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	Computable number - Wikipedia, the free encyclopedia
		The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
		Although the computable numbers are an ordered field, the set of Gödel numbers corresponding to computable numbers is not itself computably enumerable, because it is not possible to effectively determine which Gödel numbers correspond to Turing machines that produce computable reals (this problem is in Turing degree 0ˈˈ).
		Because the set of computable numbers is not closed under basic operations such as taking the supremum of a bounded sequence, this set cannot be used as a replacement for the full set of real numbers in classical mathematics.
	en.wikipedia.org /wiki/Computable_number (1268 words)

	Gödel number - Wikipedia, the free encyclopedia
		In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN).
		A Gödel numbering can be interpreted as a programming language with the Gödel numbers assigned to each computable function as the programs which calculate the values for the function in that programming language.
		Gödel numbers are constructed with reference to symbols of the propositional calculus and formal arithmetic.
	en.wikipedia.org /wiki/G%C3%B6del_number (774 words)

	Gödel Numbers and Gödel Numbering (Site not responding. Last check: 2007-11-01)
		Numbers correlated to linguistic objects are called Gödel numbers and the correlation is called a Gödel numbering, after Kurt Gödel, who introduced it in 1931.
		Gödel numbering is often used to show that the elements of some set are countable, effectively constructable (recursively enumerable), and to implement self-reference for the purpose of demonstrating that something is not constructable (incompleteness or non-computability).
		Using this scheme, the set of polynomial equations may be enumerated by systematically counting in base 14 and for each number, determining whether or not it is the number of an equation.
	cs.wwc.edu /KU/Logic/Book/book/node24.html (480 words)

	The Halting Problem, and Gödel's Theorem
		However, instead of just concatenating those numbers into a long string of digits, instead the number representing a string of symbols was the product of consecutive prime numbers, each one raised to the power representing one fo the symbols in the string.
		The rule will be that if a number begins with a digit from 5 to 9, it represents a negative number in ten's complement notation.
		Therefore, a positive number starting with a digit from 5 to 9 must be preceded by at least one leading zero, which is why 01500 was used to represent 5 above instead of 500, which represents -5.
	www.quadibloc.com /math/undint.htm (1652 words)

	PlanetMath: Gödel numbering
		A Gödel numbering is any way off assigning numbers to the formulas of a language.
		Athough anything meeting the properties above is a Gödel numbering, depending on the specific language and usage, any of the following properties may also be desired (and can often be found if more effort is put into the numbering):
		This is version 4 of Gödel numbering, born on 2002-08-23, modified 2003-08-26.
	planetmath.org /encyclopedia/GodelNumbering.html (148 words)

	20th WCP: Computational Complexity and Philosophical Dualism
		G also stands for a number — a Gödel number Gp — which results from the assignment of a code number to each sentence in the language of P that expresses metamathematical sentences.
		If the number of towns increases to a figure greater than 100 we are likely to face combinatorial explosion and a situation in which an algorithm becomes inefficient.
		Of course, one could plausibly argue that the major problems arisen by Complexity Theory, i.e., the size of N (number of steps to solve a problem) and the speed of computing might depend on the type of machine on which the algorithm is to be run.
	www.bu.edu /wcp/Papers/Cogn/CognTeix.htm (3006 words)

	Goedel Numbering
		Goedel invented GoedelNumbering as a way of assigning a unique positive integer to every possible formula in a mathematical system, so that mathematical statements could be translated into numbers, which could then appear within other statements, allowing mathematical statements to become FirstClass values, which he then used to prove his two famous theorems.
		In computing a CRC, the entire message is treated as one large number (string-o'-bits) which is then divided (using digital slight of hand) by an appropriate prime polynomial, yielding a (nearly unique) remainder.
		One is never really interested in the "value" of the message as a number, but only in whether the math (performed again later, on retrieval) produces the same remainder.
	c2.com /cgi/wiki?GoedelNumbering (945 words)

	Goedel summary
		Given any formula, one could figure out what its number is; given any number that belonged to a formula, one could tell what the formula was.
		Then he considered the sentence that essentially said, "My very own Gödel number is not in the set of numbers that belong to provable formulas of the axioms you are now using." In other words, this sentence says of itself that it is not provable from whatever axioms you are now using.
		But it says this entirely in the language of arithmetic, since it says it by referring to its own number and to the set of numbers that represent the theorems that follow from the axiom set.
	www.ilstu.edu /~kfmachin/phi281/goedel_summary.htm (1037 words)

	Peter Suber, "Gödel's Proof"
		The essence of Gödel numbering is to assign numbers to the symbols of a formal language.
		In the numbering scheme that Gödel used to prove his incompleteness theorem, numbers were assigned to (1) the symbols, (2) finite strings of symbols (wffs and nonwffs), and (3) finite sequences of strings (proofs and nonproofs) of the formal language.
		Here, instead of saying that there is no number which is the Gödel number of the proof of G, we are making the separate denials for each natural number: wff-sequence(0) is not the proof of G, wff-sequence(1) is not the proof, wff-sequence(2) is not the proof, and so on.
	www.earlham.edu /~peters/courses/logsys/g-proof.htm (4313 words)

	Mathematics from Goedel to Zeno
		The "rules of the game" are properties of numbers and the way in which the operations (e.g., addition, subtraction, multiplication and division are operations) are legitimately performed.
		That is to say properties of numbers as well as other types of mathematical relationships were viewed as being correct if they seemed to coincide with the physical world.
		Thus numbers themselves generally were regarded as being meaningful only when they represented physical quantities (e.g., numbers of stars, sheep, apples,grains of sand5, distance, temperature, etc.).
	www.nevada.edu /~coheng/goedel.htm (5949 words)

	Gödel and sound numbers Metalogic A2/4 - abelard (Site not responding. Last check: 2007-11-01)
		With numbers, there also exist a series of rules to be followed, often termed an algorithm, (see also lists of instructions) and there is a human who carries out the process and writes down or ‘thinks’ of the various real numbers.
		We may produce a number greater than the number of ‘grains’ of sand on all the shores of this planet, or even a number that is greater than the number of ‘atoms’ in the known universe.
		The integers are the natural numbers plus the negative integers -1,-2,-3 etc. In both cases, zero is taken as a ‘number’, this inclusion of zero as a ‘number’ undifferentiated from other integers, I regard as dubious (see the error of ‘zero’,...
	www.abelard.org /metalogic/metalogicA2.htm (5313 words)

	The New York Review of Books: The Dream of Mind and Machine
		Each string was matched with a particular number determined by its signs, and, since a proof is nothing but an array of strings, it too could be represented in the code with its own "Gödel number." Hofstadter, using a different code from Gödel's, maps each sign of TNT onto a three-digit number.
		If one were to code letters of the alphabet into numbers, for example, one could code any message, and, conversely, given a string of numbers in code, one would be able to tell if they "made sense," if they could be decoded, and what the message was.
		Such a coding would divide all strings of numbers into two classes: those that could be translated into a message and those that couldn't, either because the numbers did not correspond to coded letters or because they did but made no sense.
	www.nybooks.com /articles/7598 (5252 words)

	BBC - h2g2 - Gödel’s Theorem applied to Physics
		Whereas Gödel's theorem applies to the natural numbers, physics deals with observables such as distance, time, mass, energy, which are usually represented by real or complex numbers, in combination with physical constants involving units of distance, time and mass.
		The argument that leads to the contradiction needs some modification, to the extent that a Gödel number must be an integer, then, because the argument depends on demonstrating the existence of contradictions, these contradictions also occurring in any extension of arithmetic, the conclusion follows.
		The first step is to eliminate the real numbers by representing each by a symbol, as is normally the case.
	www.bbc.co.uk /dna/h2g2/A822250 (909 words)

	A Simple exposition of Gödel's Theorem
		But he managed it, and was able to define a relation between numbers which obtained just in case the first (even) number was the Gdel number of a proof-sequence which was in fact a valid proof of the well-formed formula whose Gödel number was the second number in the relation.
		That is to say, there is a very complicated relation between numbers, Pr(x,y), which can be defined in terms of addition and multiplication, and holds when y is the Gödel number of a particular well-formed formula, and x is the Gödel number of a sequence of well-formed formulae which constitutes a proof of y.
		These two manoeuvres enable Gödel to refer to well-formed formulae by numbers, and to represent provability as a property of numbers definable in terms of the simple arithmetical operations of addition and multiplication, though the definitions are themselves very complicated.
	users.ox.ac.uk /~jrlucas/simplex.html (1899 words)

	Godel and Godel's Theorem: Math
		So if you made the statement "every even number greater than two is the sum of two primes," you would be able to prove strictly and mechanically, from the axioms, that it is either true or false.
		In other words, saying "this number has theoremhood" is kind of like saying "this number is a perfect square" or "this number is prime" that we discussed earlier: it's a complicated mathematical construct, which can be built out of simpler operations.
		Similarly, the Gödel number for 10 is 123123123123123123123123123123666 (ten "S"s and a 0).
	www.ncsu.edu /felder-public/kenny/papers/godel.html (4488 words)

	Glenn's Presentation Outline
		In 1899, Giuseppe Peano axiomatized the arithmetic of cardinal numbers.
		We agree to associate with the formula the unique number that is the product of the first ten primes in order of magnitude, each prime being raised to a power equal to the Gödel number of the corresponding elementary sign.
		In a similar fashion, a unique number, the product of as many primes as there are signs (each prime being raised to a power equal to the Gödel number of the corresponding sign), can be assigned to every finite sequence of elementary signs and, in particular, to every formula.
	www.physics.ucla.edu /~chester/CES/november/glenn.html (1210 words)

	Gödels Incompleteness Theorem, A Critical Examination, Revision 2005/11/17 by Don W. Stoner
		The number 97 is the ASCII code for the lower case letter "a." The computer sees an "a" as the binary number "01100001b".
		If the number "n" is the Gödel number for a proof of a statement whose Gödel number is "m", then "n" and "m" can be said to constitute a proof pair.
		We can restate this as: construct "m" by inserting the Gödel number for the number "n" into every location of "n" which is marked by the hexadecimal code "67" for "g".
	geocities.com /stonerdon/godel.html (4684 words)

	A Critical Examination of Gödels Incompleteness Theorem, Revision 2004/12/24 by Donald Wayne Stoner
		We can go a little bit farther into layman's territory by saying that a formal system of number theory must either be inconsistent (make inconsistent decisions about whether or not some statements are true) or be incomplete (be unable to demonstrate the truth of every true statement).
		The number 97 is the ASCII code for the lower case letter "a." The computer sees an "a" as the number "01100001" (binary for 97).
		This means the right side of the "∧" operator in Gödel's string must be true when "m" is the Gödel number for "this string." Therefore, if the whole string is not a theorem, the failure must be in the other part of it.
	www.geocities.com /stonerdon/godel3.html (4443 words)

	[No title]
		This denial of the role of the Gödel code, besides being uncalled for, is also not very carefully stated: discussing the relationship between arithmetical and meta-arithmetical meaning, Sloman sometimes represents the meta-arithmetical meaning as being additional to the arithmetical one, sometimes as being separate from it, and non-existent, and sometimes as excluding the arithmetical one.
		Models of natural numbers can be multiple if they are isomorphic, and even if they are not, provided they are non-standard in the standard sense of the term.
		The non-finite numbers which appear in those models of F in which G(F) is false are not natural according to this informal specification, since they are not successors of 0 in any intuitive, but only in a formal sense.
	nl.ijs.si /~damjan/wi-txt.html (1931 words)

	Reviewing Gödel’s and Rosser’s meta-reasoning of “undecidability”
		I also argue that Goedel’s reasoning can be constructively interpreted as implying that, from the PA-provability of [(Ex)H(x)], we may not always assume the existence of some numeral [h] such that [H(h)] is provable in PA.
		In Anand [An02], I outline in general terms a constructive, and intuitionistically unobjectionable, formalisation PP of our Intuitive Arithmetic M of the natural numbers in which the Axioms and Rules of Inference are recursively definable.
		The non-terminating series of numerals “[0], [0+1], [(0+1)+1], [((0+1)+1)+1],...”, is taken as formally representing, in PA, the various non-terminating, intuitive, natural number series such as “0, 1, 10, 11,...” (in binary format), or “0, 1, 2, 3,...” (in the more common decimal format).
	alixcomsi.com /Constructivity_consider.htm (1900 words)

	[No title]
		We define the computable function p: p(n,m) = 1 if n is the Goedel number of a proof of the formula with Goedel number m, = 0 otherwise.
		Consider the formula ~T(d(a,a)) Let's assume that the Goedel number of this formula is n.
		Observe (1) phi(G) = d(n,n) (because G is obtained from formula n by substituting [n] for the free variable a), and (2) G says "The formula with Goedel number d(n,n) is not a theorem", which is equivalent to "Formula G is not a theorem", which is equivalent to "I am not a theorem".
	www.cs.sfu.ca /~kabanets/308/lectures/10.txt (559 words)

	General Setting for Incompleteness (Site not responding. Last check: 2007-11-01)
		The function satisfies the condition that for every predicate H and every natural number n, the expression H(n) is a sentence; H(n) expresses the proposition that n belongs to the set named by H.
		Assume that every number is the Gödel number of some expression (this is convenent but not necessary).
		For any number set A, by A* we mean the set of all numbers n such that d(n)∈A.
	cs.wwc.edu /~aabyan/Logic/General.html (1232 words)

	Joshua Helston
		This number is defined as the symbol’s “Gödel number.” Using these primitive symbols and their Godel numbers, it is then possible to define what numerical properties a number must have in order to be the Gödel number of some formula.
		Another way to summarize this consequence is to say that any consistent axiomatization of any system rich enough to include natural numbers and their properties fails to capture all truths about the natural numbers, for in any such system it is possible to construct Gödel numbers.
		The argument is essentially an attempt to “list the real numbers” and then construct a real number along the diagonal of the list that differs from each number in the list in at least one spot.
	www.gustavus.edu /academics/philosophy/Helston.html (5499 words)

	A Model-Based Gödel Numbering Scheme
		The basic idea here is to regiment a scheme for associating a natural number with a first-order formula, and vice versa -- where this association must be completely mechanical.
		It's a trivial matter to have our monkey obtain the Gödel number of a given formula (or to have him work in the opposite direction), using the the two tables.
		with their corresponding numbers in the second table, writing these numbers down, and continuing, left to right.
	www.rpi.edu /~faheyj2/SB/SELPAP/MBR/mbr1/node5.html (270 words)

	Incompleteness theorem
		Gödel numbering things is often thought of as an abstract mathematical idea, but it is both extremely common and immensely practical.
		Once we have Gödel numbered the statements in a formal system we can think of the system as a recursive process for enumerating the numbers that correspond to provable statements in the system.
		The essential new ideas are Gödel numbering of statements and modeling a formal system within itself as a computer program to enumerate theorems (or Gödel numbers of theorems).
	www.mtnmath.com /book/node56.html (936 words)

	Goedel's Incompleteness Theorem. Liar's Paradox. Self Reference. By K.Podnieks
		Goedel's proof remains valid for any extensions of PA. An extension of PA is nevertheless some formal theory T (in the language of PA).
		Therefore, Goedel's method allows to prove that a perfect axiom system of natural number arithmetic is impossible: any such system is either w-inconsistent, or it is insufficient to solve some natural number problems.
		Goedel soon discovered that truth in number theory is undefinable — he later went on to prove a combinational form of the Incompleteness Theorem.
	linas.org /mirrors/www.ltn.lv/2005.01.29/~podnieks/gt5.html (6664 words)

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