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 ("Es liegt daher die 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.
The metalevel operation in Gödel’s First Incompleteness Theorem is illogical. Quine [p.82] praised the Incompleteness theorems as ‘ground-breaking, bond-breaking, road-breaking, epoch-making’. He. also stated in another context that .the metalevel proceeds "as if the classes were not there until specified, and hence not present among the values of variables used in specifying them", [ ibid p.95].
To Tarski’s `paradigmatic reference to truth, his denotation of truth in formalized ‘algebraically’ constructed and rule following’ metalanguages, where the sense of the sentences are given by their structures, he himself added a footnote, "...it is seen in some cases having succeeded in constructing an adequate definition of truth for a theory T in its metatheory [M], we may still be unable to show that all provable sentences of T are true in the sense of this definition, and hence we may be unable to carry out the consistency proof for T in M", [Benacerraf fn. p.237]. M. A. Cherian Username MACherianMACherian (talk) 19:21, 16 November 2010 (UTC) W.V. Quine, Quiddities, Penguin, London 1990 P. Benacerraf and H. Putnam, eds. Philosophy of Mathematics, Selected Readings, CUP, Cambridge, 1983
User MACherian There has been no edits nor objections to my article On Godel’s Conjecture. I present valid objections to his second theorem. Could those concerned, now incorporate it to the main article on Godel's Incompleteness Theorem, in the section Limitations of Gödel's Theorems. Thank you. MACherianMACherian (talk) 12:52, 25 March 2009 (UTC)
- I don't think it is suitable for the article, sorry. It's confusingly written and seems to miss the point of the second incompleteness theorem. You might like the presentation in Torkel Franzén's book about the incompleteness theorem, which is supposed to be pretty clear (I've only glanced at the book briefly, but it has gotten many favorable reviews). 69.228.170.24 (talk) 02:21, 1 May 2010 (UTC)