Responses to the demonstration of the flaw in Gödel's proof

A number of people have raised objections to the demonstration of the flaw in Gödel's proof. As yet there is no objection that shows any attempt to give a reasoned logical argument against the demonstration of the flaw.

Why?

There are several articles that claim to disprove Gödel's proof, but for anyone who has a thorough understanding of Gödel's original proof, it's not actually very difficult to find the errors in these articles – as shown in Flawed Disproofs. It is most odd that persons who are supposedly well-versed in the application of logic are adamant that my demonstration is incorrect, while at the same time they refuse to give any logical argument to indicate why that demonstration might be incorrect. A Mr Russell O'Connor has published a 'review' where he asserts that my paper has several errors, but fails to identify any actual error in the demonstration of the flaw in Gödel's proof. His review and my response are given here.

Below I give the arguments that I have received according to type.

Common Objection 1

The most common objection is the claim that Gödel's proof cannot be incorrect because there are numerous other 'versions' of proofs of Gödel's theorem that logicians accept as being correct. (Update: there are detailed papers demonstrating the errors in many such proofs, see Errors in incompleteness proofs and Analysis of incompleteness proofs.)

This is simply a deflection of the argument from the main issue which is: Is the demonstration of the flaw in Gödel's original paper correct? It introduces another matter of contention which is: How are we to define what constitutes a 'version' of Gödel's original argument? One has to wonder why a logician would attempt to deflect the argument concerning Gödel's proof in this way instead of simply pointing out an error in my demonstration of a flaw in Gödel's original paper. If logicians believe that the argument is incorrect, why do they try to deflect attention from the main argument? That approach appears to suggest that logicians have not been able to find any error in the argument.

If they are refusing to even examine a demonstration of a flaw in Gödel's original proof, why would they examine a demonstration of a flaw in some 'version' of Gödel's proof? But many other incompleteness proofs have been found to have fundamental errors that render these proofs mathematically invalid. See Errors in incompleteness proofs and Analysis of incompleteness proofs

Common Objection 2

Another common objection is that my demonstration of the flaw in Gödel's proof must be wrong since so many logicians have examined Gödel's proof and have been unable to find any flaw in it. Again, this is a deflection from the main issue: Is my demonstration of the flaw in Gödel's original paper correct?

It is illogical to claim that the mass of accumulated wisdom of hundreds of logicians cannot ever be mistaken. It is illogical to claim that one individual could never perceive anything of importance if hundreds of logicians have not already perceived it; that assumes that such examination by logicians is absolutely perfect.

Common Objection 3

Another common objection is the claim that Gödel did not actually prove in his original paper that for every consistent formal language, there will always be statements that are 'true' but unprovable in that formal language.

Several persons have, for some reason, assumed that my demonstration of the flaw in Gödel's proof relies on disputing this aspect of the proof. Is it really too much to ask such people to actually examine my paper before they start criticizing it? If they had actually read my paper, they would find that my demonstration of the error in Gödel's proof does not depend in any way on whether the proof produces a statement that might be said to be 'true' but unprovable in that formal language.

That said, there is an ongoing disagreement as to whether Gödel's proof produces a statement that is 'true' but unprovable in that formal language - even among the most ardent supporters of the proof. This position that it does not actually produce a 'true' statement is currently fashionable in some quarters; it is a position which conveniently ignores the facts of the matter. It is the head in the sand approach - if Gödel's theorem appears to lead to a contradiction, then - let's deny that Gödel's proof says what even Gödel thought it said - and ask everyone else to put their heads in the sand as well in the hope that the problem will disappear. More details regarding this are given at True but unprovable.

Gödel's original paper produces a statement that Gödel referred to in his paper as 'the proposition which is undecidable in the [formal language] turns out to be decided by [other] considerations'. Gödel also states that the statements that are unprovable in a formal language are not absolutely unprovable, which means that, according to Gödel, there is some way in every such statement can be proved to be 'true' or 'false'.

And it is not the case that it is only those who believe that there are problems with Gödel's proof that assert that he proved that "for every consistent formal language, there will always be statements that are 'true' but unprovable in the that formal language." In fact, most supporters of Gödel's theorem also make such an assertion - this includes the most ardent supporters of Gödel's proof, for example Gregory Chaitin, Douglas Hofstadter and John Dawson.

Again, this objection is a deflection of the argument from the main issue which is: Is my demonstration of the flaw in Gödel's original paper correct? and it is completely irrelevant to this question.

Common Objection 4

Another common objection is the claim that if someone did not study pure mathematics in a university course, then that person cannot have a sufficient grounding in mathematics to be able to understand and make pronouncements about Gödel's proof.

These people assume, entirely without foundation, that there are no people with a capacity for self-study, that no-one is able to reach a sufficient level of knowledge and understanding by intensive study of a subject, without following a university course. Where do people get these notions? The fact that I have two University degrees is deemed irrelevant.

In fact, not following the traditional study route can actually be of great benefit. By not following the beaten track, but trekking alongside it, one is not inculcated with the conventional viewpoint of all those who follow the conventional path.

And, again, this is a deflection from the main issue which is: Is my demonstration of the flaw in Gödel's original paper correct?

And the more bizarre responses

One person (Russell O'Connor) sent me a long criticism of my paper, saying that it was full of errors, and it had to be wrong because he had devised a formalized version of Gödel's proof. When it was pointed out to him that he had failed to find any error whatsoever in my argument that demonstrated the flaw in Gödel's theorem, his response was this:

Even in Gödel did make an error in his proof, it's only a historical curiosity.

A truly astonishing response. This person thinks it unimportant that a proof that has been hailed as the most significant proof in the entire history of logic is wrong. He doesn't even stop to consider that we might be able to learn something from the discovery that it is wrong. And in any case, his 'formalized version' of Gödel's proof contains an obvious error. See An Error in a Proof of Incompleteness by Russell O'Connor.

Fallacious Arguments

As Wilfrid Hodges^[1] has pointed out, 'to attack an argument, you must find something wrong in it. Several authors believed that you can avoid [that] by simply doing something else.'

As a point of interest, the reader might be interested in the various ways that people avoid actually giving a logical response to a demonstration of a flaw in Gödel's proof - and various other results.

There are several good web pages on fallacious arguments at

• A List of Fallacious Arguments - Don Lindsay Archive
• Fallacies - The Internet Encyclopedia of Philosophy
• List of common fallacies at 'No Beliefs'
• Wikipedia entry - List of Fallacies

An excellent book was written many years ago (in fact, 1930, a year before Gödel published his Incompleteness theorem) that deals with fallacious reasoning and shows common ways in which arguments are deflected by those unwilling to engage in fair argument. It is still very relevant today. It is called "Straight and Crooked Thinking", by Robert H. Thouless (unfortunately out of print at present). A section of the book is viewable on-line: Thirty-eight dishonest tricks which are commonly used in argument, with the methods of overcoming them

[1] Wilfrid Hodges, An Editor Recalls Some Hopeless Papers, The Bulletin of Symbolic Logic, Vol 4, Number 1, March 1998.