In a message dated Thu, 20 Dec 2001 8:58:00 PM Eastern Standard Time, "Richard S. Crawford" <[EMAIL PROTECTED]> writes:
> More precisely, the self-evolving system that The Fool describes will > encounter statements which are true within its internal logic but which > cannot be proven within that logic. > > In mathematics, there are statements which are true but which cannot be > proven mathematically. > > In any system symbolic logic (S), there are axioms which are true in S but > which cannot be demonstrated within S. However, those statements *can* be > proven within meta-logic, S^2. But then we find axioms within S^2 which > cannot be proven within S^2, but which can theoretically be proven true by > using S^3. Unfortunately, S^3 has proven too complex to be easily > understood by mere graduate students. > >Richard Hofstadter has joined the list.
