> No, Godel's proof says you can not construct a system
> which completely describes ordinary arithmetic with any
> finite number of axioms; in fact, if I'm not mistaken, not
> even with any countable number of axioms.
You are mistaken. It says that a finite axiom system is necessarily incomplete. Your description of how he proceded to prove this is accurate:
> Godel's proof does this by a chasing-your-tail technique of mapping
> logical proofs onto arithmetic itself; any proof can be rewritten, using
> a special notation, as an equation, either true or untrue, in
> arithmetic; then whether that numerical equation is true or false is the
> axiomatically "unprovable" arithmetic assertion.
This proof takes the same logical form as
(1) "assume N is the largest integer" (2) N+1 is an integer (3) N+1 > N (4) Therefore (1) is false.
which proves that the number of integers is infinite, NOT that it is uncountable.
> Godel's proof does not assert that there will necessarily be
> contradictions in any axiomatic system, though of course it is simple
> enough to construct a series of axioms which are mutually contradictory;
> it just says that any system of axioms will not completely describe the
> arithmetic system. That is, given any axiomatic system, there are true
> statements about arithmetic which can not be proven using those axioms.
It's important to note that this came as a shocking surprise, not simply because of ignorant prejudice, but because Euclidian Geometry had already been completely axiomatized.
NONE of this bears much relationship to Constitutional Law, however. It's never been claimed that ANY kind of law is a formal system in the sense that branches of mathematics can be so construed.
Barkley writes:
> It is one thing to know that a system must be
> incomplete and contain contradictions. It is quite
> another to actually spot some contradictions.
A good point resting on a confused premise.
Goedel's Theorem has NOTHING to do with contradictions, except that it was assumed a complete axiomatization of arithmetic could then lead to proof of non-contradiction.
Proving that a a complete axiomatization is impossible doesn't prove the existence of a contradiction. It proves the existence of statements whose truth cannot be determined from the axioms.
In fact, it was not until the late 1970s that an example was found of such an unprovable statement, thus illustrating the basic thrust of Barkley's argument, with "unprovable statement" substituted for "contradiction".
> Thus arithmetic must have contradictions but nobody has
> spotted any yet.
Not so. You are confusing "unprovable statement" and "contradiction".
> Apparently Godel spotted at least
> one in the US Constitution, but I do not know what it
> or they are or were.
The analogy is faulty, since the US Constitution is not the kind of thing which can be formally axiomatized.
Furthermore, the existence of contradictions is hardly hidden from ordinary mortal eyes. The very notion of unrestricted rights is itself self-contradictory, as has been widely acknowledged. That's why we have things like the balancing tests in the application of 14th Amendment protections.
Rights are articulated through historical practice. They aren't fixed statements in the eternal framework of an axiomatic system.
Einstein -- who had considerable experience in the world of politics -- certainly recognized the difference between the two, though as a trickster he enjoyed playing the foolish wiseman who could not. Goedel really was such a foolish wiseman.
-- Paul Rosenberg Reason and Democracy rad at gte.net
"Let's put the information BACK into the information age!"