Ken
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At 01:11 PM 3/18/96 -0500, you wrote:
>On Thu, 14 Mar 1996, Glenn Morton wrote:
>
>> 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
>
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Ken W. Smith, Professor of Mathematics
Interim Director, Office of Institutional Research "In the future
Central Michigan University, Mt. Pleasant, MI 48859 computers may weigh
Work phone: 517-774-7222, fax: 517-774-4250 as little as 1.5 tons."
Home phone & FAX: 517-772-5042 Popular Mechanics, 1949