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http://blog.plover.com/math/Gdl-Smullyan.html
Or, in words of one syllable,
http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf
--
Darren New, San Diego CA, USA (PST)
Human nature dictates that toothpaste tubes spend
much longer being almost empty than almost full.
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Darren New wrote:
> http://blog.plover.com/math/Gdl-Smullyan.html
>
> Or, in words of one syllable,
>
> http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf
>
The second one made me literally laugh out loud... that's classic!
...Chambers
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Darren New <dne### [at] san rrcom> wrote:
> http://blog.plover.com/math/Gdl-Smullyan.html
This is a great explanation! Finally I can stand up from the shame of never
me being dumb... :P
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On 12/14/2009 12:55 AM, Darren New wrote:
> http://blog.plover.com/math/Gdl-Smullyan.html
>
> Or, in words of one syllable,
>
> http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf
>
"...there is some hidden truth,
computer can't produce" - G:odel
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Ah yes, Godel. It's right up there with the Halting Problem and
NP-Completeness in that it's almost impossible to correctly comprehend
what its consequences actually are.
(For example, both these explanations neatly omit the fact that Godel's
theorum only applies to "sufficiently powerful systems".)
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Invisible wrote:
> (For example, both these explanations neatly omit the fact that Godel's
> theorum only applies to "sufficiently powerful systems".)
Yes, except "sufficiently powerful" includes "anything that can do integer
arithmetic." And Godel indeed does apply to insufficiently powerful systems
because they can't, by definition, prove everything that is true.
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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>> (For example, both these explanations neatly omit the fact that
>> Godel's theorum only applies to "sufficiently powerful systems".)
>
> Yes, except "sufficiently powerful" includes "anything that can do
> integer arithmetic." And Godel indeed does apply to insufficiently
> powerful systems because they can't, by definition, prove everything
> that is true.
I can't remember which article it was, but I was reading earlier today
about some system or other, and there was a remark about "there exists
an algorithm which can prove or disprove any possible statement in this
language. This does not contradict Godel since this language is
insufficiently powerful." (No, I can't remember what the language was...)
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Invisible wrote:
> This does not contradict Godel since this language is
> insufficiently powerful." (No, I can't remember what the language was...)
Yes, it's not difficult to set up such a language. Lots of, for example,
sufficiently simple regular expressions are easy to write a machine for that
will always either halt with acceptance or reject their input.
There are also non-Godel proofs that you can prove are unprovable. I.e.,
there are statements about integers that you can prove are true (using
mathematical systems more powerful than integers) that you also can prove
you can't prove using only integers, and those statements are actually in
some ways useful rather than arbitrarily constructed just to prove Godel's
point, which I think is even *more* cool.
--
Darren New, San Diego CA, USA (PST)
Forget "focus follows mouse." When do
I get "focus follows gaze"?
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>> This does not contradict Godel since this language is insufficiently
>> powerful." (No, I can't remember what the language was...)
>
> Yes, it's not difficult to set up such a language.
It was something pre-existing. (First-order logic?)
> There are also non-Godel proofs that you can prove are unprovable. I.e.,
> there are statements about integers that you can prove are true (using
> mathematical systems more powerful than integers) that you also can
> prove you can't prove using only integers, and those statements are
> actually in some ways useful rather than arbitrarily constructed just to
> prove Godel's point, which I think is even *more* cool.
My brain hurts...
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