[Corpora-List] PS:minimal changes in a paragraph (based on a corpus it appeared) ... (2nd attempt (after first one was deleted))

[Angus Grieve-Smith]

On 8/9/2011 12:38 PM, Patrick Juola wrote:
>
>
> On Tue, Aug 9, 2011 at 12:31 PM, Angus Grieve-Smith <grvsmth at panix.com 
> <mailto:grvsmth at panix.com>> wrote:
>
>
>         Godel showed that a formal system capable of doing arithmetic
>         perfectly cannot be both complete and consistent.
>
>        Gödel showed that a formal system cannot be both complete and
>     consistent.  His results were not specific to arithmetic.
>
>
> He did not.   Aristotelian logic is a formal system that is both 
> complete and consistent; anything true and expressible in the system 
> is also provable in the system -- anything (expressible) and provable 
> in the system is also true.
>
> It just happens to be a very weak system, since you can't express 
> basic arithmetic in it.
>
> Kleene's description of Godel's results (quoted in Wikipedia) is 
> relevant here: "Any effectively generated theory /capable of 
> expressing elementary arithmetic/ cannot be 
> both consistent and complete. In particular, for any consistent, 
> effectively generated formal theory that proves certain basic 
> arithmetic truths, there is an arithmetical statement that is true but 
> not provable in the theory."   (Italics mine.)   That's one of the key 
> things that a lot of people don't understand about Godel.
>

     I didn't mention systems that were too weak to express basic 
arithmetic truths, because I thought they were irrelevant to the 
discussion, but yes, Gödel did use arithmetic to weed out complete, 
consistent formal systems that were not powerful enough, and to encode 
the syntax of the formal systems.  That doesn't mean that his theorem 
was all about arithmetic.  The theorem makes a general point about 
formal systems, and his genius was using arithmetic to make that point.

     Saying this is about arithmetic is like saying that because he 
wrote in German, his work only applies to formal systems when they're 
being discussed in German.

-- 
				-Angus B. Grieve-Smith
				grvsmth at panix.com

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