[FOM] 18 Word Proof of the Godel, Rosser and Smullyan Incompleteness Theorems

Charlie V axiomsandrules at yahoo.com
Thu Jul 15 08:23:31 EDT 2010

> From: Panu Raatikainen <panu.raatikainen at helsinki.fi>
> Subject: Re: [FOM] 18 Word Proof of the Godel, Rosser and Smullyan Incompleteness Theorems
> To: axiomsandrules at yahoo.com
> Date: Thursday, July 15, 2010, 6:03 AM
> "Charlie V" <axiomsandrules at yahoo.com>:
> It does not provide a concrete example of a true but unprovable statement.

Incompleteness requires only that an undecidable (unprovable and unrefutable) sentence be shown to exist.  Nonetheless, consider this variation of the proof:

“The sets of untrue, unprovable and refutable sentences all differ w.r.t. whether they are representable or expressible.”

And again truth, provability and unrefutability are distinct, proving the 3 incompleteness theorems.

But now let us again exploit the fact that unprovability and refutability are expressible, and refutability is representable, and add to the proof:

“When any one of these sets, P, is expressible or representable, the sentence that expresses or represents, respectively, ‘This is in P.’ is undecidable.”

> More adequately, one could call this  a version of Tarski's theorem.

Exactly what is a version of what?  My 18 word proof proves 3 theorems.  Are you saying that all three theorems are versions of Tarski’s Undecidability Theorem?

Tarski’s Undecidability Theorem is that truth is not expressible.  This statement and all of the premises from the Theory of Computation used in my (first) proof are statements regarding how the system can characterize various sets, to wit:

1. Truth is not expressible.  (Tarski)
2. Truth is not r.e.
3. Provability is r.e. (2+3 = Godel)
4. Unrefutability is not r.e. (3+4 = Rosser)
5. Untruth is not r.e.
6. Refutability is r.e. (5+6 = Smullyan)

So Tarski’s theorem is a variation of the premises, not the conclusions, of my proof.  In fact, add,

7. Provability is expressible. (1+7 = Godel)

and we see clearly that Tarski’s result is simply a premise of incompleteness.  It is no more related to the conclusions than the other 6 premises.  The three conclusions, on the other hand, are that two of the three sets differ:

1. Godel: Truth and provability differ.
2. Rosser: Provability and unrefutability differ.
3. Smullyan: Truth and unrefutability differ.

Charlie Volkstorf
Cambridge, MA

> Best, Panu
> --Panu Raatikainen
> Ph.D., Academy Research Fellow,
> Docent in Theoretical Philosophy
> Department of Philosophy, History, Culture and Art Studies
> P.O. Box 24  (Unioninkatu 38 A)
> FIN-00014 University of Helsinki
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