This is essentially what I learned in my symbolic logic course about
Godel's theorem. How can Godel's theorem be just about the natural
numbers when it is discussed in deductive logic courses as if it applied
to systems of extensional logic, which involves propositions and their
relations, of which the theorems of science are a subset? Are logic
professors misusing Godel when they do not talk about the arithmatic of
the natural numbers, but rather of axiomatizable systems of extensional
logic?
Jeff
+++++++++++++++
It may sound incredible, but no. Systems of extensional logic (involving
propositions and their relations, and including theorems of science) are
actually a subset of the natural numbers. So you have it backwards.
Godel's theorem no doubt was proved on the natural numbers, but it can be
extended to systems of logic by encoding the logic system into natural
numbers. This is a rather standard technique in Math, logic, and
theoretical computer science.
For instance, any computer (hardware or software) can be encoded as a Turing
machine, but a turing machine consists of an infinite tape of 0's and 1's,
with a switching mechanism that causes it to move left and right based on
the patterns of 0's and 1's. By reducing complex algorithms/systems into a
pattern of 0's and 1's, one can deduce certain fundamental properties of the
system.
A Turing machine is purely conceptual, and of little practical use. A
physical Turing machine would be monsterously slow, and useless for real
world problems. On the other hand, computer systems whose components are
logically equivalent to Turing machines, are tremendously usefull, and
billions of times faster.
Same thing holds for Godel's construct. Encoding logical systems as Natural
numbers are theoretically usefull, but are of little practical value in,
say, deducing properties of supernova.
Dennis Sweitzer