>> 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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