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World’s shortest explanation of Gödel’s theorem?

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


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):


(A) PX


(C) ~PX

(D) ~PNX

  1. We define PX as true if and only if this machine prints X.
  2. We define PNX as true if this machine prints XX.
  3. We define ~PX as true if and only if this machine does not print X.
  4. We define ~PNX as true if this machine does not print XX.
  5. We define ~X to mean the negation of X, and for all sentences, either that sentence or its negation is true.
  6. The machine is completely accurate in that all sentences printed by the machine are true.
  7. The machine never prints any false sentences.



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.

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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?.

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The Lighter Side


Paper on the diagonal proof

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.

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Other recently added pages

The Myths of Platonism


Goodman’s Paradox


The Platonist Rod paradox


The Balls in the Urn Paradox


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Lebesgue Measure

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

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Easy Footnotes

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).

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O’Connor’s “computer checked” proof

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.

Previous Blog Posts

Moderate Platonism

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room


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:

Gödel Links


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

Gödel, Minds, and Machines

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