> Warren Weaver, "The Imperfections of Science", in Samuel Raport and Helen
> Wright, eds. Science: Method and Meaning, (Washington Square Press, 1964), p.
> 25 said,
>
> "But apart from this inherent limitation on deductive logic, which has of
> course been long recognized, there have rather recently been discovered, by
> Godel, wholly unsuspected and startling imperfections in any system of
> deductive logic. Godel has obtained two main results, each of which is of the
> most massive importance. He proved that it is impossible- theoretically
> impossible, not just reasonably difficult--to prove the consistency of any set
> of postulates which is, so to speak, rich enough in content to be interesting.
> the question 'Is there an inner flaw in this system?' is a question which is
> simply unanswerable.
> "He also proved that any such deductive logical system inevitably has a
> further great limitation. Such a system is essentially incomplete. Within
> the system it is always possible to ask questions which are undecidable."
>
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