[Marxism] Re: [PEN-L] More Godel
rholt at planeteria.net
Tue May 17 08:23:24 MDT 2005
Perhaps. But if Gödel's biographers are correct, and he was in fact a
Platonic idealist where Truth, Whiteness and Beauty all existed first
off and forever independently of the human mind and we poor mortals
could at best but dimly grasp them, then it would be repugnant to him to
admit that a final proof of consistency would be forever beyond man's
Remember, Gödel had just proven the Completeness Theorem, which states
simply that every valid statement of the predicate calculus is a theorem
(has a proof). Rebecca Goldstein misses this by a wide margin. As is
clear on page 153 of her book, she fails to even understand the
difference between the predicate calculus and the statement calculus.
For the statement calculus (also called the propositional calculus) the
Completeness Theorem states that every tautology is a theorem, i.e.,
there exist no undecidable propositions. This was proven in the 1920s,
but I forget now who did it.
Look at where Gödel was going: 1.) every valid statement is a theorem,
i.e., has a proof. 2.) there exists a statement in number theory where
neither it nor its negation is provable, so therefore 3.) this statement
is neither valid nor invalid. Then, some years later, in his proof of
the independence of the axiom of choice and the continuum hypothesis, he
gets around the internal contradictions of naïve set theory (i.e., the
set theory of "Classical Mathematics") with the device of excluding
statements whereby a logic system can talk about itself. The implication
of this whole approach is that Gödel believed that the ONLY statements
that lay outside formal number theory (and stronger systems) were
statements that were self-referential. Or, to put it another way; ONLY
the statement "I am consistent" is unprovable. But both the consistency
and completeness of number theory is close to an article of faith for a
materialist, because, for any particular problem, the real world is a
model (because, among other reasons, the real world is ennumerable).
This Gödel never proved for set theory.
And, it is on the borderline between the "realness" of the real world
model and the undecidability of certain problems with ennumerable models
that Turing gets most interesting.
Carlos A. Rivera wrote:
> ----- Original Message ----- From: "Rod Holt" <rholt at planeteria.net>
>> For example, at the very end of his proof, Gödel states, "It must be
>> expressly noted that Proposition XI represents no contradiction of
>> the formalist standpoint of Hilbert. For this standpoint only
>> presupposes only the existence of a consistency proof effected by
>> finite means, and there might conceivably be finite proofs which *
>> cannot * be stated in P (or in set theory M or in classical
>> mathematics A)."
> Yet I have always taken this protestation somewhat a kind to
> Einstein's objection to quantum physics, as the contradiction of
> someone one doesn't *want* to grasp the significance of what they have
> just formulated.
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