such that A(n) is a non-empty set for every n ∈ N, then there exists a function f with domain N such that f(n) ∈ A(n) for every n ∈ N.
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Description numbers play a key role in many undecidability proofs, such as the proof that the halting problem is undecidable. In the first place, the existence of this direct correspondence between natural numbers and Turing machines shows that the set of all Turing machines is denumerable, and since the set of all partial functions is uncountably infinite, there must certainly be many functions that cannot be computed by Turing machines. By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable. First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine.
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The British philosopher Roy Bhaskar, who is closely associated with the philosophical movement of critical realism writes: :"I differentiate the 'ontic' ('ontical' etc.) from the 'ontological'. I employ the former to refer to :# whatever pertains to being generally, rather than some distinctively philosophical (or scientific) theory of it (ontology), so that in this sense, that of the ontic1, we can speak of the ontic presuppositions of a work of art, a joke or a strike as much as a theory of knowledge; and, within this rubric, to :# the intransitive objects of some specific, historically determinate, scientific investigation (or set of such investigations), the ontic2. :"The ontic2 is always specified, and only identified, by its relation, as the intransitive object(s) of some or other (denumerable set of) particular transitive process(es) of enquiry. It is cognitive process-, and level-specific; whereas the ontological (like the ontic1) is not.
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In his 1975 article "Outline of a Theory of Truth", Kripke showed that a language can consistently contain its own truth predicate, something deemed impossible by Alfred Tarski, a pioneer in formal theories of truth. The approach involves letting truth be a partially defined property over the set of grammatically well-formed sentences in the language. Kripke showed how to do this recursively by starting from the set of expressions in a language that do not contain the truth predicate, and defining a truth predicate over just that segment: this action adds new sentences to the language, and truth is in turn defined for all of them. Unlike Tarski's approach, however, Kripke's lets "truth" be the union of all of these definition-stages; after a denumerable infinity of steps the language reaches a "fixed point" such that using Kripke's method to expand the truth-predicate does not change the language any further.
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