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Topic: Definable number

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	Science Fair Projects - Definable number
		However, most real numbers are not definable: the set of all definable numbers is countably infinite (because the set of all logical formulas is) while the set of real numbers is uncountably infinite (see Cantor's diagonal argument).
		The field of definable numbers is not complete; there exist convergent sequences of definable numbers whose limit is not definable (since every real number is the limit of a sequence of rational numbers).
		In fact, all theorems of calculus remain true if the field of real numbers is replaced by the field of definable numbers, sequences are replaced by definable sequences, sets are replaced by definable sets and functions by definable functions.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Definable_number (532 words)

	Encyclopedia: Definable number (Site not responding. Last check: 2007-10-26)
		A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V).
		However, most real numbers are not definable: the set of all definable numbers is countably infinite (because the set of all logical formulas is) while the set of real numbers is uncountably infinite (see Cantor's diagonal argument).
		The field of definable numbers is not complete; there exist convergent sequences of definable numbers whose limit is not definable (since every real number is the limit of a sequence of rational numbers).
	www.nationmaster.com /encyclopedia/Definable-number (452 words)

	Definable number - Wikipedia, the free encyclopedia
		While every computable number is definable, the converse is not true: Chaitin's constant is definable but not computable.
		The related concept of "standard" numbers, which can only be defined within a finite time and space, is used to motivate axiomatic internal set theory, and provide a workable formulation for illimited and infinitesimal number.
		For example, we can enumerate all such definable numbers by the Gödel numbers of their defining formulas, and then use Cantor's diagonal argument to find a particular real that is not definable in the sense of this article.
	en.wikipedia.org /wiki/Definable_number (787 words)

	Encyclopedia Search (Site not responding. Last check: 2007-10-26)
		A computable normal number was constructed by Becher and Figuera; an example of...normal number is given by Chaitin 's constant Ω.
		Although both Skewes numbers are big...compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as...
		He defined a computable number to be a real number for which there is an algorithm that,...all numbers are uncomputable.
	www.encyclopedian.com /search.php?searWords=Computable+number (435 words)

	Talk:Definable number - Wikipedia, the free encyclopedia
		A number is definable if there exists a proposition capable of being conceived by a mind of at least human-level intelligence which is true for that number and false for every other number.
		A number x is definable if, for some formal language F and associated model M, there exists a statement S(x) in F such that S(x) is true in M and for all numbers y S(y) is true in M iff x=y.
		All _real numbers_ are definable and countable --- just pick any irrational number (say, pi) which is definable and any real number can be defined by how it differs digit-for-digit with pi with their place-value-positional-numeral-system (say, decimal or binary) expansion point aligned.
	en.wikipedia.org /wiki/Talk:Definable_number (2518 words)

	Definable number
		There are however many real numbers which are not definable: the set of all definable numbers is countable (because the set of all logical formulas is) while the set of real numbers isn't (see Cantor's diagonal argument).
		The field of definable numbers isn't complete; there exist convergent sequences of definable numbers whose limit isn't definable (since every real number is the limit of a sequence of rational numbers).
		While every computable number is definable, the converse isn't true: Chaitin's constant is definable (otherwise we couldn't talk about it) but not computable.
	www.findword.org /de/definable-number.html (592 words)

	PlanetMath: computable number
		There are however many real numbers which are not computable: the set of all computable numbers is countable (because the set of algorithms is) while the set of real numbers is
		Every computable number is definable, but not vice versa.
		This is version 7 of computable number, born on 2003-04-08, modified 2004-09-28.
	planetmath.org /encyclopedia/ComputableNumber.html (286 words)

	Definable number - Wikipedia
		A definable number is a real number which can be unambigously defined by some mathematical statement.
		The definable numbers form a field containing all numbers that have ever been or can be unambigously described.
		The field of definable numbers is not complete; there exist convergent sequences of definable numbers whose limit is not definable.
	nostalgia.wikipedia.org /wiki/Definable_number (279 words)

	Definable number (Site not responding. Last check: 2007-10-26)
		The definable numbers form a field containing allnumbers that have ever been or can be unambiguously described.
		In fact, all theorems of calculus remaintrue if the field of real numbers is replaced by the field of definable numbers, sequences are replaced by definable sequences,sets are replaced by definable sets and functions by definable functions.
		While every computable number is definable, the converse isnot true: Chaitin's constant is definable (otherwise wecouldn't talk about it) but not computable.
	www.therfcc.org /definable-number-74419.html (264 words)

	computable number@Everything2.com
		We call a real number x computable, if it may be described by a Turing machine T, in the sense that when T is input a number n, it outputs the first n digits (say decimal, or whatever you like) of x.
		Computable numbers are also called recursive numbers (when defined through recursive functions rather than TMs) or effectively definable (as distinct from definable numbers, of which there are many more).
		It's a hop and a skip to deciding that all algebraic numbers are computable, and throwing e and pi into your expressions does nothing for leaving the computable numbers.
	everything2.com /index.pl?node_id=47526 (924 words)

	Personal identification system - Patent 4376279
		Thus, the generation of the Offset Number is dependent upon the PAN which is assigned by the institution, the IN, which is secretly selected for a one-time entry by one or preferably more officers of the institution, and the PIN which is secretly selected by the customer.
		Furthermore, for many values of m the number of binary registers required is identical for the decomposed and nondecomposed versions of the modulo m FSR as is the case for m=10.
		To determine the PIN from the PAN and Offset Number is impossible, particularly due to the focusing and defocusing effects which is produced by the incorporation of FSR B, as heretofore described.
	www.freepatentsonline.com /4376279.html (16338 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Not all definable real numbers are computable: the real number whose n-th digit is 0 iff the n-th Turing machine stops and 1 otherwise is definable but it is not computable.
		One last concept, but probably the most important, is that of constructibility: a number is constructible iff it belongs to the smallest class containing the ordinals and closed under the Godel operations, the class L (the constructible universe).
		Now all constructible real numbers are ordinal definable, and the converse is consistent (but it is also consistent that there exists a real number that is ordinal definable but not constructible).
	www.math.niu.edu /~rusin/papers/known-math/98/definable (316 words)

	Godel's Theorems
		Definability, the relation "the number n is defined by the (English) sentence s" not definable in English.
		Since sentences of number theory can be coded up as numbers (the ASCII coding your computer uses does just fine), the set of true sentences can be identified with the set TRUTH of numbers which encode true sentences.
		TRUTH, the set of numbers which encode true sentences of number theory, is not definable in number theory.
	www.math.hawaii.edu /~dale/godel/godel.html (2115 words)

	Cryptographic method using modified fractional fourier transform kernel - US Patent 6718038 (Site not responding. Last check: 2007-10-26)
		It is another object of the present invention to encrypt and decrypt a signal using at least one component of a modified fractional Fourier transform kernel a user-definable number of times with at least one encryption key and at least one decryption keys.
		The cryptographic strength of the encryption method of the present invention is proportional to the number, type, and combination of components of the modified fractional Fourier transform kernel selected in the third step 3.
		The cryptographic strength of the encryption method of the present invention is proportional to the number and diversity of encryption keys established in the second step 2.
	www.patentstorm.us /patents/6718038.html (2497 words)

	Richard's Paradox
		The solution to these paradoxes is usually explained as follows: they depend on being imprecise about what is meant by defining a number, or being interesting.
		We are assuming that the rules T specify which strings T-define which numbers.
		Let X be the set of all real numbers that can be defined, not necessarily according to some formal system, but just in some unambiguous way that any mathematically trained person could in principle understand.
	www.dpmms.cam.ac.uk /~wtg10/richardsparadox.html (1040 words)

	Marked orthodonic aid and method of manufacturing - US Patent 5326259 (Site not responding. Last check: 2007-10-26)
		In order to so improve a metallic orthodontic aid having an optically visible marking area that the marking is easy to apply and is also visible up to a definable number of recycling process cycles, it is proposed that the marking area be a surface of an additionally produced layer with a remelt structure.
		In this case, the number of recycling process cycles after which the layer with a remelt structure is worn away is fixable by the thickness alone.
		In accordance with the invention, the thicknesses d, d 1, d 2 of the layers with a remelt structure 30, 30' are chosen in the range of approximately 0.1 µm to 0.5 µm.
	www.patentstorm.us /patents/5326259.html (2128 words)

	Definable number (Site not responding. Last check: 2007-10-26)
		In April, no doubt, something may drift on the still surface you hear the constant sweet plash of the wants re-stocking, and the burns in winter need watching, in the of, to take trouble and incur expense; and free fishing, by the all.
		The best we can say for it is that vast numbers of persons hope.
		Even solitude is no longer to be found in the scene which Here, have I thought, 'twere sweet to dwell, But no longer does So stilly is the solitude." tourists, and with yells from nymphs and swains disporting Ages and revolutions must pass before the ancient peace returns;.
	www.explainthis.info /de/definable-number.html (517 words)

	Voice Recorder \| Call Processing \| Call Logging \| Emergency Predictive Dialers
		Definable number of attempt to dial and get a positive contact:
		Definable time to wait to retry the same contact alternate number:
		Definable time to wait to retry the same contact number:
	www.voicegate.com /features_list.html (225 words)

	[No title]
		Obviously, the larger the number of such potential classifications, the lower the predictability that is possible on the basis of a partial assignment ("loading of training data") already achieved.
		In fact, it is possible to define a fixed analytic function fi, with m = 1, for which Sk consists of all sequences of k distinct elements of R, not merely dense in Rkm, for all k; see [Sontag 1992].
		The Appendix recalls the definition and basic facts about ("expra") definable functions--informally, these are functions that can be defined in terms of any 3 first-order logic sentence which is built out of the standard propositional connectives, existential and universal quantification, and which involves rational operations and exponentiation.
	www.math.rutgers.edu /~sontag/generic.html (3848 words)

	solution (Site not responding. Last check: 2007-10-26)
		And that leads to the representation of every real number as not just the would-be limit of a sequence of finite decimals, but also a limit which is actually reached, so that a real number is identical with a certain infinite decimal ([26], pp189, 191).
		If we were to define 'real numbers' not in terms of Platonic limits, but merely convergent sequences of rationals, as the Intuitionists have done, then we would be identifying 'real numbers' with certain functions, since sequences are functions from the natural numbers.
		Even if fn(x) determines a 'real number', which function it is is only determinable from its ordinal place amongst all computable functions, not from its ordinal place amongst the real-number functions, with the result that, if the latter is 'n', then fn(x) is not a calculable function of n.
	www.arts.uwa.edu.au /philoswww/Staff/solution.html (5675 words)

	Three beliefs that lend illusory legitimacy to Cantor’s diagonal argument
		Whatever the reason, he offered an alternative argument that was, essentially, based on defining an uncomputable number-theoretic function, rather than on non-constructively postulating that a period, followed by a non-terminating sequence of the digits 0 and 1, necessarily defines a Dedekind real number.
		Turing defined the “oracle” purely mathematically as an uncomputable function, and said, “We shall not go any further into the nature of this oracle apart from saying that it cannot be a machine.” The essential point of the oracle is that it performs non-mechanical steps.
		By Platonically we mean it is not necessary that, given any natural number n, we have access to an effective method for calculating the n’th term of the sequence in the definition that follows.
	alixcomsi.com /Three_beliefs.htm (4352 words)

	United States Patent: 5,937,422
		After a topic description is generated for a number of documents, the present invention may be used to search for documents similar to a particular document or for sorting a number of documents.
		The dictionary salience score may be defined as D.sub.j =.beta..sub.j (S.sub.j log(d.sub.max/d.sub.j)) 0.5, where d.sub.j is the number of dictionary terms that use the word w.sub.i in their definition, and where d.sub.max is the maximum value of d, (i.e., the number of dictionary terms that use the most frequently used dictionary term in the dictionary).
		The user may define the number of words that will be used to describe the topic of the input text.
	www.fas.org /irp/program/process/990810-nsa-patent.htm (4933 words)

	Download Fleet Maintenance Pro 7.0 - Fleet Maintenance Pro Free Download - tracks and organizes preventive maintenance ... (Site not responding. Last check: 2007-10-26)
		When maintenance is performed, a detailed entry is recorded to the vehicle's history report.
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	www.maxxdownload.com /Fleet_Maintenance_Pro.html (457 words)

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