A statement is "unprovable within a system" if any effort to prove it within that system inevitably leads to a contradiction. You are lecturing the wrong person on the work of Kurt Godel. (See "Godel Theorems for Nonconstructive Logics," J. Barkley Rosser, Journal of Symbolic Logic, 1937, vol. 2, pp. 129-137). J. Barkley Rosser, Jr. -----Original Message----- From: Paul Henry Rosenberg <rad at gte.net> To: lbo-talk at lists.panix.com <lbo-talk at lists.panix.com> Date: Monday, May 24, 1999 3:30 PM Subject: Re: incompleteness theorem (was: gun control)
>W. Kiernan wrote:
>
>
>> 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!"
>