James R. Meyer claims to have found an error in Gödel’s proof of the incompleteness theorem. Last June, Meyer gave Henk Barendregt a free copy of his book and invited him to read about the flaw he discovered. Henk regularly gets emails from people claiming to have found errors in the incompleteness theorem. Henk wrote a polite response and mentioned that there are computer checked formal proofs of the incompleteness theorem and included me in the reply in case I might wish to respond.
A formal proof ought to assuage almost all doubts about the validity of a theorem. Essentially, the only ways that an error can occur is if the proof checking software is in error in such a way as to allow invalid proofs, or if the theorem is stated incorrectly.
Meyer seemed like he had the competence necessary to understand the incompleteness theorem. Having formalized the theorem, I have quite a good understanding of the technicalities encountered in the proof. I hoped I could settle any doubts he had about the incompleteness theorem.
I wrote up a review of his paper pointing out several errors he makes. Then we engaged in a moderate email exchange where I would explain something and he would find some reason to dismiss what I had to say.
Eventually it became clear that Meyer is only interested in finding a flaw in Gödel’s original paper. He does not seem to care about whether the theorem is valid or whether there are other proofs of the theorem. Thus, my formal proof does not interest him.
Of course, it much more difficult to argue that Gödel’s original paper is correct. I am not as familiar with Gödel’s original presentation. The original paper uses old notation that is quite different from modern presentations. Also, Meyer chooses to attack Gödel’s proposition V, whose proof is only vaguely sketched in Gödel’s original paper. Let me quote Martin Hirzel.
Gödel’s famous proof is highly interesting, but may be hard to understand. Some of this difficulty is due to the fact that the notation used by Gödel has been largely replaced by other notation. Some of this difficulty is due to the fact that while Gödel’s formulations are concise, they sometimes require the readers to make up their own interpretations for formulae, or to keep definitions in mind that may not seem mnemonic to them.
I gave up arguing with Meyer at this point. Defending’s Gödel’s original argument would require too much work. Furthermore, finding an error in the original proof is uninteresting (from a mathematical perspective). Even if Gödel’s orginal proof contained a minor error, there are plenty of modern (and computer verified) proofs that establish the theorem.
I was recently reminded about Meyer, and I have been thinking him, wondering if he is a hopeless crank or if I might still be able to convince him that he is wrong. A little research turned up the answer to this question.
I found a sci.math thread from last July that discusses Meyer’s argument. The thread is initiated by thirdmerlin@hotmail.com—who is likely a puppet of Meyer himself—and eventually Meyer “joins” in. In particular, MoeBlee correctly picks out the specific errors Meyer makes, and MoeBlee probably does a better job arguing the case than I ever could.
Alas, Meyer remains an inconvincible crank. Despite MoeBlee’s pointing out the flaws in Meyer’s argument, he still claims that no one has found a flaw in his argument. At least Meyer’s website no longer proudly claims that he is the first person to understand (and refute) Gödel’s incompleteness proof
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That’s my story, but what bothers me most was how upset I felt last summer in my inability to convince him of his error. Because of the time zone difference, we could only exchange one round of email per day. Each day I would put forth my best argument I could at the time given my limited understanding of his perspective. Each day I knew that no matter what I said, he would presumably find it insufficient. I would be anxious at about what way he would dismiss my argument. It also upset me a little that he took snippets of my email conversation with him and placed them out-of-context on his web-page (and occasionally completely rewording my statements).
However, I found it a great relief today to read that sci.math thread. Everyone there was saying the same things I said. Tim Chow even brought up my formal proof. I’m not sure why I felt so anxious when emailing him, nor am I entirely sure why I am relieved now.
I think my concern is that someone stumbles upon his website and gets the impression that this guy is on to something and has not received fair criticism. Damage is done by conspiracy theories that deny the moon landing (for example, high school teachers have a hard time countering these claims when they are brought up by students). Actual harm is done by bogus claims that there is a link between MMR vaccines and autism. Simply dismissing these people plays into their argument that they are being ignored, and engaging them is expensive and largely futile.
Granted, no one is going to die if people falsely believe that Gödel’s theorem is wrong, but I still do not know what is a good policy for dealing with cranks.