[Marxism] Re: [PEN-L] More Godel
Rod Holt
rholt at planeteria.net
Fri Mar 18 12:30:13 MST 2005
A short note. Gödel proved what is called "The Completeness Theorem."
Kleene independently proved it also. (See his book, "Introduction to
Metamathematics") What system was it he proved complete? The Predicate
Calculus, which as we all know, is a much stronger system than number
theory; i.e., number theory can be embedded in the Predicate Calculus.
Gödel went further than this by developing what is now known as
Gödel-Zermelo set theory in his paper on the consistency of the Axiom of
Choice. I mentioned this briefly in 4\29\04 & 5\13\04.
--rod
====================
Jim Farmelant wrote:
>On Thu, 17 Mar 2005 14:49:31 -0500 Les Schaffer <schaffer at optonline.net>
>writes:
>
>
>
>
>>Both Goldstein and Yourgrau discuss extensively the Vienna school,
>>and i
>>am hoping Jim Farmelant kicks in here with some further insights.
>>[Jim:
>>i'll get to Dumain's questions hopefully tmw]
>>
>>
>
>Well, Gödel as a young mathematician was associated with
>the Vienna Circle, but he apparently never subscribed to their
>doctrines. The basis of this association with them seems
>to have been a shared interest in mathematical logic and
>the foundations of mathematics. Gödel, unlike the positivists,
>of the Vienna Circle, held to a Platonic view of mathematics,
>and indeed of human knowledge in general, and unlike
>the positivists, he was a confirmed theist. Among his
>papers that were discovered after his death, was one
>in which he advanced an ontological proof of the existence of God,
>using modal logic.
>
>
>BTW I pointed out in a post on the Thaxis list that Stephen
>Hawking has come around to the postion that I know that
>Gödel's work does have implications for physical theory.
>In his lecture, "Gödel and the end of physics,"
>(http://www.damtp.cam.ac.uk/strtst/dirac/hawking/), he contends that
>physical theory, like Arithmetic, may necessarily be incomplete in
>Gödel's sense
>because for one thing, our physical theories are formulated in
>terms of the very mathematics that Gödel had proven to be incomplete.
>Therefore, there will be physical problems that cannot be predicted
>on the basis of our physical theories. Furthermore, Hawking
>goes on to contend that :
>
>"Although this is incompleteness of sort, it is not the kind of
>unpredictability I mean.gIven a specific number of blocks,
>one can determine with a finite number of trials, whether they
>can be divided into two primes. But I think that quantum theory
>and gravity together, introduces a new element into the discussion,
>that wasn't present with classical Newtonian theory. In the standard
>positivist approach to the philosophy of science, physical theories
>live rent free in a Platonic heaven of ideal mathematical models.
>That is, a model can be arbitrarily detailed, and can contain an
>arbitrary amount of information, without affecting the universes they
>describe.
>But we are not angels, who view the universe from the outside. Instead,
>we and our models, are both part of the universe we are describing.
>Thus a physical theory, is self referencing, like in Goedels theorem.
>One might therefore expect it to be either inconsistent, or imcomplete.
>The theories we have so far, are both inconsistent, and incomplete."
>
>Hawking's lecture represents a reversal of his previously long held
>position that a final physical theory is possible (and that we were
>indeed close to attaining it). Now, Hawking holds that such a thing may
>never be possible even in principle. Hawking's new position is
>thus not unlike the one that David Bohm took in writings such as
>*Causality and Change in Modern Physics* (London, 1957)
>in which, in the course of presenting a philosophy
>of nature to support of his hidden-variable interpretation
>of QM, he drew upon Friedrich Engels'
>philosophy of nature as presented in writings like *Anti-Durhing*
>and *The Dialectics of Nature*, to argue that no final physical
>theory was possible.
>
>
>
>
>>
>>les schaffer
>>
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