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

[Patrick Juola]

On Tue, Aug 9, 2011 at 12:31 PM, Angus Grieve-Smith <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.





>
>   Since human beings are not formal systems, this is of limited
>> application.   In particular, we know for independent reasons that a) humans
>> can't do arithmetic perfectly, b) humans aren't consistent, and c) humans
>> aren't "complete" (as they have limitations like finite attention spans and
>> finite lifetimes).
>>
>
>    You know that, and I know that.  But linguistics and cognitive science
> are full of people who claim that human beings contain complete, consistent
> formal systems.
>
> --
>                                -Angus B. Grieve-Smith
>                                grvsmth at panix.com
>
>
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