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Would you please consider reviewing the attached article, Thank you MACherian m.a.cherian@btinternet.com

On Gödel’s Conjecture Abstract: ‘Not (proved or disproved)’ does not exhaust all reference to ‘proved’, or ‘disproved’.

Gödel presents his Incompleteness Theorems as proof that in natural numbers, inductively (recursively) generated as a ‘denumerably infinite’ set large enough for his numbering procedure, there is no consistent and complete formalization of elementary arithmetic. His proof is conditional on the axioms of Principia Mathematica [PM], with the added axiom of infinity (in the form he wants it, viz. ‘there are exactly denumerably many individuals’), the axiom of choice, and Zermelo-Fraenkel-von Neumann axioms of set theory appended to the Peano Postulates. [Collected Works Vol.1, OUP 1986 p.124]. He says, "…all methods of proof used in mathematics today have been formalized in them, i.e. reduced to a few axioms and rules of inference. It may therefore be conjectured (Vermutung) that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned", (p.145). For his conjecture to hold he also needs to have shown that only valid formulae follow from the rule-following inferences he relies on of PM.

If ‘p’is taken as true, and ‘-p’ false, the logical and the formalist equivalence and truth of: |p| < = > |(- -p)| < = > |(p or -p)| < = >|-(p and -p)|, viz. the laws of double negation, excluded middle and non-contradiction follow. Any one statement taken as true implies implies the truth of any and all the others. Based on the same axioms and rules of inference, on which Gödel [p.145] claims that in a formally deductive system, an arithmetical statement cannot be 'proved or disproved', i.e. -(p or -p), and hence is undecidable from within that system; he could have added that it is also not |-(p and -p)| i.e. ‘proved and disproved'; and ‘-p’, i.e. ‘disproved’. ‘Not (proved or disproved)’ does not exhaust all reference to ‘proved’, or ‘disproved’.

The law of Double Negation is |-|-p| < = > |p|. There are only two ways about it, either p or else -p, viz. the law of the excluded middle |p or -p|. |-|-p| is another way of writing |p|, and |p or –p|. Against Gödel, it is only necessary to show that the law of excluded middle |p or -p| entails that of non-contradiction |-(p and –p)|. When only |p or -p| is true, |p and -p| is false, |-|p and -p| is true viz. the law of non-contradiction. Equivalent steps of deduction are used in PM. The equivalence of |p or -p| and |-|p and -p|, also follows by De Morgan’s Rules (included in the PM) starting from either side. MACherian ||||


Archives: 1 (December 2006 to January 2008), 2 (First half of February 2008), 3 (Second half of February 2008), 4 (March 2008), 5 (April and May 2008), 6 (June 2008), 7 (July, August, September 2008), 8 (October 2008 - January 2009; left)

I'll miss you. Garden. 21:48, 4 January 2009 (UTC) Ahh, don't go. ~EDDY (talk/contribs/editor review)~ 18:38, 10 January 2009 (UTC)[reply]

The Military history WikiProject Newsletter : Issue XXXIV (December 2008)

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Delievered by SoxBot II (talk) at 00:00, 18 January 2009 (UTC)[reply]

Stop right there!

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You are leaving? Seriously? Oh'man!

Well, you are bound to come back. Even if you don't contribute, I know you are checking your page and watchlist regularly!

Cheers mate!

Λuα (Operibus anteire) 09:57, 22 January 2009 (UTC)[reply]

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Help!

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Hey Nousernamesleft, we met during the FAR for Holkham Hall‎ when you rewrote the lead and did a fine job ce'ing the page. I never forget a good copyeditor, so I'm wondering if could point your attention towards The Lucy poems. Its an article on a poem that might be taken to FAC in 2 or 3 weeks, and as a lith article the prose just need to be just so. There is already a fine copyeditor working on it, but goods as she is she cannot be SuperLiz so help would be appreciated. If you are busy, no worries. Ceoil (talk) 14:04, 14 February 2009 (UTC)[reply]

I am afraid he is taking some time off Wikipedia! :(
Cheers mate!
Λuα (Operibus anteire) 20:27, 16 February 2009 (UTC)[reply]

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