81. Wikinfo | Gödel's Incompleteness Theorem Gödel s incompleteness theorem. from Wikinfo, an internet encyclopedia. Theseresults do not require the incompleteness theorem. http://www.internet-encyclopedia.org/wiki.php?title=Gödel's_incompleteness_theo

82. Homage To Kurt Godel. folly). Gödel s theorem. Gödel s contribution in D, or; incompleteness,if either n or m is in neither P nor D. Maintained by Eddy. http://www.chaos.org.uk/~eddy/math/Godel.html

The Berry Paradox (a cleaner variant on the `smallest non-interesting number' folly). could not be both consistent and complete; and could not prove itself consistent without proving itself inconsistent. The crucial technical terms of the discussion: Peano's axioms provide a formal description of the process of counting. They can be constructed in any logical system capable of the variety of counting in which any number has a successor - so that there is no `last' number - and distinct numbers have distinct successors. Consistency (of which the petty variety is the hobgoblin of small minds) is that desirable property of a logical system which says that there are no statements which the system regards as both true and false. Completeness is the desirable property of a logical system which says that it can prove, one way or the other, any statement that it knows how to address. ie it cannot be proven either true or false; in particular that it cannot be proven true. But `that it cannot be proven true' is Consequently, any logical system which can make up its mind about its consistency can prove itself inconsistent (provided it can count -

83. Gödel's Theorem My quotations above are from p. 95; Dale Myers, Gödel’s IncompletenessTheorem A very nice web page that builds slowly to the proof; http://www.santafe.edu/~shalizi/notebooks/godels-theorem.html

Notebooks 06 Aug 2002 17:30 A much-abused result in mathematical logic consistent if, given the axioms and the derivation rules, we can never derive two contradictory propositions; obviously, we want our axiomatic systems to be consistent. (The trick is to replace each symbol in the proposition, including numerals, either the system is inconsistent (horrors!), or incomplete, and the truth of those propositions is undecidable (within that system). Such undecidable propositions are known as or models of Peano arithmetic.) It follows that these systems, too, contain undecidable propositions, and are incomplete. deduction from axioms, all arithmetical propositions. And that is all. quod erat demonstrandum. This would actually be a valid demonstration, were only the pentultimate sentence true; but no one has ever presented any evidence that it is true, only vigorous hand-waving and the occasional heartfelt assertion. (Thanks to Jakub Jasinski for politely pointing out an embarrassing error in an earlier version.) Recommended: Michael Arbib

84. Gödel's Incompleteness Theorem - Wikipedia, The Free Encyclopedia http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorem

Gödel's incompleteness theorem From Wikipedia, the free encyclopedia. In mathematical logic Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in . Somewhat simplified, the first theorem states: In any consistent formalization of mathematics that is sufficiently strong to define the concept of natural numbers , one can construct a statement that can be neither proved nor disproved within that system. This theorem is one of the most famous outside of mathematics, and one of the most misunderstood. It is a theorem in formal logic , and as such is easy to misinterpret. There are many statements that sound similar to Gödel's first incompleteness theorem, but are in fact not true. We will explore this further in Misconceptions about Gödel's theorems Gödel's second incompleteness theorem, which is proved by formalizing part of the proof of the first within the system itself, states: Any consistent system cannot be used to prove its own consistency. This result was devastating to a philosophical approach to mathematics known as Hilbert's program David Hilbert proposed that the consistency of more complicated systems, such as

85. Account Disabled Your account is currently unavailable. Please contact your support representative http://www.myrkul.org/recent/godel.htm

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86. How Definitive Is The Standard Interpretation Of Gödel's Incompleteness Theorem How definitive is the standard interpretation of Gödel s IncompletenessTheorem? Bhupinder Singh Anand. (A .pdf file of this paper http://alixcomsi.com/How_definitive_is_the_standard.htm

Index Main essay How definitive is the standard interpretation of Gödel's Incompleteness Theorem? Bhupinder Singh Anand A .pdf file of this essay before the current update is available at http://arXiv.org/abs/math/0307074 and at http://www.mathpreprints.com/math/Preprint/anandb/20030610.1/3 Standard interpretations of Gödel's “undecidable” proposition, [(A x R x )], argue that, although [~(A x R x )] is PA-provable if [(A x R x )] is PA-provable, we may not conclude from this that [~(A x R x )] is PA-provable. We show that such interpretations are inconsistent with a standard Deduction Theorem of first order theories. Contents Introduction An overview A standard Deduction Theorem A number-theoretic corollary ... Conclusion Introduction In his seminal 1931 paper , Gödel meta-mathematically argues that his “undecidable” proposition, [(A x R x , is such that (cf. iv If [(A x R x )] is PA-provable, then [~(A x R x )] is PA-provable. Now, a standard Deduction Theorem of an arbitrary first order theory states that ( , p61, Corollary 2.6): If T is a set of well-formed formulas of an arbitrary first order theory K, and if [

87. ¤à¼wº¸¤£§¹³Æ©w²z The summary for this Chinese (Traditional) page contains characters that cannot be correctly displayed in this language/character set. http://episte.math.ntu.edu.tw/articles/mm/mm_15_4_11/

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