User talk:MACherian
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I have replaced your Kurt Gödel's Incompleteness Theorem by a redirect, as we already have an article about that theorem. If you like you can check that article and incorporate whatever is still missing. However the article you posted looked a bit like you copied it from somewhere (no offence intended), which is not allowed here, we can only accept contributions you create on your own, or which have the explicit allowance of the copyright owner. But anyway, welcome and keep on editing. andy 08:35, 21 Feb 2005 (UTC)
February 2009
[edit]
Please refrain from making test edits in Wikipedia pages, such as those you made to Gödel's incompleteness theorems, even if your ultimate intention is to fix them. Such edits appear to be vandalism and have been reverted. If you would like to experiment again, please use the sandbox. Thank you. Welcome, etc. Please read up on how to write and edit articles in Wikipedia. After you have done that, if you would like to add a large amount of content, first make sure you're not doing WP:OR and it's not WP:FRINGE, and then bring it up on the discussion page for the article. Thanks. Quaeler (talk) 13:15, 19 February 2009 (UTC)
On Gödel’s Conjecture
[edit]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 in his second theorem 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.
Copied from User_talk:Tango: I'm afraid formal logic is among my weakest areas, so I'm probably not the best person to review this. Try Wikipedia_talk:WikiProject Mathematics. Good luck and happy editing! --Tango (talk) 14:09, 25 February 2009 (UTC)
Request for inclusion
[edit]Your request for inclusion is basically out of place. Your logic seems to be if you shout "The moon is made of cheese" in a room of people and no one has the energy to point out what's wrong with that statement, then your statement must be valid. If there has been feedback and discussion about your proposed edits, please point people to these discussions; otherwise, there is nothing to do. Additionally, your User page is for your own personal writing. If you want to make a discussion (like "Request for inclusion"), you must do it on your talk page. Quaeler (talk) 13:47, 25 March 2009 (UTC)