Godels incompleteness theorem are invalid ie illegitimate
in [Math]
|
Prev: unit vector extrapolation
Next: PRECISE VALAUTION OF RADIAN VALUE BY EQUATION VERSUS PI MEASUREMENT using -1 tangent value
From: melsi on 16 Jun 2010 06:04 Godels incompleteness theorems are invalid ie illegitimate It is argued that Godels incompleteness theorems are invalid ie illegitimate for 5 reasons: he uses the axiom of reducibility- which is invalid ie illegitimate,he constructs impredicative statement which is invalid ie illegitimate ,he cant tell us what makes a mathematic statement true, he falls into two self-contradictions,he ends in three paradoxes http://www.scribd.com/doc/32970323/Godels-incompleteness-theorem-invalid-illegitimate First of the two self-contradictions Godels first theorem ends in paradox –due to his construction of impredicative statement Now the syntactic version of Godels first completeness theorem reads Proposition VI: To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r). But when this is put into plain words we get http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem Gödel's first incompleteness theorem states that: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250). Now truth in mathematics was considered to be if a statement can be proven then it is true Ie truth is equated with provability http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics ”…from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system. The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system” http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250) For each consistent formal theory T having the required small amount of number theory … provability-within-the-theory-T is not the same as truth; the theory T is incomplete.” Now it is said godel PROVED "there are true mathematical statements which cant be proven" in other words truth does not equate with proof. if that theorem is true then his theorem is false PROOF for if the theorem is true then truth does equate with proof- as he has given proof of a true statement but his theorem says truth does not equate with proof. thus a paradox THIS WHAT COMES OF USING IMPREDICATIVE STATEMENTS GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE GODEL CAN NOT TELL US WHAT MAKES A STATEMENT TRUE Now truth in mathematics was considered to be if a statement can be proven then it is true Ie truth was s equated with provability http://en.wikipedia.org/wiki/Truth#Truth_in_mathematics ”…from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's theorem and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system. The works of Kurt Gödel, Alan Turing, and others shook this assumption, with the development of statements that are true but cannot be proven within the system” Now the syntactic version of Godels first completeness theorem reads Proposition VI: To every ω-consistent recursive class c of formulae there correspond recursive class-signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg(c) (where v is the free variable of r). But when this is put into plain words we get http://en.wikipedia.org/wiki/G%C3%B6del ... ss_theorem “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250) For each consistent formal theory T having the required small amount of number theory … provability-within-the-theory-T is not the same as truth; the theory T is incomplete.” In other words there are true mathematical statements which cant be proven But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless Ie if Godels theorem said there were gibbly statements that cant be proven But if godel cant tell us what a gibbly statement was then we would say his theorem was meaningless Now at the time godel wrote his theorem he had no idea of what truth was as peter smith the Cambridge expert on Godel admitts http://groups.google.com/group/sci.logi ... 12ee69f0a8 Quote: Gödel didn't rely on the notion of truth but truth is central to his theorem as peter smith kindly tellls us http://assets.cambridge.org/97805218...40_excerpt.pdf Quote: Godel did is find a general method that enabled him to take any theory T strong enough to capture a modest amount of basic arithmetic and construct a corresponding arithmetical sentence GT which encodes the claim 'The sentence GT itself is unprovable in theory T'. So G T is true if and only if T can't prove it If we can locate GT , a Godel sentence for our favourite nicely ax- iomatized theory of arithmetic T, and can argue that G T is true-but-unprovable, and godels theorem is http://en.wikipedia.org/wiki/G%C3%B6...s_theorems#Fir... Quote: Gödel's first incompleteness theorem, perhaps the single most celebrated result in mathematical logic, states that: For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed.1 That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. you see godel referes to true statement but Gödel didn't rely on the notion of truth now because Gödel didn't rely on the notion of truth he cant tell us what true statements are thus his theorem is meaningless |