Dear Russell:

Yes we did indeed have a similar debate some time ago.
At that time I was still trying to express this point of view correctly and 
admittedly made a number of mistakes back then [and still do].
Our debate helped me considerably and I thank you.

In response:

I just posted a response to Juergen that may help.  But here are a few 
comments.

At 4/13/01, you wrote:

>Basically, Hal believes a finite FAS by definition implies that each
>theorem is constrained to be no more than N-bits in length.

Well more precisely that the shortest possible proof chain of any theorem 
of a finite FAS can not be more complex than the limit which Chaitin 
provides.  Is this not Chaitin's position?  Given that, then there are only 
a finite number of theorems that satisfy this limit on the complexity of 
their shortest proof chains.

>So by his
>definition, number theory is not a finite FAS.

That of course is one possible view if we allow theorem cascades.  However, 
I prefer a more conservative view that this is an unresolved incompleteness 
that theorem cascades force towards completeness.

>By contrast, almost everyone else believes finiteness in FASes refers
>to a finite number of axioms, not that the theorems are bounded in any
>fashion.

Actually I believe that all the components of a finite FAS - axioms, rules, 
alphabet, and number of theorems must be finite.

>Whilst I can appreciate diversity of viewpoints, I fail to see how
>Hal's position actually yields a useful mathematical object. In
>Juergen's chain below, what is the use of a system where a+1=b
>(lets say) is a valid theorem, but b+1=c (where c=b+1=a+2) is an
>invalid theorem because of an arbitrary cutoff rule?

As I tried to show in my latest response to Juergen it seems to introduce a 
need for a random oracle which I believe you fell is an essential 
ingredient of SAS supporting universes.

Hal

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