|To:||Avril Styrman <Avril.Styrman@xxxxxxxxxxx>|
|From:||Pat Hayes <phayes@xxxxxxx>|
|Date:||Thu, 31 Jan 2008 13:59:35 -0600|
At 9:04 PM +0200 1/31/08, Avril Styrman wrote:
> > This is the idea
Choose your formal arithmetic, and I'll give you the proof (which will be very trivial). But look: what you say is obviously wrong, because a formal proof does not use any "ability". It (the proof) is a syntactically defined entity satisfying certain formal constraints, and merely by existing it establishes that its conclusion is a theorem. Ability does not enter into this discussion.
Um.. do you want me to explain the Goedel proof? There are many intuitive accounts already written,and I don't have the energy to write out yet another one.
observation of the proof.
And this does not mention self-reference anywhere, right?
Honestly, you have no idea what you are talking about. I see where you are coming from (that in order to understand a the formal proof one needs the same cognitive or intuitive abilities as one needs to simply understand that the conclusion is true.So the formal proof does not itself add anything to our pre-mathematical intuition. Do I have that more or less right?) but this is NOT what the Goedel theorem is about: its conclusion is that certain things that are, in a sense, clearly true cannot be proved *at all*, even from the obvious premises; and it establishes this by showing that one can always construct a sentence which has the logical structure similar to the liar paradox: this sentence is unprovable. If this were provable, the whole system would be inconsistent: so (assuming that the system is consistent) it cannot be: but then what it says is correct, so it is in fact true. True, but unprovable. The really clever part of the proof was to show that one can do this is any axiomatic system for arithmetic, so arithmetic is itself undecideable. This last part was done using Goedel numbering, which is what the above quote from Wikipedia is all about.
No, of course not.
Why do you? After you have proved it, do you
I don't. You miss the point altogether. The purpose of the formalization is not to provide better evidence for obvious truths. It is to support highly non-obvious reasoning with some confidence that it will be correct.
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