Re: Godel's theorem

ken.w.smith@cmich.edu
Mon, 18 Mar 1996 17:00:55 -0500

Hi Jeff,
About Godel's theorem: Godel's theorem is a statement about any
axiomatic system which *includes* the axioms of the natural numbers.
Therefore it is true about *any* larger system. This describes any
"reasonable" system of logic.
So it is not just about the natural numbers. It is about any
environment where the natural numbers can be conceived!

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