Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com
You may find on the web various claims of the ‘shortest’ explanation of Gödel’s proof of incompleteness, which is based on an idea in Raymond Smullyan’s book ‘Gödel’s Incompleteness Theorems’ (Footnote: Raymond M Smullyan. Gödel’s Incompleteness Theorems. Oxford University Press, 1992. ISBN: 0195046722. See Smullyan, Gödel’s Incompleteness Theorems: Details.) - such as http://blog.plover.com/math/Gdl-Smullyan.html and http://www.sikharchives.com/?p=1417.
You will find Smullyan’s ideas in Chapter 1 of his book. To be fair to Smullyan, he himself doesn’t claim that it is an incompleteness proof. You can also see a form of it in an article by Selmer Bringsjord Is Gödelian model-based deductive reasoning computational? PDF. (Footnote: Note that in Smullyan (and in Bringsjord) there is a variation of the above where there is a machine that refers to its own statements, and the difference is that it now refers to its own statements in a coded form. This makes absolutely no difference to the argument, since it also does not explain how every formal mathematical system might be able to produce such self-referential statements.)
Under the heading ‘A Gödelian Puzzle’, Smullyan’s book basically says:
Suppose we have a machine, and the specification for the machine is as follows:
This machine prints out various sentences composed of the following symbols:
~ P N
The sentences that the machine can print are all one of the following four forms (where X is any expression):
That completes the specification for the machine
Now, consider the symbol combination that is ~PN~PN.
Suppose the machine prints ~PN~PN. If the machine prints it, then by rule 2 above, PN~PN is true. And by rule 5, the negation of this, which is ~PN~PN must be false. By rule 7, the machine never prints any false sentences, so it could not print ~PN~PN. Therefore our supposition that it does print ~PN~PN is incorrect, and it must be the case that the machine does not print ~PN~PN. By rule 4, ~PN~PN is true if the machine does not print ~PN~PN.
Therefore, ~PN~PN is true, but the machine does not print it.
And that’s it. A specification for a machine that cannot print certain combinations of symbols. Note that it is only a specification for a machine, not a description of an actual machine.
But it’s not, by any stretch of the imagination, an explanation of a proof of incompleteness. Nor, despite many claims to the contrary, does it contain anything of deep significance. All it tells us is that if you have a system that can produce certain types of sentences that refer to themselves, then you will end up with this sort of conclusion. As such, it doesn’t tell us anything more than the simple Liar statement, “This statement is false”. Like the Liar statement, it is entirely reliant on the system being able to reference itself.
Furthermore, as noted above, the given specification for the machine tells us nothing about whether a machine can be constructed that can satisfy all the specifications. It doesn’t give any information as to how such a machine can be constructed, or even if it can be constructed.
It would only be an explanation of a proof of the incompleteness of every formal mathematical system if it also showed how every formal mathematical system of a certain type (such as a formal system that includes numbers and arithmetic) can produce self-referential statements like the above machine - statements that refer to what that formal system can and cannot prove. Since this ‘World’s shortest explanation’ doesn’t even start to explain that, then it simply isn’t an explanation of an incompleteness proof at all.
Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked (comments will be checked before appearing on this site). 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 - this will only be used to notify you of replies to your comments - it will never be used for any other purpose, will never be displayed and does not require verification. Comments are common to the entire website, so please indicate what section of the site you are commenting on.
If you cannot see any comments below, it may be that a plug-in on your browser is blocking Disqus comments from loading. Avast anti-virus in particular is known to do this, especially with Internet Explorer and Safari. See Disqus Browser plug-in/extension conflicts or Why isn’t the comment box loading?.
Please wait for comments to load …
There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
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.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
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:
All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.
Comments on this site are welcome, please see the comment section.
Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.
If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.
Feedback about site design would also be appreciated so that I can improve the site.
Copyright © James R Meyer 2012 - 2016