[FOM] 18 Word Proof of the Godel, Rosser and Smullyan Incompleteness Theorems
Panu Raatikainen
panu.raatikainen at helsinki.fi
Wed Jul 21 09:32:23 EDT 2010
"Charlie V" <axiomsandrules at yahoo.com>:
> But you can get three concrete undecidable sentences utilizing a lot
> less proof by noting more of my post, viz,
>
> "When any one of these sets, P, is expressible or representable, the sentence
> that expresses or represents, respectively, 'This is in P.' is undecidable."
>
> This requires only the Recursion Theorem and includes:
>
> 1. Since unprovability is expressible: The sentence that expresses
> "This is not provable." is undecidable.
>
> 2. Since refutability is expressible: The sentence that expresses "This is
> refutable." is undecidable.
>
> 3. Since refutability is representable: The sentence that represents "This is
> refutable." is undecidable.
Very well. But this was not what was said in your first posting, which
(and only which) I was commenting.
>> This I would happily call equivalent with Gödel's first incompleteness
> theorem.
>
> And what would you call it if you get three undecidable sentences?
Three different theorems... Why should they have a common name?
Cheers, 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
Finland
E-mail: panu.raatikainen at helsinki.fi
http://www.mv.helsinki.fi/home/praatika/
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