>> (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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