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 https://blog.plover.com/math/Gdl-Smullyan.html.
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?. (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):
- We define PX as true if and only if this machine prints X.
- We define PNX as true if this machine prints XX.
- We define ~PX as true if and only if this machine does not print X.
- We define ~PNX as true if this machine does not print XX.
- We define ~X to mean the negation of X, and for all sentences, either that sentence or its negation is true.
- The machine is completely accurate in that all sentences printed by the machine are true.
- 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, nor a blueprint for creating one. It does not tell you if a machine with that specification can actually be constructed.
Hence 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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