Mohsen Ravanbakhsh
>
>*Jesse,
>I definitely don't think the two systems could be complete, since
>(handwavey
>argument follows) if you have two theorem-proving algorithms A and B, it's
>trivial to just create a new algorithm that prints out the theorems that
>either A or B could print out, and incompleteness should apply to this too*
>**
>They're not independent systems.putting that aside, I can't find the
>correspondence to my argument. It would be nice if you could clarify your
>point.
I didn't say they were independent--but each has a well-defined set of
theorems that they will judge to be true, no? My point was just that they
could not together be complete as you say, since the combination of the two
can always be treated as a *single* axiomatic system or theorem-proving
algorithm which proves every theorem in the union of the two sets A and B
prove individually, and this must necessarily be incomplete--there must be
true theorems of arithmetic which this single system cannot prove (meaning
that they don't belong to the set A can prove or the set B can prove).
Jesse
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Received on Fri May 25 2007 - 09:27:04 PDT