Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com

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A number of people have raised objections to the demonstration of the flaw in Gödel’s incompleteness proof. As yet there is no objection that shows any attempt to give a reasoned logical argument against my 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 the demonstration of the flaw in Gödel’s proof 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 the paper on the flaw in Gödel’s proof 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 O’Connor Review: here.

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

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 the 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.

Another common objection is that the 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 the 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.

It might be noted that the incompleteness paper is not the only paper where Gödel made a completely unproven assertion which almost everyone assumed to be correct because of Gödel’s esteemed reputation. In that other paper Gödel made a similar intuitive assertion, but it was eventually proved (some 50 years later) that Gödel’s intuitive assumption stated precisely the * opposite* of what was actually the case. For details of this see Gödel’s Intuitive Error No.2. Not many mathematicians are aware of this - Gödel’s reputation is still so honored today that most discussions of Gödel’s work skip over this inconvenient detail.

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 the 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 the paper before they start criticizing it? If they had actually read the paper, they would find that the 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 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 the demonstration of the flaw in Gödel’s original paper correct?* - and it is completely irrelevant to this question.

Another common objection is the claim that if I did not study pure mathematics in a university course, then I 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 I have no capacity for self-study, that I am unable to reach a sufficient level of knowledge and understanding by intensive study of a subject, unless I follow a university course. Where do people get these notions?

So, partly in response to those people who claim that I don’t understand Gödel’s proof at all, and partly to assist anyone who wishes to get to grips with Gödel’s original paper, I have created a walk-through guide to Gödel’s original incompleteness proof, and which is intended to be read alongside the paper. If you are speculating that I don’t understand Gödel’s proof, I suggest that you take a look at it; it may help you to decide whether it is reasonable to suggest that I do not understand the proof. When I decided to make that guide, I did an intensive search to see if there was anything similar already published anywhere, but I found no detailed guide at all - so I constructed my guide from scratch - which I would claim is hardly indicative of someone who does not understand Gödel’s proof.

And yes, I did not follow the traditional study route, but this was actually very beneficial, because I was not inculcated with the conventional viewpoint of all those who follow the conventional path. In a university course, there are so many things to learn within a short time that it is impossible to analyze everything in your course, so you have to assume that much of what you are being taught is correct. By the time you have finished the university course, you have reached the point where you have accepted the majority viewpoint to the extent that you will argue that a particular proof is correct, even though you have not actually verified it yourself.

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

Some people appear to believe that by adopting a Platonist standpoint they can circumvent the absurdities that are inherent in Gödel’s proof. They argue that certain expressions in Gödel’s paper are not actually themselves number relationships, but that they * correspond* to number relationships that “exist” independently of any language. It is easily shown that this notion makes no difference whatsoever to the demonstration that there is an irredeemable flaw in Gödel’s proof, see The Platonist response to the flaw in Gödel’s proof.

One person (Russell O’Connor) sent me a long criticism of the paper on the flaw in Gödel’s proof, 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 if 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 own ‘formalized version’ of Gödel’s proof contains an obvious error. See An Error in a Proof of Incompleteness by Russell O’Connor.

As Wilfrid Hodges (Footnote: Wilfrid Hodges, An Editor Recalls Some Hopeless Papers, The Bulletin of Symbolic Logic,
Vol 4, Number 1, March 1998.) 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). Thouless gives a summary of the dishonest tricks used in arguments, see Dishonest tricks used in Arguments.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

There is now a new page on a contradiction in Lebesgue measure theory.

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

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Copyright © James R Meyer 2012 - 2018

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