Re: [Marxism] Re: [PEN-L] More Godel

[Jim Farmelant <farmelantj@xxxxxxxx>]

On Fri, 18 Mar 2005 11:15:15 -0800 Rod Holt <rholt@xxxxxxxxxxxxxx>
writes:
> Friends one & all:
> Any body who says "... Gödel kicked the whole [formalist program]
> over,"
> and then goes on to Hofsteader make me stop reading.
> I have a question: Why don't people read GÖDEL in stead of reading
> about
> him?


Kurt Gödel's paper, "On formally  undecidable propositions
of Principia Mathematica and related systems," can
be found online at:
http://home.ddc.net/ygg/etext/godel/godel3.htm
It probably won't hurt to have a good grasp of
predicate logic, set theory, and the theory of recursive
functions to understand Gödel's proof, although
the underlying idea is not that hard to grasp, which
is that just as Russell and Whitehead were able to
show that the axioms of Arithmetic could be formulated
in the notation of mathematical logic, so the propositions
of Russell & Whitehead's mathematical logic could
be formulated in terms of the arithmetic of natural
numbers.  Given this, it is possible to formulate
in Arithmetic a proposition of the G, which asserts
that it is not provable in terms of the axioms of
the arithmetic of natural numbers.  This leads
to a liars-type of paradox, since if G is provable
then it must be false, but since we know it is
true, then it cannot be provable. Therefore,
the axiom set of arithmetic cannot be complete
since there are propositions in arithmetic that
are true but are not provable in terms of those
axioms. And since we are always free to add
these Gödelian propositions to the axioms
of arithmetic and since we can prove that even
with the addition of those propositions, the axiom
set remains incomplete, therefore there must
be an infinite number of Gödelian propositions.

Also, see:
http://www.math.hawaii.edu/~dale/godel/godel.html
http://planetmath.org/encyclopedia/GodelsIncompletenessTheorems.html
http://www.apronus.com/math/goedel.htm

> Why don't Marxists take a little time off and learn a little
> mathematics, or at least enough to know the difference between
> "True"
> and "derivable," between logic and metalogic??
> Does anybody here know what a Gödel number is? If not, then avoid
> propounding on Gödel, the dialectic, etc. ad nausium.
>        --rod
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