Talk:Gödel's incompleteness theorems/Arguments/Archive 1

message: User:Biedermann, unless you can give a reputable source for there being a gap in Gödels proof, please don't write that in the article. You might wan't to read this for the official policy on the subject.
Rasmus (talk) 11:42, 14 November 2005 (UTC)
replies: text: By the way, I submitted this stuff to many of those peer reviewed papers.
Instead of trying to ridicule me like you as a Wanna-be-Gödel they just tell me. 'That does not fit into the program of our paper'! They never try to give me any ARGUMENTS ! And when I insist,they just don't give any further answers! Biedermann 10:54, 8 January 2007 (UTC)
thread_title:  Gap in Gödel's proof 

message: :The following was copied by me from my talk page. Paul August ☎ 12:45, 4 May 2006 (UTC)
replies: text: Hello. The proof outline you inserted is copied verbatim from Infinity and the Mind by Rudy Rucker, which is copyrighted text, publishing it in Wikipedia could be a copyright violation. Your "refutation" is an apparent violation of WP:NOR. Paul August ☎ 13:11, 3 May 2006 (UTC)

text: see User_talk:Trovatore#Godel and User_talk:Aleph4#Your_reversion_of_G.C3.B6del for other places this discussion is taking place (might be nice if all three discussions were moved to the talk page). -lethe talk [ +] 02:56, 4 May 2006 (UTC)
replies: text: Wikipedia is not a place for you to carry out a conversation about your refutations of an author's work.  Your refutation is original research, and so is not allowed.  If you want to get comments on why your refutation is wrong, we'd be happy to help you at Wikipedia:Reference Desk/Mathematics. -lethe talk [ +] 02:58, 4 May 2006 (UTC)

text: Good. Paul August ☎ 12:23, 4 May 2006 (UTC)
thread_title:  Your revertion... Gödel Incompleteness Theorem 

message: The absence of the quality of impermanence in pure mathematics is the reason for incompleteness in Gödel's Incompleteness Theorem. -- PCE 01:58, 8 May 2006 (UTC)
thread_title: To Summarize

message: (moved here from main article)
’’’Concern’’’, sorry, Sir, yet I have to insert my concern at this place, else nobody will ever take notice of it. To my understanding your ‘’’Proove Sketch’’’ is very successfully concealing the essential problems in the construction of what you designate by ‘’’SU( )’’’, in Gödel’s paper (following Nagel-Newman to get it on a single line of typing) the formula (statement form) (x)~Dem(x,sub(y,13,Z(y))). In the long sequence of some 40 definitions Gödel defines under Nr. 16 and 17 Z(n) to be the Gödel-number of the presentation of the number n as it results from ‘putting the sign ‘f’ n times in front of 0’, so Z(4) is defined as the Gödel-number of  ffff0, and Z(y) would have to be the Gödel-number of the symbol sequence that results from ‘putting the sign f  ‘’’y’’’ times in front of 0’. But such a symbol sequence does not exist, nobody can do something ‘’’y-times’’. Evidently, ‘’’SU( )’’’, as pretty as it appears as abbreviation, does not exist as a symbol sequence in the Formal System, nor does your pretty ‘’’SU(G(SU))’’’!!!    Biedermann 10:44, 6 January 2007 (UTC)
replies: text: Given a number y, it is possible to produce the Godel number for the sequence of y copies of f followed by 0.  Here Z is a function that does this. CMummert 12:50, 6 January 2007 (UTC)

text: It is still true that (x)~Dem(x,sub(y,13,Z(y))) is a formula in which y is a free variable.  That is all that is required in the remainder of the proof. CMummert 21:07, 7 January 2007 (UTC)

text: So please just tell me of what symbol sequence is Z(y)  the Gödel number ! Biedermann 23:35, 7 January 2007 (UTC)
replies: text: Actually, I am going to stop participating in this conversation.  I would recommend that, instead of looking at Godel's original paper, you read a modern textbook in which the proof will be written more clearly. An excellent book is "Computability and Logic" by Boolos, Burgess, and Jeffries. This recommendation is not because Godel's paper is faulty, but only because it is cryptic, leading to the type of confusion that you describe. CMummert 02:51, 8 January 2007 (UTC)

text: Well, CMummert, I much regret, that you are stopping in this conversation. Sorry, but once you have read Gödel carefully it is not cryptic at all, it is just more specific than any modern proof sketches! I insist on sticking to Gödels original!  First step 'in the remainder of the proof' is to determine the Gödel-number n of above formula, and to that purpose you have to know what Z(y) is according to Gödels definition 16 - 17 in the Symbols of the SYSTEM.  Please continue! I have no library and no books at my disposal.  Biedermann 10:37, 8 January 2007 (UTC)
thread_title:  Comment by Biedermann 

message: This article explains that there ∃ statement F such that neither T ⊢ F nor T ⊢ ~F in a "sufficiently powerful" T. A Gödel sentence G which claims that G cannot be proven is proven to be an example of such statement. J. R. Lucas, as explained in , claims that people can see that G is ture while machines cannot do that concluding that people's mind is superior and AI is not possible. I cannot see why the liar's sentence "this sentence is a lie" is ture. Am I a computer? --Javalenok 15:17, 11 January 2007 (UTC)
thread_title:  Gödel sentence is an objection to AI. Is it true? 

message: In these terms, the Gödel sentence is a claim that there does not exist a natural number with a certain property. A number with that property would be a proof of inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to hypothesis. So there is no such number, and the Gödel statement is true, but the theory cannot prove it.
Ok, so let's say, "There does not exist a number that is prime, higher than 2 and is divisible by 2."
IF we found such a number, then the statement is false.  However, since we can't "prove a negative", thus we can't ever *prove* this to be true.  Tell me if I am sorta right so far (this might be a retarded example, I'm not a mathematician, just trying to muddle through the logic)?
SO, why can't we just prove the exact reverse of this?  i.e. Given an infinite amount of time, we look at ALL the prime numbers greater than 2 and show that all of them are not divisible by 2.
It seems to my logic that when you've got infinity to play with, showing that *all* possible numbers do not have the characteristic is the same as saying that a possible number *cannot* have that characteristic. —The preceding unsigned comment was added by 66.159.227.60 (talk • contribs)  19:36, 1 April 2007   (UTC)
:The first big problem in the above is the canard that you can't "prove a negative". In mathematics we prove negative statements all the time, and proving that there is no number with the property you give is in fact very easy. --Trovatore 23:21, 1 April 2007 (UTC)
thread_title: I don't get it, this seems simple to fix

message: There is a simple paradox in Godel's incompleteness theorem, pointed out by the Australian philosopher Colin Leslie Dean, that invalidates it. 
Godel claims that:
:undecidability "does not depend upon the special nature of the constructed systems [PM and ZF] but rather holds for a very wide class of formal systems"
and yet in his proof he states undecidability is dependent on formal system P which includes PM"hence in every formal system which satisfies assumptions 1 and 2 [ from system P which includes PM] and is w - consistent there exist undecidable propositions". Thus Godel's incompleteness theorem ends in paradox and is meaningless or invalid. 
Godel’s incompleteness theorem. Ends in absurdity or meaninglessness
Quote:
Godel makes the claim that there are undecidable propositions in a formal system that dont depend upon   the special nature of the formal system
Quote
It is reasonable therefore to make the conjecture that these axioms and  rules of inference are also sufficent to decide all mathematical questions which can be formally expressed in the given systems. In what  follows it will be shown .. there exist relatively simple problems of ordinary whole numbers which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR] This situation does not depend upon the special nature of the 
constructed systems [PM and ZF] but rather holds for  a very wide class of formal systems   (K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.6).( K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965, p.6)
Thus Godel says he is going to show that undecidability is not dependent on the 
axioms of a system or the speacial nature of PM and ZF
Also 
Godels refers to PM and ZF AS FORMAL SYSTEMS 
"the most extensive formal systems constructed .. are PM ZF" ibid, p.5 
so when he states that 
"This situation does not depend upon the special nature of the constructed 
systems but rather holds for  a very wide class of formal systems" 
he must be refering to PM and ZF as belonging to these class of formal systems- further down you will see this is true- 
as well 
thus he is saying 
the undecidability claim is independent of the nature of the formal system  but PM is a formal system
Godel says he is going to show undecidabilitys by using the system of PM (ibid)
he then sets out to show that there are undecidable propositions in PM (ibid. p.8)
where Godel  states 
"the precise analysis of this remarkable circumstance leads to  surprising results concerning consistence proofs of formal systems  which will be treated in more detail in section 4 (theorem X1) ibid p. 9 note this theorem comes out of his system P
he then sets out to show that there are  undecidable propositions in his system P -which uses the axioms of PM and Peano axioms.
at the end of this proof he states
"we have limited ourselves in this paper essentially to the system P and have only indicated the applications to other systems" (ibid p. 38)
now
it is based upon his proof of undecidable propositions in P that he draws out broader conclusions for  a very wide class of formal systems
After outlining theorem V1 in his P proof - where he uses the axiom of choice- he states
"in the proof of theorem 1V no properties of the system P were used other than the following
1) the class of axioms and the riles of inference- note these axioms include reducibility 
2) every recursive relation is definable with in the system of P 
hence in every formal system which satisfies assumptions 1 and 2 [ which uses system PM] and is w - consistent there exist undecidable propositions ”. (ibid, p.28)
CLEARLY GODEL IS MAKING SWEEPING CLAIMS JUST BASED UPON HIS P PROOF
 
Godel has said that undecidability is not dependent on the 
axioms of a system or the special nature of PM and ZF
There is a contradiction here
He   says  every formal system which satisfies assumption 1 and 2  ie 
based upon axioms - but he has said  undecidablity is  independent of axioms 
Also there is a contradiction here
Godel has  said  undecidablity is not dependent on PM yet says it is hence” in every formal system which satisfies assumptions 1 and 2 [ which uses system PM] and is w - consistent there exist undecidable propositions “
 
Thus the paradox  undedciablity is not dependent of the axioms of a system or PM but is dependent on the axioms of the system and PM 
In the above Godel  must be referring to PM and ZF as they are formal systems 
but he has said 
"This situation does not depend upon the special nature of the constructed systems [PM ZF] but rather holds for  a very wide class of formal systems" 
now P  is constructed with the axioms of PM and Peano axioms 
"P is essentially the system which one obtains by building the logic of PM around Peanos axioms..."   K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,, p.10) 
so clearly undecidability is dependent on the quirky nature of PM-which is a formal system
but 
he has told us undecidable propositions in a formal system are not due to the nature of the formal system  but he is making claims about a very wide range of formal  systems based upon the nature of  formal system P
QUOTE
[undecidability]does not depend upon the special nature of the
constructed systems [PM and ZF] but rather holds for a very wide class of formal systems 
contradict this
hence in every formal system which satisfies assumptions 1 and 2 [depending on the special nature of formal system P WHICH USES PM ] and is w - consistent there exist undecidable propositions
HE HAS SAID UNDECIDABILITY DOES NOT DEPEND UPON THE NATURE OF PM YET SAYS UNDECIABILITY IN FORMAL SYSTEMS- OF WHICH PM- IS ONE IS DEPENDENT ON PM
put simply
Undecidability is independent on nature of PM, yet is dependent on the nature of PM.
thus undecidability is not dependent on the nature of the [PM and ZF] but he has said undecidability is dependent upon the nature of  formal system P which uses PM 
thus 
“[undecidability] does not depend upon the special nature of the 
constructed systems [PM and ZF] but rather holds for  a very wide class of formal systems  “
Contradicts this
“hence in every formal system which satisfies assumptions 1 and 2 [ depends upon the special nature  of formal  system PM] and is w - consistent there exist undecidable propositions ”.
Thus when Godel states
"hence in every formal system [PM example] which satisfies assumptions 1 
and 2 and is w [Dependent on the special nature of P and thus PM ] - 
consistent there exist undecidable propositions" 
he is creating paradox and circularity of argument 
he says undecidability is independent of formal system PM and  ZF yet 
deriving assumptions dependent on  this formal system PM  he says those 
formal systems that have these assumption have undecidability and he 
states ZF has these assumptions (ibid, p.28) 
put simply
Undecidability is independent on nature of PM, yet is dependent on the nature of PM.
clearly Godel is in paradox and invalid due to meaninglessness
There is another paradox in Godels incompleteness theorem
As we have seen undecidability in a formal system is dependent on the system PM but the system PM has  undecidability
Godel tells us that among those  very wide range of formal systems that have undecidability are to be included those systems which arise from PM by the addition finitely many axioms  
As he states
“It is reasonable therefore to make the conjecture that these axioms and  rules of inference are also sufficent to decide all mathematical questions which can be formally expressed in the given systems. In what  follows it will be shown that this is not the case but rather that in both systems cited [PM  and ZF]  there exist relatively simple problems of ordinary whole numbers which cannot be decided on the basis of the axioms. [NOTE IT IS CLEAR] This situation does not depend upon the special nature of the constructed systems [PM and ZF] but rather holds for  a very wide class of formal systems   among which are included in particular all those which arise from the given systems [PM and ZF] by addition of finitely many axioms”
In other words PM is included in those systems which have undecidability
Thus we have the paradox that while PM is used to find if a formal system is undecidable  it is undecidable itself
i.e. 
hence in every formal system which satisfies assumptions 1 and 2 [ from P  which uses system PM] and is w - consistent there exist undecidable propositions
In other words the very system which is used to find undecidability is included in the set of undecidable systems
Thus we have the situation overall that clearly Godel is in paradox and invalid due to meaninglessness
1) there is circularity/paradox  of argument he says his consistency proof is independent of the nature of a formal system yet he bases this claim upon the very nature of a particular formal system P- which includes PM which is itself undecidable
2) he is clearly basing his claims for his consistency theorems upon the systems PM and P
P and PM are the meta-theories/systems  he uses to prove his claim that there are undecidable propositions in a very wide range of  formal systems
We have a dilemma
1)either Gödel is right that his claims for undecidability of formal systems
are independent of the nature of a formal system 
and thus he is in paradox when he  makes claims about formal systems based
upon the special nature of P - AND THUS PM
OR 
2) he makes claims about formal systems based upon the special nature of P
and PM
that would mean that PM and P are the meta-systems/meta-theory through
which he is make undecidable claims  about formal systems 
thus indicating the axioms of PM and P  are central to these meta claims
there by when I argue s these axioms are invalid then Godels
incompleteness theorem is invalid and a complete failure.
Thus either way Godels incompleteness theorem are invalid and a complete failure :either due to the paradox in his theorem or the invalidity of his axioms.
::I just wonder at these long discussions about Gödels work and nobody takes a closer look at the mathematical aspect of gödels construct, i.e the inherent mathematical inconsistancy in his famous Formula G. Gödel himself named the LIAR paradox as basis for his formula. But this 'paradox' is based on the mathematically non-sensical equating  2 = 4 ;  the two words the subject 'this sentence'  is identified with the 4 words of the whole sentence 'This sentence is false' !  Consequently, Gödels construct suffers from the same inconsistency.  This is evident in the so highly praised DIAGONALIZATION, that is based on setting   x = Gödelnumber of F(x),  so x is used as free variable within a formula F( ) and at the same time it is equated not only with that entire formula that may consist of some hundreds of symbols, but even with the so more complicated Gödel number of this formula. This is a serious violation of the PM - rule for the proper aplication of a variable: that a variable has to preserve one and the same identity throughout a given context. 83.223.161.184 (talk) 11:16, 28 June 2010 (UTC)
thread_title:  Paradox in Godels incompleteness theorem that invalidates it 

message: In regard to system P The Australian philosopher Colin Leslie Dean points that godels system P is invalid 
http://gamahucherpress.yellowgum.com/books/philosophy/GODEL5.pdf
Godel constructs an artificial system P made up of Peano axioms  and axioms including the  axiom  of reducibility- which is not in the edition of  PM he says he is  is using. This system is invalid as it uses the invalid axiom of reducibility. Godels theorem has no value out side of his system P and system P is invalid as it uses the invalid axiom of reducibility
"P is essentially the system which one obtains by building the logic of PM around Peanos axioms..."   K Godel , On formally undecidable propositions of principia mathematica and related systems in The undecidable , M, Davis, Raven Press, 1965,, p.10)
system P contain the axiom of reducibility
Godel uses the axiom of reducibility axiom 1V of his system is the axiom of reducibility “As Godel says “this axiom represents the axiom of reducibility (comprehension axiom of set theory)”
http://www.math.ucla.edu/~asl/bsl/1302/1302-001.ps.
"The system P of footnote 48a is Godel’s
streamlined version of Russell’s theory of types built on the natural
numbers as individuals, the system used in [1931]. The last sentence ofthe footnote allstomindtheotherreferencetosettheoryinthatpaper;
KurtGodel[1931,p. 178] wrote of his comprehension axiom IV, foreshadowing
his approach to set theory, “This axiom plays the role of [Russell’s]
axiom of reducibility (the comprehension axiom of set theory).”
system P is the system from which he derives his incompleteness theorem 
quote from the van Heijenoort translation
”Theorem XI. Let κ be any recursive consistent63 class of FORMULAS; then the SENTENTIAL FORMULA stating that κ is consistent is not κ-PROVABLE; in particular, the consistency of P is not provable in P,64 provided P is consistent (in the opposite case, of course, every proposition is provable [in P])". (Brackets in original added by Gödel “to help the reader”, translation and typography in van Heijenoort 1967:614)
ALSO
IT MUST BE NOTED THAT GODEL IS USING 2ND ED PM BUT RUSSELL TOOK THE AXIOM OF REDUCIBILITY OUT OF THAT EDITION – which Godel must have known.
The Cambridge History of Philosophy, 1870-1945- page 154
 
http://books.google.com/books?id=I09hCIlhPpkC&pg=PA154&vq=Russell+repudiated+Reducibility&dq=taken+out+2nd+ed+principia+russell+axiom+of+reducibility&source=gbs_search_r&cad=1_1&sig=-LmJ1voEsKRoWOzml_RmOLy_JS0
Quote
“In the Introduction to the second edition of Principia, Russell repudiated Reducibility as 'clearly not the sort of axiom with which we can rest content'…Russells own system with out reducibility was rendered incapable of achieving its own purpose”
quote page 14
http://www.helsinki.fi/filosofia/gts/ramsay.pdf.
“Russell gave up the Axiom of Reducibility in the second edition of
Principia (1925)”
THUS 
Godel constructs an artificial system P made up of Peano axioms  and axioms including the  axiom  of reducibility- which is not in the edition of  PM he says he is  is using. This system is invalid as it uses the invalid axiom of reducibility. Godels theorem has no value out side of his system P and system P is invalid as it uses the invalid axiom of reducibility
THAT AR IS INVALID IS SEEN FROM
the standford encyclopdeia of philosophy  says of AR
http://plato.stanford.edu/entries/principia-mathematica/
“many critics concluded that the axiom of reducibility was simply too ad hoc to be justified philosophically”
From Kurt Godels collected works vol 3  p.119
http://books.google.com/books?id=gDzbuUwma5MC&pg=PA119&lpg=PA119&dq=godel+axiom+of+reducibility&source=web&ots=-t22NJE3Mf&sig=idCxcjAEB6yMxY9k3JnKMkmSvhA#PPA119,M1
“the axiom of reducibility is generally regarded as the grossest  philosophical expediency “
Godel would have know all these criticism by Russell Wittgenstein and Ramsey but still used the axiom. Russell Witgenstein and Ramsey would have know Godel used this invalid axiom in his artificial system P but said nothing  —Preceding unsigned comment added by Xamce (talk • contribs) 12:16, 2 March 2008 (UTC)
thread_title: Godel system P is invalid -thus his incompleteness theorem is invalid

message: There is a famous recipe for a cake called a Gödel cake that absolutely no one has succeeded in baking. To every experienced culinary eye, the recipe appears self-evidently to be one that when followed should produce a cake, and yet all attempts to make the cake have been frustrated.
This had led some to question whether the recipe really is a recipe for a cake: if a recipe for a cake has never produced a cake, then there is no evidence that it is a recipe for a cake. The creator of the recipe and his adherents argue however that it is clear from everything we know about the philosophy of cooking that the recipe is a recipe for a cake (in fact he claims to have proved on sound philosophical grounds that, in theory at least, the cake can be baked), and that this means that there can be recipes that can never be cooked until someone invents a presently unimagined new cooking utensil that would enable the successful baking of the cake. Gödel goes on to argue, though, that if such a cooking utensil were in fact to be invented, there is nothing to preclude the possibility that he could create a new cake recipe that included the use of the new utensil in its preparation yet which still no one would be able to succeed in baking.
This is what has come to be known as the fundamental uncookability of all reasonably useful recipe systems, which essentially states that there are recipes that no one can cook using just the currently accepted practices, and that even if new practices were introduced, this would allow new recipes to be described that still nobody can cook.
Vociferous opponents of Gödel argue however that a cake isn’t a cake until you have your cake and eat it, and that Gödel’s culinary thinking is so absurd that it’s not even half-baked.
However, for today at least, it is still generally accepted that the Gödel cake is a true cake.
--Vibritannia (talk) 10:58, 28 April 2008 (UTC)
thread_title:  Gödel Cake 

message: Consider the following sentence:
:This sentence is not provable because it's absurd.
It sounds reasonable, but it can't be true. The sentence says it's absurd, so how can it be true? Now consider the following sentence:
:X is false.
It's an assertion, but we can't say whether it is true or not because we don't know what X is. But we can see that X itself has to be an assertion because if, for instance, X were "2+2", the sentence wouldn't assert anything. If X were "2+2=5", the sentence would be true. If X were "2+2=4", the sentence would be false. But what if X were "This"? The sentence would be
:This is false.
What does "This" assert? The sentence looks like an assertion, but is it really? And if it doesn't in fact assert anything, it's not logical to say that it is true or false. All you can say is whether the spelling, punctuation, and grammar are correct, but that's not the same thing. If that sentence isn't an assertion, then this one can't be either:
:This is not provable.
What isn't provable? Exactly! We haven't asserted anything. Or if we have, then what? If you replace the word "This" with the whole sentence surrounded by quotes, you still cannot discover an assertion. And if you keep substituting for occurrences of "This", you're still none the wiser.
In English if you say "This is not provable", you really mean "That is not provable", or you mean something you're pointing to, or you mean something you're about to say next. So if the sentence on its own doesn't assert anything even in English, how can it possibly assert anything in a formal language?
:In the formal language, the assertion is about numbers. By agreeing that those numbers can be taken (by the reader) to refer to strings of symbols, the 'interpreted content' of the assertion about numbers says that the string of symbols that forms the assertion about numbers in the formal system cannot be obtained by transforming the starting strings of the formal system using the allowed rules of transformation.
:Say we added extra starting strings to the formal system so that the string of symbols that forms the assertion about numbers could now be obtained by applying the allowed rules of transformation. The interpreted content of the assertion about numbers would not now be incorrect because it does not say that the string cannot be obtained from the starting strings whatever they happen to be; it says the string cannot be obtained from precisely the strings that were explicitly identified during the definitions that were required to express that particular assertion about numbers.--Vibritannia (talk) 11:25, 24 July 2009 (UTC)
thread_title: Tricky Natural Language

message: Consider the following sentence which is intended to refer to itself:
:This sentence is not provable.
If the sentence were provable, then what the sentence says would be false. But a proof is only a proof if it proves something that is true, therefore any proof that we might consider could not by definition actually be a proof. Therefore the sentence is not provable.
Therefore what the sentence states is correct.
So we have just deduced by the above argument that the sentence is true. In other words, we have proved it. But if we have proved it, then what the sentence says is false. And we cannot have proved true something that is false. So our proof cannot be a proof after all.
:But Gödel's sentence doesn't say that it is not provable. Rather it says that a particular string of symbols cannot be obtained by transforming certain starting strings according to certain rules of transformation. When Gödel proves the Gödel sentence, he doesn't do it by transforming starting strings, so he doesn't contradict what his sentence says.
:The Gödel sentence is a string of symbols in the formal system. The formal content of that string is an assertion about numbers. The interpreted content of that string is what we understand that assertion about numbers to mean when we accept that the numbers can be taken to refer to strings of symbols. Gödel's argument was that if the interpreted content and the formal content were consistent, the negation of the Gödel sentence could not be true because the interpreted content of that assertion about numbers implied that the unnegated sentence should be true as well. Therefore the negated Gödel sentence must be false, and therefore the Gödel sentence must be true (which was the proof).
:The Gödel sentence did not say it was not provable; it said that string could not be obtained by string manipulation. Gödel did not decide the Gödel sentence by string manipulation. He decided it by an argument that relied on the informal interpreted content of the string and certain assumptions about the consistency of the formal system in question.--Vibritannia (talk) 12:05, 24 July 2009 (UTC)
replies: text: This is false.
Could we regard this sentence as being true? The sentence says it is false. But the sentence is true, therefore saying that the sentence is false is saying something that is false, which is what the sentence says it says. Therefore the sentence is true. It is true that the sentence is false.
On the other hand, if the sentence is false, then what it says is true. Therefore it is false that what the sentence says is false. So the sentence is false.
We have just considered the case in which the sentence is true and the case in which the sentence is false and found in both cases that the truth of the sentence and that of what the sentence says differ. Therefore if we assume that the sentence is either true or false, it is not the case that the truth of the sentence and the truth of what the sentence says are one and the same thing. And in that case, our premise is not an acceptable one. But on the other hand, if we were to reject our assumption that the sentence must be either true or false, then our premise would become acceptable but inapplicable to the sentence in question.
So by our consideration of a sentence similar to the original one, we have discovered that a premise of our argument presents a significant possibility that it may be either unacceptable given our other premises or inapplicable to the sentence we have applied it. For this reason, it does not make a good choice for a premise from which to start our argument. And for that reason, our argument is not a convincing one.
If our argument is unconvincing, and it is our only argument, then we cannot claim to have a conviction about the argument's conclusion. Therefore we cannot claim to have a rational conviction that the sentence is true.
--Vibritannia (talk) 09:36, 17 August 2008 (UTC)
thread_title: What is a Rational Conviction?

message: A sentence is constructed within an axiomatic system. It is then shown that if that sentence were an immediate consequence of the axioms of the system, then an inconsistency would be implied. Furthermore it is shown that if instead the negation of that sentence were an immediate consequence of the axioms of the system, then that too would imply an inconsistency. It is therefore deduced that neither the sentence nor its negation can be an immediate consequence of the axioms if the system is to be consistent. Since it is believed and desired that the system is consistent, it is accepted that neither the sentence nor its negation is an immediate consequence of the axioms.
Now when it is said that a sentence is an immediate consequence of the axioms, what is meant is that if you start with just the things that you have assumed are true beyond question (the axioms), you can reason with those axioms without introducing any new assumptions and eventually reach a conclusion which is the sentence in question. Gödel said this was what was meant by saying a sentence was provable in a particular system. But note that this notion of proof does not include proof by reductio ad absurdum (because a reductio ad absurdum proof must be begin by assuming the negation of the sentence to be proved, but for an immediate consequence, the only allowable assumptions are the axioms).
And for a sentence to be provable, in the sense that it is an immediate consequence of the axioms, it must be a sentence that is either true or false, i.e. if the sentence is an immediate consequence of the axioms, then the negation of that sentence cannot be an immediate consequence of the axioms (or else we say the system is inconsistent, and nothing is proved by any immediate consequence of the axioms if the system is inconsistent). Now if a sentence must be true or false, we say that sentence asserts something, i.e. it is an assertion.
So an assertion had been constructed within the axiomatic system, a sentence that said something that must either be true or false, but for which neither, Gödel said, could ever be proved in the axiomatic system – because proof of either would imply an inconsistency, and systems containing inconsistencies prove nothing.
:Wrong! It was not the case that neither could ever be proved in the axiomatic system. Rather the strings of symbols for those assertions could not be obtained from the specific starting strings explicitly identified in the definitions that were required to express those assertions. The strings of those assertions might be obtainable from other starting strings (other axioms), and if such were the case, no inconsistency would be implied by it (because an inconsistency would only result if the sentence were an immediate consequence of the originally identified axioms, not if it were to turn out to be an immediate consequence of any axioms we might subsequently add).--Vibritannia (talk) 12:26, 24 July 2009 (UTC)
thread_title:  Why the First Incompleteness Theorem is Absurd 

message: I have pondered Gödel's Incompleteness Theorem for over ten years, since I first learned about it at age 14 as an advanced mathematics student.
One idea essential to the proof is that the theory ZFC, considered as an object within Zermelo-Fraenkel set theory itself, contains its own theory, ZFC´, which contains its own theory again, ZFC´´, and so forth. The undecidable statement G does not quite say, "I am unprovable," but, "My next reflection, G´, in the next theory, ZFC´, is unprovable." In doing so, it creates an endless chain of semantical reference, and we end up with something like this:
:G --states unprovable--> G´ --states unprovable--> G´´ --states unprovable--> ...
or,
:The following cannot be proven: The following cannot be proven: The following cannot be proven: ...
The addition of ~G as an axiom results in omega-inconsistency, which means that there is a property P such that while
:P(1), P(2), P(3), ...
are all provable, the statement
:(Ex)(x is a natural number and ~P(x))
is also provable. A number x that satisfies ~P might be called a supernatural number.
You may ask, "How is this even possible? Why can't mathematical induction 'reach' x? If induction applies to 1, 2, 3, and so forth, every step of the way, shouldn't x inherit the property P? Isn't that just what induction states?"
The answer turns out to be very simple. What we can do is add new numbers that, while unreachable outside the theory, act as if they were reachable inside the theory by functioning logically as unknown variables. The supernatural number x, even though you can't actually count to it, is made to inherit inductive properties by playing the role of 'some number,' in the way that any variable would. In such a case, you could say that the new theory would be made 'blind' to the property of x's actual unreachability.
It is a bit like a drop of colored water falling into a clear pond. The drop begins as an entity distinct from the rest of the pond (the supernatural is 'unreachable'), but after the splash is made, the color dilutes the rest of the pond and loses its distinguishing property.
Of course, a proof x of G´ in ZFC + ~G would translate into the very proof of G´´ whose nonexistence it would purport to prove. The proof x can't both exist and prove itself nonexistent in ZFC´, so ZFC´ (and thus ZFC´ + ~G´) would become inconsistent with the addition of ~G. In other words, mathematics would not be able to refer to itself as a consistent theory. The theory ZFC + ~G, however, would remain consistent.
Scott7261 (talk) 03:30, 6 December 2008 (UTC)
replies: text: I'd never heard of supernatural numbers before. Your use of primes (the notation marks) has given me a useful idea. --Vibritannia (talk) 10:04, 15 December 2008 (UTC)
thread_title:  Supernatural numbers